Introduction to Accelerators - US Particle Accelerator School

57 Pages·2008·1.55 MB·English

Introduction to Accelerators
Unit 1 – Lectures 3
Preliminaries continued:
Maxwell’s Equations & Special Relativity
William A. Barletta
Director, United States Particle Accelerator School
Dept. of Physics, MIT
US Particle Accelerator School
Reminder from Registrar
(cid:1) This is to remind all that ALL classes that meet on Tuesdays WILL
NOT meet on Tuesday, February 19th.
(cid:1) Classes that have a regular Monday scheduled class, WILL meet on
Tuesday, February 19th.
(cid:1) http://web.mit.edu/registrar/www/calendar.html
(cid:1) This includes ALL classes - lectures, recitations, labs, etc. - all day
long , including evening classes.
(cid:1) If conflicts are reported to this office, we will do the best we can to
solve problems, but I would appreciate your reminding all of your
classes of the Institute wide schedule situation.
US Particle Accelerator School
The Basics - Mechanics
US Particle Accelerator School
Newton’s law
(cid:1) We all know
d
=
F p
dt
(cid:1) The 4-vector form is
(cid:3) dm dp(cid:5) dpμ
Fμ = (cid:7)(cid:1) c ,(cid:1) (cid:8) =
(cid:4) dt dt (cid:6) d(cid:2)
2 = 2 2
(cid:1) Differentiate p m c with respect to (cid:1)
o
dpμ d(mc2)
p = p Fμ = (cid:1) F v = 0
o
μ μ
d(cid:2) dt

(cid:1) The work is the rate of changing mc2
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Harmonic oscillator
(cid:1) Motion in the presence of a linear restoring force
=
F (cid:1)kx
k
+ =
˙x˙ x 0
m
= k
x A sin(cid:1) t where (cid:1) =
o o m
(cid:1) It is worth noting that the simple harmonic oscillator is a
linearized example of the pendulum equation
3
˙x˙ +(cid:3)2 sin(x) (cid:1) ˙x˙ +(cid:3)2(x (cid:2) x ) = 0
o o 6
that governs the free electron laser instability
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Solution to the pendulum equation
(cid:1) Use energy conservation to solve the equation exactly
(cid:1) Multiply ˙x ˙ +(cid:1)2 sin(x) = 0 by x˙ to get
o
1 d d
x˙ 2 (cid:1)(cid:2)2 cos x = 0
o
2 dt dt
(cid:1) Integrating we find that the energy is conserved
1
x˙ 2 (cid:1) cos x = constant = energy of the system = E
2(cid:2)2
o
With x= (cid:1)
US Particle Accelerator School
Stupakov: Chapter 1
Non-linear forces
(cid:1) Beams subject to non-linear forces are commonplace in
accelerators
(cid:1) Examples include
(cid:2) Space charge forces in beams with non-uniform charge
distributions
(cid:2) Forces from magnets high than quadrupoles
(cid:2) Electromagnetic interactions of beams with external structures
• Free Electron Lasers
• Wakefields
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Properties of harmonic oscillators
(cid:1) Total energy is conserved
p2 m(cid:1)2x2
= +
U o
2m 2
(cid:1) If there are slow changes in m or (cid:1), then I = U/(cid:1) remains
o
invariant
(cid:1)(cid:2) (cid:1)U
=
o
(cid:2) U
o
This effect is important as a diagnostic
in measuring resonant properties of structures
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Hamiltonian systems
(cid:1) In a Hamiltonian system, there exists generalized positions q , generalized
i
momenta p , & a function H(q, p, t) describing the system evolution by
i
{ }
dq (cid:1)H dp (cid:1)H q (cid:1) q ,q ,....,q
i = i = (cid:2) 1 2 N
{ }
dt (cid:1)p dt (cid:1)q p (cid:1) p , p ,...., p
i i 1 2 N
(cid:1) H is called the Hamiltonian and q & p are canonical conjugate variables
(cid:1) For q = usual spatial coordinates {x, y, z} & p their conjugate momentum
components {p , p , p }
x y z
(cid:2) H coincides with the total energy of the system
= + = +
H U T Potential Energy Kinetic Energy
Dissipative, inelastic, & stochastic processes are non-Hamiltonian
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Lorentz force on a charged particle
(cid:1) Force, F, on a charged particle of charge q
in an electric field E and a magnetic field, B

(cid:2) 1 (cid:4)
= +
F q (cid:6)E v (cid:1) B(cid:7)

(cid:3) c (cid:5)
(cid:1) E = electric field with units of force per unit charge,
newtons/coulomb = volts/m.
(cid:1) B = magnetic flux density or magnetic induction, with
units of newtons/ampere-m = Tesla = Weber/m2.
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