Light Source Accelerator , Lecture 1

Introduction to Accelerator Physics of Storage Rings

James Safranek Stanford Linear Accelerator Center

Outline

• Transverse • Longitudinal optics • Synchrotron Radiation

N.B. Lecture borrows heavily from David Robin’s USPAS’03 & USPAS’07 lectures (www…)

Light Source Accelerator Physics, lecture 1 James Safranek, Indore, January 14-18, 2008 1 Concepts

Want to touch on a number of concepts including:

• Linear optics • Equation of motion • Transfer matrix • Twiss parameters and phase advance • Betatron tune • Dispersion • Momentum compaction • Chromaticity • Energy spread • Emittance

Light Source Accelerator Physics, lecture 1 James Safranek, Indore, January 14-18, 2008 2 Particle Storage Rings

In a particle storage rings, charged particles circulate around the ring in bunches for a large number of turns.

Optics elements Particle bunches dipoles s R e F l o c p a u v r i d ty a u q Light Source Accelerator Physics, lecture 1 James Safranek, Indore, January 14-18, 2008 3 Coordinate System

Change dependent variable from time to longitudinal position, s

Coordinate system used to describe the motion is usually locally Cartesian or cylindrical y x s

Typically the coordinate system chosen is the one that allows the easiest field representation

Light Source Accelerator Physics, lecture 1 James Safranek, Indore, January 14-18, 2008 4 Integrate

Integrate through the elements

Use the following coordinates* dx dy∆ p∆ L xx, '==== , yy , ' , δτ , ds ds p0 L

y x s

*Note sometimes one uses canonical momentum rather than x’ and y’

Light Source Accelerator Physics, lecture 1 James Safranek, Indore, January 14-18, 2008 5 Equations of Motion in a Storage Ring

The motion of each charged particle is determined by the electric and magnetic forces that it encounters as it orbits the ring:

• FmaeEvB==(), +× m is the relativistic mass of the particle, e is the charge of the particle, v is the velocity of the particle, ρ a is the acceleration of the particle, E is the electric field and, B is the magnetic field. ρ • Lorentz force in practical units (B-field only) dx' = ds/ρ → x''=1/ρ ρ 1/ [m] = 0.3 B(T)/E[GeV] or 1/ [m] = B(T)/(B ) where Bρ[Tm] = E[GeV]

Light Source Accelerator Physics, lecture 1 James Safranek, Indore, January 14-18, 2008 6 Typical Magnet Types

There are several magnet types that are used in storage rings: Dipoles Æ used for guiding

Bx = 0 By = Bo Quadrupoles Æ used for focusing

Bx = Ky By = Kx Sextupoles Æ used for chromatic correction

Bx = 2Sxy 2 2 By = S(x –y)

Light Source Accelerator Physics, lecture 1 James Safranek, Indore, January 14-18, 2008 7 Magnets that make up a storage ring

Dipole (bends e- beam) Quadrupole (focuses e- beam)

Assembled magnet lattice Sextupole (corrects aberrations) Light Source Accelerator Physics, lecture 1 James Safranek, Indore, January 14-18, 2008 8 Equations of motion – Linear fields

‰ The equations of motion are:

‰ Inhomogeneous equations with s-dependent coefficients ‰ Note that the term 1/ρ2 corresponds to the dipole weak focusing ‰ The term ∆p/(pρ) is present for off-momentum particles.

Light Source Accelerator Physics, lecture 1 James Safranek, Indore, January 14-18, 2008 9 Hill’s equation

Light Source Accelerator Physics, lecture 1 James Safranek, Indore, January 14-18, 2008 10 Two approaches

There are two approaches to introduce the motion of particles in a storage ring

1. The traditional way in which one begins with Hill’s equation, defines beta functions and dispersion, and how they are generated and propagate, …

2. The way that our computer models actually do it – transfer matrices/linear algebra, plus transfer maps.

It is worthwhile to become proficient in both approaches.

Light Source Accelerator Physics, lecture 1 James Safranek, Indore, January 14-18, 2008 11 Transfer matrix of a drift space

Light Source Accelerator Physics, lecture 1 James Safranek, Indore, January 14-18, 2008 12 Transfer matrix in a thin-lens quadrupole

Light Source Accelerator Physics, lecture 1 James Safranek, Indore, January 14-18, 2008 13 Motion in finite-length quadrupole = harmonic oscillator

Light Source Accelerator Physics, lecture 1 James Safranek, Indore, January 14-18, 2008 14 Transfer matrix of a finite-length quadrupole

Light Source Accelerator Physics, lecture 1 James Safranek, Indore, January 14-18, 2008 15 Sector Dipole

Light Source Accelerator Physics, lecture 1 James Safranek, Indore, January 14-18, 2008 16 Rectangular dipole

Light Source Accelerator Physics, lecture 1 James Safranek, Indore, January 14-18, 2008 17 Transfer Matrix Formalism

Light Source Accelerator Physics, lecture 1 James Safranek, Indore, January 14-18, 2008 18 4x4 matrices

Light Source Accelerator Physics, lecture 1 James Safranek, Indore, January 14-18, 2008 19 Quadrupoles

Light Source Accelerator Physics, lecture 1 James Safranek, Indore, January 14-18, 2008 20

Light Source Accelerator Physics, lecture 1 James Safranek, Indore, January 14-18, 2008 21 Alternate Gradient Focusing

