Fundamentals of Accelerators - U.S. Particle Accelerator School

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US Particle Accelerator School Fundamentals of Accelerators Lecture - Day 2 - Beam properties William A. Barletta Director, US Particle Accelerator School Dept. of Physics, MIT Economics Faculty, University of Ljubljana 1

Transcript of Fundamentals of Accelerators - U.S. Particle Accelerator School

US Particle Accelerator School

Fundamentals of AcceleratorsLecture - Day 2 - Beam properties

William A. BarlettaDirector, US Particle Accelerator School

Dept. of Physics, MITEconomics Faculty, University of Ljubljana

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Ions - either missing electrons (+) or with extra electrons (-)

Electrons or positrons

Plasma - ions plus electrons

Source techniques depend on type of beam & on application

Beams: particle buncheswith directed velocity

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Electron sources - thermionic

Heated metals Some electrons have energies above potential barrier

# of

ele

ctro

ns

Cannot escape

Enough energyto escape

Work function = φ

!

dn(E)dE

= A E 1e(E"EF ) / kT +1[ ]

+ HV

Electrons in a metal obey Fermi statistics

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Electrons with enough momentum can escapefrom the metal

Integrating over electrons going in the z direction with

yields

some considerable manipulation yields the Richardson-Dushmanequation

!

I" AT 2 exp #q$kBT%

& '

(

) *

A=1202 mA/mm2K2

!

pz2 /2m > EF + "

!

Je = dpx"#

#

$ dpy"#

#

$ dpxpz , free

#

$ (2 /h3) f (E)vz

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Brightness of a beam source

A figure of merit for the performance of a beam source isthe brightness

Typically the normalized brightness is quoted for γ = 1

!

B = Beam currentBeam area o Beam Divergence

=Emissivity (J)

Temperature/mass

=JekT

"moc2

#

$ %

&

' (

2 =Je"

kTmoc

2#

$ %

&

' (

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Other ways to get electronsover the potential barrier

Field emission Sharp needle enhances electric field

Photoemission from metals & semi-conductors Photon energy exceeds the work function These sources produce beams with high current densities & low

thermal energy This is a major topic of research

+ HV

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Anatomy of an ion source

Gas in

Energy in to ionize gas

Plasma

Container for plasma

Electron filter

Extraction electrodes

Beam

out

Electron beams can also be used to ionize the gas or sputter ions from a solid

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What properties characterize particle beams?

5 minute exercise

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Beams have directed energy

The beam momentum refers to the average value of pz ofthe particles

pbeam = <pz>

The beam energy refers to the mean value of

For highly relativistic beams pc>>mc2, therefore

!

Ebeam = pz2c 2 +m2c 4[ ]

1/ 2

!

Ebeam = pz c

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Measuring beam energy & energy spread

Magnetic spectrometer - for good resolution, Δp one needs small sample emittance ε, (parallel particle velocities) a large beamwidth w in the bending magnet a large angle ϕ

!

"p =#w$

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Beam carry a current

I ~ ne<vz>

Imacro

I

τpulse

Q

!

I peak = Q" pulse

!

Iave =QtotT

Τ

!

Duty factor =" pulse#T

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Measuring the beam current

Examples: Non-intercepting: Wall current monitors, waveguide pick-ups Intercepting: Collect the charge; let it drain through a current meter

• Faraday Cup

Beam current

Return current

Voltage meter

Βeam pipe

Beam current

Stripline waveguide

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Collecting the charge: Right & wrong ways

Proper Faraday cupSimple collector

The Faraday cup

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Beams particles have random (thermal) ⊥ motion

Beams must be confined against thermal expansion duringtransport

Thermal characteristics of beams

!

" x =pxx

pz2

1/ 2

> 0

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Beams have internal (self-forces)

Space charge forces Like charges repel Like currents attract

For a long thin beam

!

Esp (V /cm) =60 Ibeam (A)Rbeam (cm)

!

