Programmable and tunable cylindrical deflector analyzers
An electrostatic analyzer includes a cylindrical body having an inner cylinder and an outer cylinder that are coaxial with one another along a longitudinal axis of the cylindrical body. An inner cylindrical electrode is positioned on an exterior face of the inner cylinder. An outer cylindrical electrode is positioned on an interior face of the outer cylinder. A first azimuthal electrode positioned on a face of a first azimuthal plane that passes through the longitudinal axis. A second azimuthal electrode is positioned on a face of a second azimuthal plane that passes through the longitudinal axis. A first end electrode is positioned on a first end face of the cylindrical body. A second end electrode is positioned on a second end face of the cylindrical body.
1 . An electrostatic analyzer comprising:
a cylindrical body comprising an inner cylinder and an outer cylinder that are coaxial with one another along a longitudinal axis of the cylindrical body;
an inner cylindrical electrode positioned on an exterior face of the inner cylinder;
an outer cylindrical electrode positioned on an interior face of the outer cylinder;
a first azimuthal electrode positioned on a face of a first azimuthal plane that passes through the longitudinal axis;
a second azimuthal electrode positioned on a face of a second azimuthal plane that passes through the longitudinal axis;
a first end electrode positioned on a first end face of the cylindrical body; and
a second end electrode positioned on a second end face of the cylindrical body.
2 . The electrostatic analyzer of claim 1 , wherein the inner cylinder and the outer cylinder are spaced apart to form a gap therebetween, the gap having an entry slit positioned at a first end thereof and an exit slit positioned at a second end thereof.
3 . The electrostatic analyzer of claim 2 , further comprising a detector positioned adjacent the exit slit configured to receive a particle beam that enters the gap via the entry slit and exits the gap via the exit slit.
4 . The electrostatic analyzer of claim 2 , wherein the entry slit and the exit slit are positioned such that a particle beam is configured to enter the gap at a radius of the electrical center of the electrostatic analyzer.
5 . The electrostatic analyzer of claim 4 , wherein the inner cylindrical electrode and the outer cylindrical electrode have equal and opposite voltages applied thereto and the electrical center of the electrostatic analyzer is located at r 0 =√{square root over (r 1 r 2 )}, where r 1 is the radius of the inner cylindrical electrode and r 2 is the radius of the outer cylindrical electrode.
6 . A method of determining the electrostatic field distribution in an electrostatic analyzer, comprising:
providing an electrostatic analyzer comprising:
a cylindrical body comprising an inner cylinder and an outer cylinder that are coaxial with one another along a longitudinal axis of the cylindrical body;
an inner cylindrical electrode positioned on an exterior face of the inner cylinder;
an outer cylindrical electrode positioned on an interior face of the outer cylinder;
a first azimuthal electrode positioned on a face of a first azimuthal plane that passes through the longitudinal axis;
a second azimuthal electrode positioned on a face of a second azimuthal plane that passes through the longitudinal axis;
a first end electrode positioned on a first end face of the cylindrical body; and
a second end electrode positioned on a second end face of the cylindrical body;
grouping the boundary conditions of each of the electrodes into six simplified problems;
defining the electric potential in the electrostatic analyzer by applying the Dirichlet's problem to the six simplified problems;
expressing the electric potential as a product of three potential functions in terms of Laplace's equation for cylindrical coordinates;
transferring a function to another side of the equation and separating the variables;
using the Bessel equation to build a system of equations; and
determining a general solution to the system of equations.
7 . The method of claim 6 , wherein the six simplified problems comprises:
S 1 :U (ρ 1 ,z ,φ)= U 1 =1, U 2 =U 3 =U 4 =U 5 =U 6 =0
S 2 :U (ρ 2 ,z ,φ)= U 2 =1, U 1 =U 3 =U 4 =U 5 =U 6 =0
S 3 :U (ρ ,z 1 ,φ)= U 3 =1, U 1 =U 2 =U 4 =U 5 =U 6 =0
S 4 :U (ρ ,z 2 ,φ)= U 4 =1, U 1 =U 2 =U 3 =U 5 =U 6 =0
S 5 :U (ρ ,z,φ 1 )= U 5 =1, U 1 =U 2 =U 3 =U 4 =U 6 =0
S 6 :U (ρ ,z,φ 2 )= U 6 =1, U 1 =U 2 =U 3 =U 4 =U 5 =0.
