Eric Prebys, FNAL. Let’s look at the Hill’ equation again… We can write the general solution as a linear combination of a “sine-like” and “cosine-like”

Transverse Motion 2Eric Prebys, FNAL

Some Formalism Let’s look at the Hill’ equation again… We can write the general solution as a linear combination of a

“sine-like” and “cosine-like” term where

When we plug this into the original equation, we see that

Since a and b are arbitrary, each function must independently satisfy the equation. We further see that when we look at our initial conditions

So our transfer matrix becomes

USPAS, Knoxville, TN, Jan. 20-31, 2013Lecture 4 - Transverse Motion 1 2

0)( xsKx

)()()( sbSsaCsx

1)0(;0)0(

0)0(;1)0(

SC

SC

0)()()()()()( sSsKsSbsCsKsCa

0

0

)0()0()0(

)0()0()0(

xbbSbCax

xaabSaCx

0

0

0

0

)()(

)()(

)(

)(

)(

)(

x

x

sSsC

sSsC

sx

sx

x

x

sx

sxM

Calculating the Lattice functions If we know the transfer matrix or one period, we can explicitly

calculate the lattice functions at the ends

If we know the lattice functions at one point, we can use the transfer matrix to transfer them to another point by considering the following two equivalent things Going around the ring, starting and ending at point a, then proceeding to point b Going from point a to point b, then going all the way around the ring


2cos1sin;2

1cos

sincossin

sinsincos

M

M

Tr

),(),(),(),(

),(),(),(),(1

abaaabbb

aaababbb

sssCssssCs

sCssssssCs

MMMM

MMMM

),()(),()(

),(2sin)(2cos),(2sin)(2cos

2sin)(2cos),(

1

1

abaabb

abaabb

ssssss

ssssss

ssCs

MJMJ

MJIMJI

JIM

Recall:

)()(

)()()(

ss

sss

J

Calculating the Lattice functions (cont’d) Using

We can now evolve the J matrix at any point as

Multiplying this mess out and gathering terms, we get


1121

1222

2221

1211

)()(

)()(

)()(

)()()(

mm

mm

ss

ss

mm

mm

ss

sss

aa

aa

bb

bbb

J

1121

12221

2221

1211 ),(),(mm

mmss

mm

mmss abab MM

)(

)(

)(

2

2

)(

)(

)(

222

2212221

212

2111211

2212211121122211

a

a

a

b

b

b

s

s

s

mmmm

mmmm

mmmmmmmm

s

s

s

Examples Drift of length L:

Thin focusing (defocusing) lens:


0200

0

00

0

0

0

212

1

112

010

01

1

11

01

ff

f

ff

f

f

M

Physical Implications The general expressions for motion are

We form the combination

If you don’t get out much, you recognize this as the general equation for an ellipse


sincos

)(;cos

Ax

sAx

x

'x

A

A

Area = πA2

Particle will trace out the ellipse on subsequent revolutions

Interpretation (cont’d) As particles go through the lattice, the Twiss parameters will vary

periodically:

s

x

x

x

x

x

x

x

x

x

x

β = maxα = 0maximum

β = decreasingα >0focusing

β = minα = 0minimum

β = increasingα < 0defocusing

USPAS, Knoxville, TN, Jan. 20-31, 2013 7Lecture 4 - Transverse Motion 1

Motion at each point bounded by

)()( sAsx

Conceptual understanding of β

It’s important to remember that the betatron function represents a bounding envelope to the beam motion, not the beam motion itself


Normalized particle trajectory Trajectories over multiple turns

)(sin)()( 2/1 ssAsx

s

s

dss

0 )()(

(s) is also effectively the local

wave number which determines the rate of phase advance

Closely spaced strong quads small β small aperture, lots of wiggles

Sparsely spaced weak quads large β large aperture, few wiggles

Betatron tune

As particles go around a ring, they will undergo a number of betatrons oscillations ν (sometimes Q) given by

This is referred to as the “tune”

We can generally think of the tune in two parts:

Ideal orbit

Particle trajectory

)(2

1

s

ds

6.7Integer : magnet/aperture

optimization

Fraction: Beam Stability


Tune, stability, and the tune plane

If the tune is an integer, or low order rational number, then the effect of any imperfection or perturbation will tend be reinforced on subsequent orbits.

When we add the effects of coupling between the planes, we find this is also true for combinations of the tunes from both planes, so in general, we want to avoid

Many instabilities occur when something perturbs the tune of the beam, or part of the beam, until it falls onto a resonance, thus you will often hear effects characterized by the “tune shift” they produce.

y)instabilit(resonant integer yyxx kk

“small” integers

fract. part of X tune

frac

t. pa

rt o

f Y tu

ne

Avoid lines in the “tune plane”


(We’ll talk about this in much more detail soon, but in general…)

Emittance

x

'xIf each particle is described by an ellipse with a particular amplitude, then an ensemble of particles will always remain within a bounding ellipse of a particular area:

Area = e

Either leave the out, or include it explicitly as a “unit”. Thus

• microns (CERN) and

• -mm-mr (FNAL)

Are actually the same units (just remember you’ll never have to explicity use in the calculation)


or 22 xxxx

Definitions of Emittance and Admittance Because distributions normally have long tails, we have to adopt a

convention for defining the emittance. The two most common are Gaussian (electron machines, CERN):

95% Emmittance (FNAL):

It is also useful to talk about the “Admittance”: the area of the largest amplitude ellipese which can propagate through a beam line


beam theof 39% contains ;2

x

2d

ALimiting half-aperture

Adiabatic Damping In our discussions up to now, we assume that all fields scale with

momentum, so our lattice remains the same, but what happens to the ensemble of particles? Consider what happens to the slope of a particle as the forward momentum.

If we evaluate the emittance at a point where =0, we have


0p

xp

pp 0

xp0p

px x

0

00

1

p

pxx

p

px

pp

pxx x

Normalized emittance

Printed copy has lots of typos!

Mismatch and Emittance Dilution In our previous discussion, we implicitly assumed that the

distribution of particles in phase space followed the ellipse defined by the lattice function

Once injected, these particles will follow the path defined by the lattice ellipse, effectively increasing the emittance


x

'x

Area = e

…but there’s no guarantee What happens if this it’s not?

'x

x

Lattice ellipse

Injected particle distribution'x

xEffective (increased) emittance

Beam Lines In our definition and derivation of the lattice function, a closed path

through a periodic system. This definition doesn’t exist for a beam line, but once we know the lattice functions at one point, we know how to propagate the lattice function down the beam line.


in

in

in

out

out

out

mmmm

mmmm

mmmmmmmm

222

2212221

212

2111211

2212211121122211

2

2

in

in

in

out

out

out

inout,M

Establishing Initial Conditions When extracting beam from a ring, the initial optics of the beam

line are set by the optics at the point of extraction.

For particles from a source, the initial lattice functions can be defined by the distribution of the particles out of the source


in

in

in

in

in

in

Documents

Eric Prebys, FNAL. Let’s look at the Hill’ equation again… We can write the general solution as a linear combination of a “sine-like” and “cosine-like”