Accelerator Physics Topic II Single-Particle Beam Dynamics

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Accelerator PhysicsTopic IISingle-Particle Beam
Dynamics

• Joseph Bisognano
• Synchrotron Radiation Center
• Engineering Physics
• University of Wisconsin

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Single-Particle Beam Dynamics

• For this topical area, we will be following
  closely the text, Accelerator Physics 2nd, by
  S.Y. Lee
• In particular, you should read
• Pages 533-537
• Pages 35-67
• Pages 85-105
• Pages 110-115
• Pages 129-161
• Pages 172-213
• Pages 239-263
• Also see on Web CERN School

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Page 17-45
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Accelerator Optics

• Start with one curve, the design orbit, where
  we hope the particles will circulate
• But there are always deviations, either because
  the injected particles dont have exactly the
  right momentum or they encounter fields that
  arent exactly right
• Deviations are small, at the 10-3 or 10-4 level,
  but are the whole game in whether the accelerator
  will work or not

Close, but not quite orbit
Design orbit
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A Cyclotron Orbit

• Consider a particle in a vertical magnetic field

CERN 94-01, next few figures
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Motion horizontally is geometrically
stabilized Unbounded vertically, a spiral Need
gradient magnets to provide restoring forces
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From CERN notes
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From CERN notes
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From CERN notes
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Form of Magnetic Field

• Without currents, magnetic field satisfies
• Restoring is governed by
  which by curl condition focus in one plane,
  defocus in the other

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Vector Potential Expansion
Cauchy Riemann condx for analyticity
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Expansion of Analytic Function
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Some Fields

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End Field Effects

• Longitudinal potential integrals from zerio field
  region to zero field region also satisfy
  Laplaces equation and have same expansion and
  nomenclature
• Since typically magnets are short, that
  information is usually enough

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Thin Lens Model

L
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Thin Lens Model/cont.
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Particle Motion from Newtons Laws
From CERN notes Rossbach/Schmuser
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Particle Motion from Newtons Laws-2
From CERN notes
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Particle Motion from Newtons Laws-3
From CERN notes
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Particle Motion from Newtons Laws-4
From CERN notes
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Particle Motion from Newtons Laws-5
!!!!!
From CERN notes
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Being Careful Do it with a Hamiltonian
Unsystematically taking approximations can be
hazardous e.g., numerically integrating a
harmonic oscillator
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Hamiltonian Dynamics

• Lagrangian variational principle
  
• Generating functions to preserve Hamiltonian
  character e.g.,

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Liouville Theorem

• Let
• Conservation of number of systems implies
• Then, if the system satisfies Hamiltons
  equations
• Incompressible fluid

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Particle Motion in EM Field from aHAMILTONIAN
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Frenet-Serret Coordinate System
Following SY Lees book now
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Frenet-Serret Coordinates/cont.
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A Canonical Transformation
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A Canonical Transformation
Yields same equations to lowest order homework
problem
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Second Order Equations

• Using expressions for curl, div, grad in
  curvilinear coordinates, and assuming that there
  are only transverse fields, we have

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Dipoles and Quads
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Hills Equation

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Stability
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Thin Lens Model
If L too large, or f too small, can go unstable
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General Form
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Floquet Theorem

• Consider the linear Hills equation, where K(s)
  is a periodic function
• Then, the solution is of the form

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Floquet Theorem
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Equations for Lattice Function
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Lattice Functions/cont.
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Lattice Functions/cont.
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Emittance

• For a channel, this is called admittance
  matching is having beam and channel ellipse
  similar

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A Picture
A??

• Invariant area is ?A2 ?? or ? depending on
  convention

A??
A/??
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Normalized Emittance

• Actually (x,px ) are canonical variables, not
  (x,x)
• So, area in (x,px ) is by Liouville invariant,
  even under acceleration. So, we define
  normalized emittance
• Since the normalized emittance is constant under
  acceleration, the geometric emittance ?
  decreases, so-called adiabatic damping

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Evolution of Ellipse

• c

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Examples

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Horizontal Motion and Dispersion

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Dispersion Function for a Periodic System

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FODO Lattice

D
F
B
B
L/2
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FODO Matrix

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FODO/cont.

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FODO Dispersion

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FODO Dispersion/cont.

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Momentum Compaction

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FODO Lattice

CERN Acc School
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Nondispersive Bend

CERN Acc School
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Isochronous Bend

CERN Acc School
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Eun-San Kim Kyungpook National University
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Linear Imperfections and All That

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Closed Orbit Distortion

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Quadrupole Errors
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Momentum Compaction
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Transition Energy

• Below transition, particles of higher momentum
  will have a shorter revolution period
• Above transition, particles of higher momentum
  will have a longer revolution period
• At transition, accelerator is nominally
  isochronous

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Synchrotron Motion
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Phase Stability

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Low Emittance Lattices

• As well learn later, in electron storage rings,
  the randomness of photon emission generates beam
  emittance through dispersion
• Mechanism
• In dispersive region, closed orbits are different
  for different energies
• When a particle loses energy to synchrotron
  radiation, closed orbit changes and a betatron
  oscillation is induced about new closed orbit
• Since photon emission has a random character
  about average, phase space spread is generated
• Magnitude determined by H-function
• Smaller the H-function, the smaller the
  equilibrium beam emittance

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H-Function
Game in Synchrotron Light Sources is to minimize
to make brighter photon beams Unfortunately,
easiest way to make H small is to bend gently,
which makes for big, expensive storage rings
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Dispersion in Straights Not All Bad

• For each dipole to be closer to center bending
  magnetic configuration requires dispersion
  outside of cell

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Minimum Emittance Configuration
?
?
F3
D
F1
D
Dispersion leakage
bend
bend
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Some Dispersion Tolerable

• Beam size and angle are function of both
  emittance and energy spread
• For example,
• For many machines, energy spread contribution
  small, so some dispersion can be tolerated. For
  latest machines at lowest emittance, not a good
  choice

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Damping Wigglers

• This is what NSLS II does to get sub
  nanometer-radian emittances
• Changing partition function with energy (e.g.,
  APS) is also possible

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Homework for Topic II

• Problems from S Y Lee
• 2.1.1
• 2.2.1
• 2.2.2
• 2.2.3
• 2.2.7
• 2.2.8
• 2.2.9
• 2.2.15a
• 2.4.1
• 2.4.11
• 2.4.13