Particle Motion in EM Fields
Lecture 1
Dr. Suzie Sheehy
John Adams Institute for Accelerator Science
Image: Andrew Khosravani, 2016
http://richannel.org/collections/2016/particle-accelerators-for-humanity
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Written by Dr. Suzie Sheehy at John Adams Institute for Accelerator Science, University of Oxford
Typology: Slides
2020/2021
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Particle Motion in EM Fields
Lecture 1
Dr. Suzie Sheehy
John Adams Institute for Accelerator Science
Image: Andrew Khosravani, 2016
http://richannel.org/collections/2016/particle-accelerators-for-humanity
Where are we now?
- (^) So far, you should already have learned about:
- (^) The basic types of accelerators
- (^) Some useful hints at types of magnets and basic beam
parameters
- (^) Electromagnetic fields
- (^) Maxwell’s equations
- (^) The Lorentz force
- (^) We are getting a general idea that particles are bent and focused
by magnets. Now we will explore the fundamental physics of how
particles behave when they experience electromagnetic fields.
Particle in an EM field
How can we describe the motion?
B E?
BEARTH
B B
How can we describe the motion?
- (^) In an accelerator, magnets and rf cavities are generally defined
along a trajectory (i.e. we know where they are in distance, not
time). The Hamiltonian approach is lets us use this fact.
- (^) The motion of particles in electromagnetic fields is conservative,
and similar to a harmonic oscillator with perturbations. We have lots
of mathematical tools to treat this in the Hamiltonian formalism.
- (^) Ultimately, it makes our lives easier.
If we know the Hamiltonian and
Hamilton’s equations, we can find the
equations of motion for a dynamical
system.
Our first goal: find out the Hamiltonian!
d p
dt
= F (1.1) F^ =^ q ( E^ +^ v^ ×^ B )^ (1.2)
- (^) Why not just use Newton’s laws & Lorentz force?
Hamiltonian (straight beam line)
- (^) The Hamiltonian represents the total energy of the particle
dx i
dt
=
δ H
δ p i
dp
i
dt
δ H
δ x i
Hamilton’s equations
H = H ( x i
, p i
; t )
- (^) We need a Hamiltonian that gives (1.1) and (1.2) when substituted
into (1.3) and (1.4)
(1.3)
(1.4)
H = c ( p − q A )
2
+ m
2
c
2
- q φ
We propose the following Hamiltonian
for a relativistic charged particle
moving in an electromagnetic field:
(1.5)
But this is still defined in terms of
time… so we want to change it.
Straight Beamline Hamiltonian
STEP 2: We choose new (canonical) variables for the
position & momentum that stay small as the particle
moves along the beam line and scale by reference
momentum P 0 (subscript ‘0’ denotes reference)
STEP 3: We define new (canonical) longitudinal variables.
B E?
BEARTH
B B
Blue: “reference” trajectory
Red: actual particle trajectory
We’re not going to go through all the steps…
Variables for Beam Dynamics
- (^) Using the Hamiltonian, we can get the equations of
motion for a particle in a (straight) beam line.
- (^) Usually these equations are too complex to solve
exactly, so we have to make some approximations.
H =
δ
β 0
− δ +
1
β 0
−
q φ
cP 0
⎛
⎝
⎜
⎞
⎠
⎟
2
− ( p x
− a x
)
2 − ( p y
− a y
)
2 −
1
β 0
2 γ 0
2
− a z
(1.7)
Eventually, our Hamiltonian with independent
variable ’s’ along the beamline, becomes:
a =
q
P 0
A
Scaled vector potential
δ =
E
cP
0
β 0
Energy deviation
( x , p
x
, y , p
y
, z , δ )
Co-ordinates & momenta
Particle motion is described with respect to a reference orbit in the non-
inertial frame (x, y, s). This co-ordinate system is known as Frenet-Serret
First, we convert to a non-inertial reference frame.
We use the ’Frenet-Serret’ co-ordinate system
In accelerator physics we ask: “What are the particles’ generalized
coordinates when they reach a certain point in space?”
- (^) First, we convert to ‘Frenet-Serret’ co-ordinate system
s ˆ( s ) = Tangent unit vector to closed orbit
dr
!
0 ( s )
ds
x^ ˆ( s ) = − ρ( s )
ds ˆ( s )
ds
Unit vector perpendicular to tangent vector
y^ ˆ( s ) = x ˆ( s ) × s ˆ( s ) (^) Third unit vector…
Particle trajectory: r
!
( s ) = r
!
0 ( s )^ +^ xx ˆ( s )^ +^ yy ˆ( s )
H = e φ + c m
2 c
2
( p s − eA s )
2
( 1 + x / ρ)
2
- ( p x − eA x )
2
- ( p y − eA y )
2
nb. the reference frame moves WITH the particle
And we follow pretty much the same procedure as before…
Hamiltonian looks a little different and we can see that now the factor 1/ρ starts to appear.
Remarkably, the addition of the one simple term x/ρ(s) gives all the new non-inertial dynamics
F 3 ( P
!"
; x , s , y ) = − P
!"
.[ r 0
!"
( s ) + xx ˆ( s ) + yy ˆ( s )]
Generating function for canonical transformation
So far, we have been looking in general for any
Vector potential
Scalar potential
φ
A
But in reality, we (usually) use electric fields to
accelerate particles and magnetic fields to
bend, focus and manipulate the beams.
So we need to know
“which magnetic fields can we really create?”
(We’ll come back to the Hamiltonian later…)
Where are we now?
Magnetic Fields
- (^) Maxwell’s equations, time independent, no sources, so:
∇ × B
∇ ⋅ B
J
B
0
H
- (^) We’ll “guess” that the following obeys these equations:
- (^) A constant vertical field B z, and
B
y
- iB x
= C
n
( x + iy )
n − 1
- (^) n is an integer > 0, C is a complex number
- (^) (real part understood)
Multipole fields
B
y
- iB x
= B
ref
( b n
- ia n
x + iy
R
ref
n = 1
∞
∑
n − 1
In the usual notation:
bn are “normal multipole coefficients” (LEFT)
and an are “skew multipole coefficients” (RIGHT)
‘ref’ means some reference value
n=1, dipole field
n=2, quadrupole field
n=3, sextupole field
Images: A. Wolski, https://cds.cern.ch/record/
bn an
Multipole Magnets
x
By
x
B y
0
100
-20 0 20
Images: Ted Wilson, JAI Course 2012 Image: Danfysik
Image: STFC
Image: Wikimedia commons
20