98
An Introduction to Beam Physics
with a large region of nearly constant axial fields, or in short or “thin” arrangements where the field on axis rises, reaches a peak, and then falls off
again. Thin magnetic lenses are the main staple used for focusing in electron
microscopes, while long magnetic lenses are used in various particle accelerators for guiding the beam, including applications in muon ionization cooling.
We have the magnetic field, linear in x and y, given as
B x =
1
2
V
0 (s) x, B y =
1
2
V
0 (s) y, B s = −V
0 (s) ,
which is derived from the corresponding magnetic scalar potential
V = V 0 (s) −
1
4
V
0 (s)
x
2 + y
2
.
Different from the electrostatic potential in the case of the electrostatic round
lens, the magnetic potential does not enter the equations of motion directly.
So we simplify notation by not having it appear in the equations of motion,
and rather express the fields in terms of the axial center field B s (s) as
B x = −
1
2
B
s (s)x, B y = −
1
2
B
s (s)y, B s = B s (s).
Applying this to eqs. (4.8), and linearizing in the similar process for eqs.
(4.2), we obtain
x
= a, a
= +
B s
χ m0
b +
1
2
B
s
χ m0
y,
y
= b, b
= −
B s
χ m0
a −
1
2
B
s
χ m0
x,
l
=
1
(2 + η 0 )
2 δ, δ
= 0.
(4.20)
It is immediately apparent that the motion of the two planes are coupled,
which will lead to interesting properties. The longitudinal motion to first
order is the same with that of a drift, thus
l f = Dδ i + l i =
L
(2 + η 0 )
2 δ i + l i ,
δ f = δ i
with length L.
To study the transversal motion, we first express the equations of motion
(4.20) in vector notation. Using
z =
x
y
, , c =
a
b
, ˆ
J =
0 1
−1 0
,
we have
z
= c,
, c
=
B s
χ m0
ˆ
JJ c +
B
s
2χ m0
ˆ
JJ z.
(4.21)
An Introduction to Beam Physics
with a large region of nearly constant axial fields, or in short or “thin” arrangements where the field on axis rises, reaches a peak, and then falls off
again. Thin magnetic lenses are the main staple used for focusing in electron
microscopes, while long magnetic lenses are used in various particle accelerators for guiding the beam, including applications in muon ionization cooling.
We have the magnetic field, linear in x and y, given as
B x =
1
2
V
0 (s) x, B y =
1
2
V
0 (s) y, B s = −V
0 (s) ,
which is derived from the corresponding magnetic scalar potential
V = V 0 (s) −
1
4
V
0 (s)
x
2 + y
2
.
Different from the electrostatic potential in the case of the electrostatic round
lens, the magnetic potential does not enter the equations of motion directly.
So we simplify notation by not having it appear in the equations of motion,
and rather express the fields in terms of the axial center field B s (s) as
B x = −
1
2
B
s (s)x, B y = −
1
2
B
s (s)y, B s = B s (s).
Applying this to eqs. (4.8), and linearizing in the similar process for eqs.
(4.2), we obtain
x
= a, a
= +
B s
χ m0
b +
1
2
B
s
χ m0
y,
y
= b, b
= −
B s
χ m0
a −
1
2
B
s
χ m0
x,
l
=
1
(2 + η 0 )
2 δ, δ
= 0.
(4.20)
It is immediately apparent that the motion of the two planes are coupled,
which will lead to interesting properties. The longitudinal motion to first
order is the same with that of a drift, thus
l f = Dδ i + l i =
L
(2 + η 0 )
2 δ i + l i ,
δ f = δ i
with length L.
To study the transversal motion, we first express the equations of motion
(4.20) in vector notation. Using
z =
x
y
, , c =
a
b
, ˆ
J =
0 1
−1 0
,
we have
z
= c,
, c
=
B s
χ m0
ˆ
JJ c +
B
s
2χ m0
ˆ
JJ z.
(4.21)