Review of basic electrodynamics
r r
r r
Maxwell’s equations: Electric field E(r , t ), magnetic
field B(r , t ) ,
r
r
r
charge density ρ (r , t ), current density J (r , t )
r r ρ
r r Q
∇• E =
=> ∫ E • da = enclosed Gauss' Law
ε0
ε0
S
r r
∇• B = 0
r
r
r r
r r
∂B
∂B r
∇×E = −
=> ∫ E • dl = − ∫ • da Faraday' s Law
∂t
∂t
C
S
r
r r
∂E
∇ × B = µ0 J + µ0ε 0
=>
∂t
r
r r
∂E r
∫ B • dl = µ0 Ienclosed + µ0ε 0 ∫ ∂t • da Ampere' s Law
C
S
LECTURE 2
Review of basic electrodynamics
Magnetic guide fields used in accelerators
Particle trajectory equations of motion in accelerators
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USPAS Lecture 2
1
r r
r
Electrodynamic potentials: A(r , t ),r V (r , t )
r
r r r r
∂A
B = ∇ × A E = −∇V −
∂t
Conservation of charge:
r r ∂ρ
∇• J +
=0
∂t
Ohm’s Law:
r
r
J = σE
Lorentz Force
on charge e
r Law:
r force
r r
F = e( E + v × B)
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USPAS Lecture 2
Magnetic guide fields used in accelerators
The simplest guide field for a circular accelerator is a uniform
field.
Pole
B
0
ρ
e
The Lorentz force generated by the accelerator’s guide field
determines the trajectory of particles in the accelerator. Before we
discuss the trajectory equations, we should make some remarks
about the guide field.
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orbit
v
Pole
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4
The Lorentz force provides the centripetal acceleration
v 2 evB0 1 eB0
=
, =
ρ
ρ mv
m
In a synchrotron, the guide field (including bending and focusing
fields) is achieved by a series of separate magnets.
B0 [T]
1 −1
m = 0.2998
ρ
p[GeV / c]
Using this relation, we sometimes measure momentum in units of
T-m:
p[GeV / c]
( Bρ )[T - m] =
0.2998
The (Bρ) product is called the magnetic rigidity.
[ ]
v
B0
B0
B0
B0
bending fields
More than a simple uniform bending field is required. Focusing
fields are required to insure stability to small displacements from
the orbit.
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5
USPAS Lecture 2
The focusing fields may be separate from the bending fields
(“separated function” machine) or may be combined with the
bending fields (“combined function” machine).
If the bending fields all have the same design value, the machine is
called isomagnetic.
focusing fields
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θ=
θ
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p
e
θ
x
f
Β
L
x
f
f
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USPAS Lecture 2
Magnetic lens:
Magnetic focusing fields:
Optical analogy: Thin lens, focal length f
x
e
∆p
L
∆p = F∆t = evBy   = eBy L
 v
∆p eBy L
θ≈
=
p
p
∂By
eB′xL x
If By =
x = B′x , then θ ≈
=
∂x
p
f
The field acts like a focusing lens of
focal length
1 eB′L B′L
≈
=
f
p
( Bρ )
This linear dependence of field on position can be generated by a
quadrupole magnet
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General description of the guide fields in the region of the particle
beam
For typical accelerator magnets, the magnet length is much larger
than the transverse dimensions of the field gap
(L>>G)
apply to solenoids, and is a poor approximation for wigglers or
undulators).
In the region of the beam,
r r
r
r
∇r × B = 0 so B = −∇φ
r
∇ • B = 0 so ∇2φ = 0
Pole
y
G
pole
B
x
L
z
The idealized fields extend only over the length L (“effective
length”) and are independent of z and transverse to the beam.
Idealized fields ignore the fringe fields. (Note that this is does not
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The magnetic scalar potential is a solution to LaPlace’s equation.
For the idealized fields, the magnetic potential is only a function of
x and y.
In this case, the general solution to LaPlace’s equation in Cartesian
coordinates
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USPAS Lecture 2
∂2φ ∂2φ
+
= 0 can be written as
∂x 2 ∂y 2
Bx = −
∞
∂φ
= − Re ∑ mCm ( x + iy)m −1
∂x
m =1
∞
By = −
∞
∂φ
= − Re ∑ imCm ( x + iy)m −1
∂y
m =1
φ ( x, y) = Re
∑ Cm ( x + iy)m
m=0
where Cm is a (complex) constant determined by the boundary
conditions.
These can be combined to give
∞
By + iBx = − ∑ imCm ( x + iy)m −1.
m =1
The vector potential has only one component and it is
Az ( x, y) = Re
10
∞
This is the general multipole expansion for the two-dimensional
magnetic field in a current-free region.
∑ iCm ( x + iy)m
m=0
Then
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B( 0 ) = B0 ; B˜ ( 0 ) = B˜ 0 ;
Since
−imCm =
 m −1
1  ∂ By
( m − 1)!  ∂x m −1