8 . The method of claim 6 , wherein the electric potential defined by the Dirichlet's problem comprises:
a
)
{
U
(
R
,
φ
,
z
)
=
U
(
ρ
,
φ
,
0
)
=
U
(
ρ
,
φ
,
L
)
=
U
(
ρ
,
φ
1
,
z
)
=
U
(
ρ
,
φ
2
,
z
)
,
U
(
ρ
1
,
φ
,
z
)
=
U
0
,
U
(
ρ
1
,
φ
,
z
)
=
U
(
ρ
,
φ
,
0
)
=
U
(
ρ
,
φ
,
L
)
=
U
(
ρ
,
φ
1
,
z
)
=
U
(
ρ
,
φ
2
,
z
)
,
U
(
R
,
φ
,
z
)
=
U
1
.
b
)
{
U
(
R
,
φ
,
z
)
=
U
(
ρ
1
,
φ
,
z
)
=
U
(
ρ
,
φ
,
L
)
=
U
(
ρ
,
φ
1
,
z
)
=
U
(
ρ
,
φ
2
,
z
)
,
U
(
ρ
,
φ
,
0
)
=
U
2
,
U
(
R
,
φ
,
z
)
=
U
(
ρ
,
φ
,
0
)
=
U
(
ρ
1
,
φ
,
z
)
=
U
(
ρ
,
φ
1
,
z
)
=
U
(
ρ
,
φ
2
,
z
)
,
U
(
ρ
,
φ
,
L
)
=
U
3
.
c
)
{
U
(
R
,
φ
,
z
)
=
U
(
ρ
,
φ
,
0
)
=
U
(
ρ
,
φ
,
L
)
=
U
(
ρ
1
,
φ
,
z
)
=
U
(
ρ
,
φ
2
,
z
)
,
U
(
ρ
,
φ
1
,
z
)
=
U
4
,
U
(
R
,
φ
,
z
)
=
U
(
ρ
,
φ
,
0
)
=
U
(
ρ
,
φ
,
L
)
=
U
(
ρ
,
φ
1
,
z
)
=
U
(
ρ
1
,
φ
,
z
)
,
U
(
ρ
,
φ
2
,
z
)
=
U
5
.
9 . The method of claim 6 , wherein the three potential functions in terms of Laplace's equation for cylindrical coordinates comprises:
1
R
ρ
d
d
ρ
(
ρ
dR
d
ρ
)
+
1
ρ
2
1
Φ
d
2
Φ
d
φ
2
+
1
Z
d
2
Z
dz
2
=
0.
10 . The method of claim 6 , wherein the equation after the transferring step comprises:
ρ
2
R
d
2
R
d
ρ
2
+
ρ
R
dR
d
ρ
+
1
Φ
d
2
Φ
d
φ
2
=
±
λ
2
ρ
2
ρ
R
d
d
ρ
(
ρ
dR
d
ρ
)
±
λ
2
ρ
2
=
-
1
Φ
d
2
Φ
d
φ
2
=
+
ξ
2
.
11 . The method of claim 6 , wherein the system of equations comprises:
d
2
Z
(
z
)
dz
2
=
±
λ
2
Z
(
z
)
,
d
2
R
(
ρ
)
d
ρ
2
+
1
ρ
dR
(
ρ
)
d
ρ
-
[
λ
2
-
n
2
ρ
2
]
R
(
ρ
)
=
0
,
d
2
Φ
(
φ
)
d
φ
2
=
n
2
Φ
(
φ
)
.
12 . The method of claim 6 , wherein the general solution to the system of equations comprises:
Z ( z )= A 1 exp(λ z )+ B 1 exp(−λ z ),
R (ρ)= A 2 J n (λρ)+ B 2 Y n (λρ)
Φ(φ)= A 3 cos( n φ)+ B 3 sin( n φ).
13 . An electrostatic analyzing system comprising:
an electrostatic analyzer comprising:
a cylindrical body comprising an inner cylinder and an outer cylinder that are coaxial with one another along a longitudinal axis of the cylindrical body, the inner cylinder and the outer cylinder are spaced apart to form a gap therebetween;
an entry slit positioned at a first end of the gap;
an exit slit positioned at a second end of the gap;
a first azimuthal electrode positioned on a face of a first azimuthal plane that passes through the longitudinal axis;
a second azimuthal electrode positioned on a face of a second azimuthal plane that passes through the longitudinal axis;
a first end electrode positioned on a first end face of the cylindrical body; and
a second end electrode positioned on a second end face of the cylindrical body;
wherein the gap is configured such that a particle beam can enter the gap via the entry slit and exit the gap via the exit slit; and
a detector positioned adjacent the exit slit and configured to receive the particle beam.
14 . The system of claim 13 , further comprising:
an inner cylindrical electrode positioned on an exterior face of the inner cylinder; and
an outer cylindrical electrode positioned on an interior face of the outer cylinder.
15 . The system of claim 14 , wherein the inner cylindrical electrode and the outer cylindrical electrode have equal and opposite voltages applied thereto.
16 . The system of claim 14 , wherein a radius of the electrical center of the electrostatic analyzer is located at r 0 =√{square root over (r 1 r 2 )}, where r 1 is the radius of the inner cylindrical electrode and r 2 is the radius of the outer cylindrical electrode.
17 . The system of claim 16 , wherein the entry slit is positioned such that the particle beam is configured to enter the gap at the radius of the electrical center of the electrostatic analyzer.
18 . The system of claim 16 , wherein the exit slit is positioned such that the particle beam is configured to exit the gap at the radius of the electrical center of the electrostatic analyzer.