Outline

Statistical Description of Charged ParticleBeams and Emittance Measurements

Jürgen [email protected] www.gsi.de/~struck

HICforFAIR Workshop “Aktuelle Probleme der Beschleuniger- und Plasmaphysik”

des Instituts für Angewandte Physikder Johann Wolfgang Goethe-Universität Frankfurt am Main

Riezlern, 7-13 March 2010

Outline

1. Emittance in geometrical optics2. Beam moments3. Invariant associated with Hill’s equation4. Moment equations5. Analysis of emittance measurements6. Conclusions

1. Emittance in geometrical optics

“Plane wave” from a distant light source

Light from a spatially extended light source(e.g. sun)

Sharp focal point: thiscorresponds to a zero beam emittance

Astigmatism:this correspondsto a beam with finite emittance

The emittance measures the “disorder” of the beam. An analytical description will be given in this talk.

2. Beam moments

In charged particle beam dynamics, we are commonly not interested in the phase-space location of individual particles

statistical mechanics approach is appropriate.

For a given distribution of N beam particles with coordinates xi, we define the moments of this distribution as

( )

1

1

1( ) ( ) : first moment in

1( ) ( ) ( ) : th central moment in

N

ii

Nnn

ii

x s x s xN

x s x s x s n xN

=

=

=

= -

å

å

The similar definition applies for all other coordinates.

2. Beam moments

( ) ( )4 4

2 22 2

,x x

x x

¢

¢

Of particular importance are the second beam moments:

and the ratios of the fourth moments to the square of the second beam moments:

( ) ( )

( )( )

( ) ( )

= =

= =

= =

= - = -

¢ ¢ ¢ ¢ ¢= - - = - ×

¢ ¢ ¢ ¢ ¢= - = -

å å

å å

å å

2 22 2

1 1

1 1

2 222

1 1

1 1( )

1 1( )

1 1( )

N N

i ii i

N N

i i i ii i

N n

i ii i

x s x x x xN N

xx s x x x x xx x xN N

x s x x x xN N

2. Beam moments

2

2 , : populated area of the ( , ) ellipseA

A

x dxdx

x A x xdxdx

¢

¢=¢

òò

òò

We ask: how are these moments related to physical beam properties?

To answer this question, we consider a uniform density of points in the (x,x′)-plane and calculate the second central moments.

we make the transition to a continuous description!

Assuming a centered distribution (i.e. zero first moments), the second moment in x is given by:

2

2( , )

( , )

x x x dxdxx

x x dxdx

r

r

¢ ¢=

¢ ¢

òòòò

For a uniform density within an elliptical boundary, this means

2. Beam moments

2 214

x X=

To ease the calculation, we can convert the ellipse into a circle by means of a normal form and a scaling transformation.

The shape of the distribution is maintained.

22 2 3

20 02 0

2

0 0 0

( , ) 1, cos , sin ,

cos

42

X X

r rX X

r r

x x x r x r dxdx rdrd

r rdrd r dr Xx

rdrd r dr

p

jp

j

r j j j

j j p

j p

= = =

= = =

¢ ¢ ¢= = = =

= = =ò ò ò

ò ò ò The second moment is thus proportional to the square of the

distribution’s maximum, here the beam envelope.

The factor ¼ is aThe factor ¼ is acharacteristic of the characteristic of the

uniformuniform density! density!

2. Beam moments

( )4

22

2x

x=

The 4th moment of a uniform distribution in a phase plane is obtained as

254 4

400 04

2

0 0 0

( , ) 1, cos , sin ,

3cos

482

XX

rrX X

r r

x x x r x r dxdx rdrd

r drr rdrd Xx

rdrd r dr

p

jp

j

r j j j

pj j

j p

== =

= = =

¢ ¢ ¢= = = =

= = =òò ò

ò ò òThe dimensionless ratio of the 4th moment to the square of the

corresponding 2nd moment follows as

The ratio of 2 is aThe ratio of 2 is acharacteristic of the characteristic of the

uniformuniform density! density!