+i
x = y=0
∂ m −1Bx
∂x m −1
B(1) = B′ =



x = y=0 
∂2 By
∂x 2
∞
1 (m) ˜ (m)
( B + iB )( x + iy)m
By + iBx = ∑
m = 0 m!
where
x = y=0
∂ By
∂x m
x = y=0
∂ Bx
; B˜ ( 2 ) = B˜ ′′=
∂x 2
2
m
B( m ) =
USPAS Lecture 2
∂B˜
; B˜ (1) = B˜ ′= x
∂x x = y = 0
∂x
B( 2 ) = B′′ =
we can write
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∂By
∂ Bx
; B˜ ( m ) =
∂x m
x = y=0
m
x = y=0
x = y=0
Vector potential in this language:
∞ 1
Az ( x, y) = − Re ∑ ( B( m −1) + iB˜ ( m −1) )( x + iy)m
m =1 m!
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USPAS Lecture 2
Let us write out the first few terms:
14
y
B′′
By = B0 + B′ x − B˜ ′ y + ( x 2 − y 2 ) − B˜ ′′ xy + ...
2
B˜ ′′
Bx = B˜ 0 + B˜ ′ x + B′ y + ( x 2 − y 2 ) + B′′ xy + ...
2
Iron
NI
Integration path
S
The terms without the twiddle are called “normal” terms; the terms
with the twiddle are called “skew” terms. The 0 coefficients are
pure dipole terms, the linear are quadrupole terms, the quadratic
are sextupole terms.
G
x
N
Pure dipole: NI turns/pole
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r r
Evaluate ∫ H • dl = Ienclosed around the integration path shown.
r
r B
For infinite permeability iron H = → 0 inside the iron, so in the
µ
gap
y
NI
Integration path
S
N
Iron
R
2 NI
2 NI
⇒ B = µ0 H = µ0
G
G
NI[ kA − turns]
B[T] = 2.52
G[ mm ]
This dipole bends a positive particle moving in the z-direction to
the left.
If the dipole is rotated clockwise by 90o about the z-axis, it
becomes a pure skew dipole.
H=
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r r
Evaluate ∫ H • dl = Ienclosed around the integration path shown.
r
r B
For infinite permeability iron H = → 0 inside the iron, so in the
µ
gap
r r 1 R
2 NI
B′ R 2
•
=
=
= NI ⇒ B′ = µ0 2
H
dl
B
rdr
′
∫
∫
0
µ0
2 µ0
R
NI[ A − turns]
T
B′[ ] = 2.51
m
R[ mm ]2
p
( Bρ )
=
eB′ L B′ L
Note that (for a positive particle moving in the z-direction),
this quadrupole is focusing in x, but defocusing in y.
If the quadrupole is rotated clockwise by 45o about the z-axis, it
becomes a pure skew quadrupole.
19
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x
S
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N
Pure quadrupole, NI turns/pole
USPAS Lecture 2
18
Iron
NI
N
Pure
sextupole
NI turns/pole
S
S
R
N
Quadrupole focal length f ≈
N
Integration path
S
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USPAS Lecture 2
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r r
Evaluate ∫ H • dl = Ienclosed around the integration path shown.
r
r B
For infinite permeability iron H = → 0 inside the iron, so in the
µ
gap
y
Iron
r r 1 R B′′ 2
6 NI
B′′ R3
•
=
=
= NI ⇒ B = µ0 3
H
dl
r
dr
∫
∫
0
µ0
2
6 µ0
R
NI[ A − turns]
T
B′′[ ] = 7540
m
R[ mm ]3
x
If the sextupole is rotated clockwise by 30o about the z-axis, it
becomes a pure skew sextupole.
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USPAS Lecture 2
Solenoid: produces an axial
field
y
N turns
I
Bz = µ 0
Bz [T] = 1.26 × 10 −3
z
Particle trajectory equations of motion in accelerators
NI
L
NI[ kA − turns]
L[ m ]
x
L
The transverse fringe fields at the solenoid ends are crucial to the
solenoid focusing action, so in this case the idealized fields must
include end fields. Maxwell’s equations allow the end fields to
1 ∂Bz
1 ∂Bz
x
By = −
y
have the form Bx = −
2 ∂z
2 ∂z
Solenoid focal length:
2
2
2 p 
2  ( Bρ )
f= 
=