B" (gauss) = Ibeam (A)

5 Rbeam (cm)

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Net force due to transverse self-fields

In vacuum:Beam’s transverse self-force scale as 1/γ2

Space charge repulsion: Esp,⊥ ~ Nbeam

Pinch field: Bθ ~ Ibeam ~ vz Nbeam ~ vz Esp

∴Fsp ,⊥ = q (Esp,⊥ + vz x Bθ) ~ (1-v2) Nbeam ~ Nbeam/γ2

Beams in collision are not in vacuum (beam-beam effects)

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Interaction point fields in the proposed ILC(10 minute exercise)

The International Linear Collider proposes to collide bunchesof e- & e+ with 10 nC each. Each bunch will be 3 µm long &10 nm in radius.When the bunches overlap at the Interaction Point, what self-forces will particles at the edges of the beams experience? Howlarge are the fields?What consequences might you expect?

electrons positrons

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Bunch dimensions

X

Y

Z

For uniform charge distributionsWe may use “hard edge values

For gaussian charge distributions Use rms values σx, σy, σz

We will discuss measurements of bunch size and charge distribution later

σx,

σy

σz,

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But rms values can be misleading

σ σ

Gaussian beam Beam with halo

We need to measure the particle distribution

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What is this thing called beam quality?or

How can one describe the dynamics ofa bunch of particles?

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Coordinate space

Each of Nb particles is tracked in ordinary 3-D space

Not too helpful

Orbit traces

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Configuration space:

6Nb-dimensional space for Nb particles; coordinates (xi, pi), i = 1,…, Nb

The bunch is represented by a single point that moves in time

Useful for Hamiltonian dynamics

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Configuration space example:One particle in an harmonic potential

px

x

But for many problems this description carriesmuch more information than needed :

We don’t care about each of 1010 individual particlesBut seeing both the x & px looks useful

ωb constant

!

Fx = "kx = m˙ ̇ x

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Option 3: Phase space(gas space in statistical mechanics)

6-dimensional space for Nb particlesThe ith particle has coordinates (xi, pi), i = x, y, zThe bunch is represented by Nb points that move in time

px

x

In most cases, the three planes are to very good approximation decoupled ==> One can study the particle evolution independently in each planes:

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Particles Systems & Ensembles

The set of possible states for a system of N particles is referred as anensemble in statistical mechanics.

In the statistical approach, particles lose their individuality.

Properties of the whole system are fully represented by particle densityfunctions f6D and f2D :

where

!

f6D x, px,y, py,z, pz( ) dx dpx dy dpy dz dpz

!

f2D xi, pi( )dxi dpi i =1,2,3

Ndpdzdpdydpdxf zyxD =! 6

From: Sannibale USPAS lectures

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Longitudinal phase space

In most accelerators the phase space planes are only weakly coupled. Treat the longitudinal plane independently from the transverse one Effects of weak coupling can be treated as a perturbation of the

uncoupled solution

In the longitudinal plane, electric fields accelerate the particles Use energy as longitudinal variable together with its canonical

conjugate time

Frequently, we use relative energy variation δ & relative time τ withrespect to a reference particle

According to Liouville, in the presence of Hamiltonian forces, the areaoccupied by the beam in the longitudinal phase space is conserved

!

" =E # E0

E00tt !="

From: Sannibale USPAS lectures

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Transverse phase space

For transverse planes {x, px} and {y, py}, use a modified phase space wherethe momentum components are replaced by:

where s is the direction of motion

We can relate the old and new variables (for Bz ≠0 ) by

Note: xi and pi are canonical conjugate variables while x and xi’ are not, unlessthere is no acceleration (γ and β constant)

From: Sannibale USPAS lectures

dsdx

xpxi =!"dsdyypyi =!" w

s

PROJECTIONWSp

w!

!

pi = " m0dxi

dt= " m0 vs

dxi

ds= " #m0 c $ x i i = x,y

( ) 2121 !!== "#" and

cvwhere s

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Consider an ensemble of harmonic oscillatorsin phase space

px

x

Particles stay on their energy contour.

Again the phase area of the ensemble is conserved

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Emittance describes area in phase spaceof the ensemble of beam particles

Phase space of anharmonic oscillator

Emittance - Phase space volume of beam

!

"2 # R2(V 2 $ ( % R )2) /c 2RMS emittance

kβ(x) - frequency ofrotation of a phase

volume

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What is the significance of(physical interpretation of)

the term

!

R2( " R )2

c 2 ?

5 minute exercise

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Force-free expansion of a beam

Notice: The phase space area is conserved

px

x

px

x

Drift distance L

0

00

0

0

101

xxxLxx

xxL

xx

!=!

!+="##

$

%&&'

(!##

$

%&&'

(=##$

%&&'

(!

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A numerical example:Free expansion of a due due to emittance

!