The ratio characterizes the type of beam distribution (hollow, uniform, gaussian, …)

2. Beam moments

2 4

2

141 96 41 128 519116

22

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

Type if distribution

hollow 1 1

KV 2

Waterbag

parabolic

gaussian 2.7125 (truncated at 3 )

gaussian 2.9446 (truncated at 4 )

x

Xx

x

s

s

æ ö÷ç ÷ç ÷çè ø

» »

» »

Table 1: Characteristic ratios of beam moments

2. Beam moments

We have seen that the beam moments have a physical meaning at a fixed position s along the beam axis.

Now the dynamical properties of the beam moments will be derived. To this end, we set up their equations of motion from the single particle equations of motions.

We first derive the invariant of Hill’s equation.

3. Invariant of Hill’s equation

2 21 1 2 2

( ) 0, ( ) 0x k s x x k s x¢¢ ¢¢+ = + =

1 2 2 1D xx x x¢ ¢= -

0dDds

º

Given a pair of distinct particles whose motion follows from Hill’s equation

Then

is a non-trivial invariant, i.e.

We can prove this easily by direct computation:

1 2 1 2 2 1 2 1

2 21 2 2 1

( ) ( )

0

dDxx xx x x x x

dsx k s x x k s x

¢¢ ¢¢ ¢¢ ¢ ¢= + - -

= - +

ºRemark:Remark: the particle’s the particle’s

energy is energy is notnot invariant. invariant.

3. Invariant of Hill’s equation

Interpretation:

• both particles experience the same focusing function k2(s) in a two-dimensional rectangular system, this corresponds to a circular symmetric force, i.e., a central force field

• The invariant thus corresponds to the conservation of angular momentum in central force fields.

for a system of n particles the sum D1 is also a constant

( ) ( )

( )

( )

2 2 2 2 21 2 2

1 1 1 1

2 2 2 2

1

22 2 2

,rms

1 12

2 2

12

2

square of the "rms emittance"

n n n n

i j j i i j i i j j j ii j i j

n

i i i ii

x

D xx x x x x xx x x x xn n

x x xx xx x xn

x x xx e

= = = =

=

¢ ¢ ¢ ¢ ¢ ¢= - = - +

¢ ¢ ¢ ¢= - +

¢ ¢= - º =

å å å å

å

4. Moment equations

2

2 2 2

2 ,

( )

i i ii i

i i i ii i i

dx xx

dsd

xx k s x xds

¢=

¢ ¢= - +

å å

å å å

We now set up the equations of motion for the second moments:

22 2 2 2

22 ( ) 2 0

dx k s x x

ds¢+ - =

Inserting the second equation into the first yields the second order equation for the second central moment in x:

This form is not useful as it contains two kinds of second moments.

4. Moment equations

22 2

2

2 2,rm2

2s2

2 ( ) 02 1

4x

ddx k xs x

ds dsxeé ùæ öê ú÷ç ÷+ çê ú÷ç ÷çè øê úë û

+ - =

22 2,rms2 2 2

,rms2 2

1 14

xx

xx dx x

dsx x

eeé ù¢+ æ ö÷ê úç¢ = + ÷ç ÷çê è øë û

= ú

Yet — in the linear approximation — we can express the unwanted moment in terms of the invariant of Hill’s equation D1≡ x

2,rms :

We thus obtain an equation that only contains constants and functions of the second central moment in x:

This equation can be simplified making use of the identity2 2

2 2 2

222

2

21

22

dx

d dx x x

ds dsdsx

æ ö÷ç =÷ç ÷çè ø-

4. Moment equations

2,rms

22

32( ) 0xd

x k s xs xd

e+ - =

%% %

Defining the abbreviation

we obtain the final form for the equation of motion of the variance of the set of beam particles in x:

This is the rms envelope equation. It applies to arbitrary phase-space distributions of the beam particles if the particle equations of motion are linear.

If the particle motion is non-linear, then x,rms is an unknown function of s. We can nevertheless use this equation as an approximation as long as the rms emittance is an adiabatic invariant (e.g.: transient effects).

def 2x x=%

2 2 2,rms ,rms

0 2 22 20 0

1 1, [J oules/ meter]

4x y u

d d W Wd IK w

ds ds ds w cx y

e e

pe b

-+ = - =

The quantity W and denotes the field energy per unit length of all particles of the actual distribution. Wu the describes the field energy for a uniform charge distribution of same rms size.