L  eBz 
L  Bz 
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USPAS Lecture 2
Gradient magnet (for combined function machines): provides both
a bending field and a gradient (due to changing gap dimension).
22
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23
Consider a particle of charge e, rest mass m0 , momentum
1
r
r
p = m0γv (γ 2 =
), moving under the action of the Lorentz
v 2

1−
 c
force:
The trajectory equation of motion is given by Newton’s Law:
r
r
r r r
dp d
r
= ( m0γv ) = F = e( E + v × B)
dt dt
r
Solution is this equation is the trajectory r (t )
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r
Write r (t ) in terms of the “reference trajectory”:
For a circular accelerator, this trajectory is closed (i.e., it repeats
itself exactly on every revolution)
p
0
Actual trajectory
B0
B0
e
Reference
trajectory
r
r
B0
ρ
B0
r
r0
r
δr
bending fields
focusing fields
r
The “reference trajectory” r0 (t ) is the trajectory of a “reference
r
particle” of momentum p0 that passes through the center of
symmetry of all the idealized magnetic guide fields. The reference
trajectory is an idealization.
25
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USPAS Lecture 2
From differential geometry: “natural” coordinate system of a curve
in space
bˆ = tˆ × nˆ
bˆ, tˆ and nˆ are unit vectors
bˆ = tˆ = nˆ = 1
r
r0
nˆ = − ρ (s)
r
dr
tˆ = 0
ds
dtˆ
ds
s
USPAS Lecture 2
r
r
r
r (t ) = r0 (t ) + δr (t )
We want to write the trajectory equation of motion in the “natural”
coordinate system of the reference trajectory.
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USPAS Lecture 2
r
For any space curve r0 ( s), where s is the arc length along the
r
dr
curve, the vector tˆ = 0 is a unit vector tangent to the curve. The
ds
dtˆ
vector
measures the rate at which the tangent vector changes,
ds
which is inversely proportional to ρ ( s), the radius of curvature:
ˆ
dt
nˆ
=−
, where n̂ (called the principal normal to the curve) is a
ρ(s)
ds
unit vector normal to tˆ . The vectors tˆ , n̂, and bˆ = tˆ × nˆ form an
orthogonal, right-handed coordinate system which moves (and may
rotate) as we progress along the space curve.
The change in the unit vectors with s is given by the
Reference trajectory
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Reference trajectory
27
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USPAS Lecture 2
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Frenet-Serret relations:
dtˆ
nˆ dnˆ
tˆ dbˆ
=−
= τbˆ +
= −τnˆ
ρ ds
ρ ds
ds
where τ(s) is called the torsion.
For most accelerators, the reference orbit lies in a plane: this
corresponds to τ(s)=0. For circular accelerators, since the orbit is
closed,
ds
dθ
∫ ρ(s) = ∫ ds ds = 2π
Take the space curve to be the reference trajectory, and write the
trajectory equations using the Frenet-Serret co-ordinate system:
r
(δr lies in x-y plane since coordinate system moves with the
particle)
Actual trajectory
y
Reference
trajectory
= st
˙ ˆ + xn
˙ˆ +
29
USPAS Lecture 2
r
r
r r r
dp d
r
= ( m0γv ) = F = e( E + v × B)
dt dt
d
d
d
d
x 

r
m0γv ] = [ m0γxn
m0γyb
˙ ˆ] +
˙ ˆ + m0γs˙1 +  tˆ 
[
 ρ 
dt
dt
dt
dt 
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d
d 
m γs˙2 
x
d
x   m γxs

˙˙ 
= nˆ  [ m0γx˙ ] − 0 1 +   + bˆ [ m0γy˙ ] + tˆ  m0γs˙1 +   + 0 
 ρ 
ρ  ρ 
ρ 
dt
 dt 
 dt
nˆ
bˆ
tˆ
x

r r
v × B = x˙
y˙ s˙1 + 
 ρ
Bx By
Bs
[
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USPAS Lecture 2
x
x

st
˙ ˆ + yb
˙ ˆ 1 +  + xn
˙ ˆ + yb
˙ ˆ = st
˙ˆ
 ρ
ρ
USPAS Lecture 2
30
d
m γs˙2 
x
x

m0γx˙ ] = 0 1 +  + yeB
˙ s − s˙1 +  eBy + eEx
[
 ρ
ρ  ρ
dt
]