Light Source Accelerator Physics, lecture 1 James Safranek, Indore, January 14-18, 2008 22 Back to Hill’s equation …

Light Source Accelerator Physics, lecture 1 James Safranek, Indore, January 14-18, 2008 23 Hills equation

The solution can be parameterized by a psuedo- harmonic oscillation of the form

xsβ ()=+εβ ϕ ()cos(() s ϕ s 0 )

' ββ xs()=−ε cos(()ϕϕ s +00 ) ϕϕ − sin(() s + ) β β ()αεss () where (s ) is the beta function, β '(s) α α(s) ≡ − (s ) is the alpha function, 2 ϕ γ (s ) is the betatron phase, and 1+α 2 (s) xy, (s) ≡ ε is an action variable β (s) s ds ϕ = ∫ & 2nd order differential equation for β(s). 0 β

Light Source Accelerator Physics, lecture 1 James Safranek, Indore, January 14-18, 2008 24 Example of Twiss parameters and trajectories

Light Source Accelerator Physics, lecture 1 James Safranek, Indore, January 14-18, 2008 25 Transfer matrices & β-functions

‰ Transfer matrices and β-functions came from same equation ‰ They are related:

‰ One-turn transfer map (i=f) π

β

‰ Tune is defined as number β-oscillations/turn, ν = φone-turn/2π πν α 1  Tr(R)  R R − R ν = cos−1 , = 12 , = 11 22 2  2  sin(2 ) 2sin(2πν )

‰ This is how a computer calculates optics functions – using transfer matrices

Light Source Accelerator Physics, lecture 1 James Safranek, Indore, January 14-18, 2008 26 Dispersion and momentum compaction

Assume that the energy is fixed Æ no cavity or damping • Find the closed orbit for a particle with slightly different energy than the nominal particle. The dispersion is the difference in closed orbit between them normalized by the relative momentum difference ∆p/p = 0 ∆p ∆p xD==, yD xyp p ∆p/p > 0 ∆p ∆p xD''==, yD '' xyp p

Light Source Accelerator Physics, lecture 1 James Safranek, Indore, January 14-18, 2008 27 Dispersion

Dispersion, D, is the change in closed orbit as a function of energy ∆E ∆E/E = 0 xD= x E ∆E/E > 0

xCSDx x   '  xCSDx''''=  x     δ fi 00 1 δ

Light Source Accelerator Physics, lecture 1 James Safranek, Indore, January 14-18, 2008 28 Momentum Compaction

Momentum compaction, α, is the change in the closed orbit length as a function of energy. ∆∆LE ∆E/E = 0 = α LE ∆E/E > 0

L0 D α = ∫ x ds 0 ρ Light Source Accelerator Physics, lecture 1 James Safranek, Indore, January 14-18, 2008 29 Beam Ellipse

In an linear uncoupled machine the turn-by-turn positions and angles of the particle motion will lie on an ellipse

Area of the ellipse, ε :

εγ=+xxxx α22 β2 '' +

xsβ ()=+εβ ϕ ()cos(() s ϕ s 0 )

' ββ xs()=−ε cos(()ϕϕ s +00 ) ϕϕ − sin(() s + ) β ()αεss ()

Light Source Accelerator Physics, lecture 1 James Safranek, Indore, January 14-18, 2008 30 Emittance Definition/Statistical

Light Source Accelerator Physics, lecture 1 James Safranek, Indore, January 14-18, 2008 31 Transport of the beam ellipse

Beam ellipse matrix ε x β −α = ∑ beam x  −α γ

Transformation of the beam ellipse matrix

xxT =RRxi, − f ∑∑beam,, f beam i xi, − f

Light Source Accelerator Physics, lecture 1 James Safranek, Indore, January 14-18, 2008 32 Transport of the beam ellipse

Transport of the twiss parameters in terms of the transfer matrix elements β CCSS22−2 β   α =−CC''1 + C S − SS ' α γ CCSS''22−2 '' γ fi 

Transfer matrix can be expressed in terms of the twiss parameters and phase advances  β f ()cosfi+ i sin fi f i sin fi βi ϕα ϕ ββ ϕ R fi =  1+−ifαα α α i f −+sin cosi cos − sin fi fi() fi f fi fiββ ββ ϕϕϕαϕ fi f

Light Source Accelerator Physics, lecture 1 James Safranek, Indore, January 14-18, 2008 33 β β Chromatic Aberration

Focal length of the lens is dependent upon energy

Larger energy particles have longer focal lengths

Light Source Accelerator Physics, lecture 1 James Safranek, Indore, January 14-18, 2008 34 Chromatic Aberration Correction

By including dispersion and sextupoles it is possible to compensate (to first order) for chromatic aberrations

The sextupole gives a position dependent Quadrupole

Bx = 2Sxy 2 2 By = S(x –y)

Light Source Accelerator Physics, lecture 1 James Safranek, Indore, January 14-18, 2008 35 Chromatic Aberration Correction

Chromaticity, ν’, is the change in the tune with energy dν ν ' = dδ Sextupoles can change the chromaticity

1 ∆=∆νβxxx' 2π ( SD)

1 ∆=−∆νβyyx' 2 ()SD π where

2 ∂ By ∆=S 2 *glength ()2Bρ ∂x

Light Source Accelerator Physics, lecture 1 James Safranek, Indore, January 14-18, 2008 36