R2 = Ro2 +Vo

2L2 = Ro2 +

"2

Ro2 L

2

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The rms emittance is a measureof the mean non-directed (thermal) energy

of the beam

This emittance is the phase space area occupied by the system of particles, divided by π

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Why is emittance an important concept

@ t1

@ t21) Liouville: Under conservative forces phase space

evolves like an incompressible fluid ==>

Z = λ/12

Z = λ/8Z = λ/4

Z = 0x

x’2) Under linear forces macroscopic

(such as focusing magnets) &γ =constant

emittance is an invariant of motion

3) Under accelerationγε = εn

is an adiabatic invariant

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An axial Bz field, (e.g.,solenoidal lenses) couples transverse planes The 2-D Phase space area occupied by the system in each transverse plane is no

longer conserved

Liouville’s theorem still applies to the 4D transverse phase space the 4-D hypervolume is an invariant of the motion

In a frame rotating around the z axis by the Larmor frequencyωL = qBz / 2g m0, the transverse planes decouple The phase space area in each of the planes is conserved again

Emittance conservation with Bz

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Emittance during acceleration

When the beam is accelerated, β & γ change x and x’ are no longer canonical Liouville theorem does not apply & emittance is not invariant

0yp

0zp0!

0p

energykineticT

pcmTTcTmTp zz

!

+

+= 02

0020

20

2

22

Accelerate by Ez

0yp

!

p

pz

From: Sannibale USPAS lectures

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Then…

Therefore, the quantity β γ ε is invariant during acceleration. Define a conserved normalized emittance

Acceleration couples the longitudinal plane with the transverse planesThe 6D emittance is still conserved but the transverse ones are not

cmp

pp

y y

z

y

0

0tan!"

# ===$cm

ppp

y y

z

y

000

0

0

000 tan

!"# ===$

!"!" 00

0

=#

#

yy

!

In this case"y

"y0

=# y # y 0 000 yy !"#!"# ===>

!

"n i = # $"i i = x,y

From: Sannibale USPAS lectures

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Consider a cold beam with aGaussian charge distributionentering a dense plasma

At the beam head the plasma shortsout the Er leaving only theazimuthal B-field

The beam begins to pinch trying tofind an equilibrium radius

Nonlinear space-charge fieldsfilament phase space via Landau damping

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Experimental example:Filamentation of longitudinal phase space

Data from CERN PS

The emittance according to Liouville is still conserved

Macroscopic (rms) emittance is not conserved

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Non-conservative forces increase emittance(scattering)

Scatterer!

Scatterer! <Vx> <Vx + "Vx>

Px/Pz

X

Px/Pz

X

Px/Pz

X

Px/Pz

X

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Measuring the emittance of the beam

RMS emittance Determine rms values of velocity & spatial distribution

Ideally determine distribution functions & compute rmsvalues

Destructive and non-destructive diagnostics

!

"2 = R2(V 2 # ( $ R )2) /c 2

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Example of pepper-pot diagnostic

Size of image ==> R Spread in overall image ==> R´ Spread in beamlets ==> V Intensity of beamlets ==> current density

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Wire scanning to measure R and ε

SNS Wire Scanner

Measure x-ray signal from beamscattering from thin tungsten wires

Requires at least 3 measurements alongthe beamline

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Horizontal, 0.22 pi mm mrad Vertical, 0.15 pi mm mrad

111

mra

d, fu

ll sc

ale

10.1 mm full scale 5.9 mm full scale

1 0 .75 0 . 5 0 . 3 0 .14 0 .07 0 .04 0 .02 1 0 .75 0 . 5 0 .25 0 . 1 0 .05 0 .03 0 .01311

1 m

rad,

full

scal

e

Measured 33-mA Beam RMS Emittances

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The Concept of Acceptance

Example: Acceptance of a slit

y

y’

-h/2

h/d

-h/d

-h/2h

d

ElectronTrajectories

Matched beamemittance

Acceptance atthe slit entrance

Unmatched beamemittance

From: Sannibale USPAS lectures

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Is there any way to decrease the emittance?

This means taking away mean transverse momentum,but

keeping mean longitudinal momentum

We’ll leave the details for later in the course.

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Phase-Space Cooling in Any One Dimension

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Schematic: radiation & ionization cooling

Transverse cooling:

Passage through dipoles

Acceleration in RF cavity

P⊥ lessP|| less

P⊥ remains lessP|| restored

Limited by quantum excitation

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Cartoon of transverse stochastic cooling

Divide (sample) the beam into disks

1) rf pick-up samples centroid of disks

2) Kicker electrode imparts v⊥

to center the disk

3) Mix up the particles & repeat

Van der Meer Nobel prize

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