We see that the rms emittance is directly related to physical quantities, hence it is physical on its part.

3 3 30

2,

4

eIK I Nec

mcb

pe b g= =

One source of rms emittance change is due to a reversible change of the beam’s field energy. With the scaled beam current K

4. Moment equations

this effect is described (for unbunched beams) by the equation

4. Moment equations

Mismatched beam in a quadrupole channel, s0 = 60°

4. Moment equations

Matched beam in a quadrupole channel, s0 = 60°

4. Moment equations

We have seen that the beam moments have both a static and a dynamic meaning.

It will now be shown that we can directly measure the moments.

5. Analysis of emittance measurements

Principle and setup of a “slit and collector” emittance measurement device

h

d

g

a

Schlitz Gitter

Strahlachse

Ionenstrahl


Simple graphical 3D representation of the raw data of a “slit-and-collector” emittance measurement


Calculation of beam moments from the raw data, given by the current matrix

max max

( , ) , ,

1, , , 1, ,nmi x x i x n x x m x

n n m m

¢ ¢ ¢º = ×D = ×D

= =K K

with inm the collector current as a function of n and m, and

x: step size of the slit position. This defines the spatial resolution of the device.

x′: angle between neighboring collector stripes, which defines the angular resolution.


( ) ( )

( )( )

2 22 22 2

, ,, ,

,,

( ) ( ),

nm nmnm nmnm nm

nm nm

nmnmnm

nm

x xx i n n x i m m

i i

x xxx i n n m m

i

¢D D¢= - = -

¢D D¢= - -

å åå å

åå

The first beam moments are now

The second moment follow as

, ,, ,

, ,, ,

1 1

1 1

nm nmnm nmnm nm

nm nm

nm nmnm nmnm nm

nm nm

xx i n x n i n

i x i

xx i m x m i m

i x i

= D Þ º =D

¢¢ ¢= D Þ º =

¢D

å åå å

å åå å


( )( )

( )

( )

( )

4

4 44 ,4

2 22, , 2

,

,

nmnm

nm nmnm nmnm

nm nmnm

i n nx x

x i n n ii x i n n

-D

= - Þ =æ ö÷ç ÷-ç ÷ç ÷çè ø

åå åå

å

Finally, the 4th beam moments are given by

Higher order moments can be calculated correspondingly.

The “rms emittance” follows from the “current matrix” inm as

( )( ) ( ) ( )( )

22 22 22

rms 2, , ,

,

( )nm nm nm

nm nm nm

nmnm

x xi n n i m m i n n m m

i

eé ùæ öæ ö æ ö¢D D ÷ ÷ ÷ê úç ç ç÷ ÷ ÷= - - - - -ç ç çê ÷ ÷ ÷úç ç ç÷ ÷ ÷ç ç çæ ö è øè ø è øê ú÷ç ë û÷ç ÷ç ÷çè ø

å å åå

6. Conclusions

Measuring the properties of charged particle beam dynamics, a statistical approach is appropriate in terms of beam moments.

The naïve approach to derive emittance “areas” from the raw data fails as a “boundary” is not well defined and hence so inaccurate that the results are mostly useless (see PhD thesis A. Schönlein).

To demonstrate this, we consider the phase-space evolution of a set of points representing solutions of the mathematical pendulum, with each point representing the evolution of a specific initial condition.

What would happen if we derived “areas” with a finite resolution measuring device?

6. Conclusions

Time evolution of a uniform (top) and a Gaussian (bottom) phase-space distribution for the mathematical pendulum.

6. Conclusions

Properties of the rms emittance calculation:

• no assumption is made with respect to the phase-space distribution of the beam particles

• all information from the measurement is taken into account

• there is usually no need to define a cut-off current imin, so that inm=0 for inm,raw<imin

• the calculated moments (and hence the rms emittance) can directly be compared to the moments of multi-particle simulations and to the moments of continuous model distributions.

• The method can be generalized: if we find a method to measure simultaneously the current at n(x), m(x′), j(y), k(y′) yielding a current tensor inmjk, then the entire 4×4 transverse beam matrix could be calculated.

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