 

x 
x

= nˆ  yB
˙ s  + tˆ xB
˙ y − yB
˙ s − s˙1 +  By  + bˆ s˙1 +  Bx − xB
˙ x
 ρ 

  ρ

x
Three trajectory equations:
Newton’s Law:
[
s
df ds df
df
f˙ ≡ =
= s˙
dt dt ds
ds
r
r
r
dnˆ
dr
dnˆ
r dr dr0
v=
=
+ xn
˙ ˆ + x + yb
˙ ˆ + xs˙ + yb
˙ ˆ = s˙ 0 + xn
˙ˆ
dt dt
dt
ds
ds
r r
r r
r = r0 + δr = r0 + xnˆ + ybˆ
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r
δr
r
r0
]
31
d
x

m0γy˙ ] = s˙1 +  eBx − xeB
˙ s + eEy
[
 ρ
dt
d
x
m γxs

˙˙
m0γs˙1 +   = − 0 + x˙eBy − yeB
˙ x + eEs

 ρ 
ρ
dt 
Write derivatives in terms of arc length s along the reference curve
as the independent variable:
df
(with f ′ ≡ )
ds
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USPAS Lecture 2
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df df
= s˙ = f ′s˙
dt ds
v d 
v
mγ
m0γ x ′  = 0


ρ
l ′ ds 
l′ 
and introduce l=the path length of the particle along the orbit
v d 
v
v
x
v
m0γ y ′  = 1 +  eBx − x ′ eBs + eEy
l ′ ds 
l ′  l ′  ρ 
l′
ds
v
x
m0γx ′  v  2
v
v
v d 
m
γ
1
+
=
−
+ x ′ eBy − y ′ eBx + eEs


0

l ′ ds 
l′  ρ  
l′
l′
ρ  l′
dl
Reference
trajectory
2
 v  1 + x  + y ′ v eB − v 1 + x  eB + eE

 y
s
x
 l ′   ρ 
l′
l′  ρ 
Trajectory
v
Divide by , use p = m0γv , and expand LHS:
l′
dl
dl
v
= v = s˙ = l ′s˙ ⇒ l ′ =
dt
ds
s˙
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p
d p
p 
x
x
l′

x ′′ + x ′   =
1 +  + y ′eBs − 1 +  eBy + eEx


 ρ
l′
ds  l ′  ρl ′  ρ 
v
x ′′ − x ′
l ′′ 1 
x
eB 
x  eBy
=  1 +  + y ′l ′ s −  1 +  l ′
l′ ρ  ρ 
p  ρ p
p
d p 
x
l′
y ′′ + y ′   = 1 +  eBx − x ′eBs + eEy
l′
ds  l ′   ρ 
v
y ′′ − y ′
l ′′
x  eB
eB

= l ′1 +  x − x ′l ′ s
 ρ p
l′
p
px ′
l′
x  d  p p d  x 

1+  = −
+ x ′eBy − y ′eBx + eEs
1 +    +

 ρ  ds  l ′  l ′ ds  ρ 
ρl ′
v
eBy
eB
x  l ′′ d  x  x ′

+ y ′l ′ x
1 +  − 1 +  = − x ′l ′
 ρ  l ′ ds  ρ  ρ
p
p
34
From the relation for the velocity,
Let the electric field be zero: the particle energy is then constant.
p
d 1 
l ′′
Divide by
and use
=− 2
l′
ds  l ′ 
l′
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35
2


x
v = s˙ (1 + Kx ) + x˙ + y˙ = s˙ 1 +  + x ′ 2 + y ′ 2 
 ρ 

Solve for ṡ
2
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2
2
2
2
USPAS Lecture 2
36
s˙ =
v
2
2
and l ′ =
v
x

= 1 +  + x ′ 2 + y ′ 2
 ρ
s˙
x

2
2
1 +  + x ′ + y ′
 ρ
Approximation:
“paraxial” motion=> the derivatives x ′ 2 , y ′ 2 << 1 so
x
l ′′
l ′′
terms in the trajectory
l ′ ≈ 1 + and we neglect x ′ and y ′
ρ
l′
l′
equations
(These neglected terms are called “kinematic terms”)
 s
x ( s) ≈ xmax cos 
 β
Typical values of
dx xmax 10 mm
≈
≈
≈ 10 −3
x′ =
10 m
ds
β
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37
Trajectory equations in the paraxial approximation:
x ′′ =
2
1
x 
x  eBy
eB 
x
+
1
1
−
+
+ y ′ s 1 + 

 

ρ  ρ  ρ p
p  ρ
2
x  eBx
eB 
x

− x ′ s 1 + 
y ′′ = 1 + 
 ρ p
p  ρ
x

l ′ = 1 + 
 ρ
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