Emittance and Emittance Measurements

S. BernalUSPAS 08

U. of Maryland, College Park

IREAPIREAPIntroduction: Phase and Trace Spaces, Liouville’s theorem, etc.

Liouville’s Theorem:area conserved in either Γ or μ spaces for Hamiltonian systems

Units of area in Phase Space (μ−space) are those of Angular Momentum.

Vlasov Approx.

6N dim.

3Np

3Nx

Γ-Space

3p

3x

μ-Space6 dim.

Units of area in Trace Space are those of length: “metre” (angle measure is dimensionless).

pX→ ′x : Trace Space

′3x

3x

[ ],T3X = x, y, z [ ]′ ′ ′T

3X = x , y , Δp pMatrix notation:

[ ]′ ′T T6X ≡ X = x, x , y, y , z, Δp por

IREAPIREAPReview of Courant-Snyder TheoryTransverse coordinate for paraxial particle motion in linear lattice obeys :

β(s) is a lattice function that depends on κ0 (s), but A and φare different for different particles.

,( ) ( )cos ( )x s Aw s sψ ϕ⎡ ⎤⎢ ⎥⎣ ⎦

= +

1 23

10( ) ( ) ( ) 0 , ( ) ( )

( )w s s w s w s s

w sκ β+ − = ≡′′

α, β, γ: Courant-Snyder or Twiss Parameters.

′ ′γ α X2 2 2x - 2 xx + βx = AFurther,

The ellipse equation defines self-similar ellipses of area πAX2at a given plane s. The largest ellipse defines the beam emittance through

area of largest ellipsexε πXmax

2= A =

IREAPIREAPSpace charge and emittance

(M. Reiser, Chap. 5)If linear space charge is included, then κ0(s)→κ(s) and the β function is no longer just a lattice function because it now depends on current also. β becomes a lattice+beam function.

The emittance above is the effective or 4×rms emittance (also called edge emittance) which is equal to the total (100%) emittance for a K-V distribution.

Further, adding space charge does not require changing the emittance definition…Emittance and current are independent parameters in the envelope equation:

For a Gaussian distribution, the total and RMS emittances are equal when the Gaussian is truncated at 1-σ.

′′22

0 02 30

K ε εR + k R - - = 0, a = kR Ra0: zero-current 2×rms beam radius

IREAPIREAPRMS Emittance and Envelope Equation(Pat G. O’Shea)

′′ ′ ′′′ ′′ ′

2

2

xxx xx xxa = , a = - a + + ,

a a a a

2RMSa ≡ x = x ,′′ 2x = -k x,Linear system: Define:

′′′

2 2 22

x k xxx= - a + - ,

a a a2 ′′ 2

23

xxx= - + - k a,

a a⎡ ⎤′′ ′ ′⎣ ⎦

2 2 2 23

1a = - xx - x x - k a,a

′′2

2 RMS3

εa + k a - = 0,a

( )′ ′ 22 2RMSε = x x - xx

IREAPIREAPEmittance Growth (M. Reiser, Chap. 6)

X’

x

X’

x

X’

x

X’

x

Without acceleration, rms emittance of a mono-energetic beam is conserved under linear forces (both external and internal):

nrms rmsε = εβγ

The normalized emittance is an invariant:

If initial transverse beam distribution is not uniform, rms emittance grows in a length of the order of λp/4.

If initial beam envelope is not rms-matched, rms emittance grows in a length of the order of λβ.

IREAPIREAPLaminar and Turbulent Flows

Transverse Emittance Measurement Techniques

IREAPIREAP

TechniqueTime

Resolved?

Pros Cons

Basic Optics NoSimple to implement.Linear space charge OK.Small or large beams.

RMS emittance only.No phase-space.

Pepper-pot NoSimple to implement.Space charge OK.No beam optics.

Coarse phase-space.Impractical for small beams.

Quad Scan No Different schemes.Easy computation.

Space charge limits.No phase-space.Linear optics assumed.

Tomography NoDetailed phase space info.Linear optics/sp. charge.

More beam manipulation required.Computationally intensive.

Slit-Wire YesPhase space info.Integrated or t-resolved.Space charge OK.

S/N problems.Hard to implement.Comp. intensive.

Basic Optics I IREAPIREAP

Without space charge we have:

ε⎡ ⎤⎛ ⎞′ ′= + + +⎢ ⎥⎜ ⎟

⎢ ⎥⎝ ⎠⎣ ⎦

122

2 2 20 0 0 02

0

( ) 2 .R z R R R z R zR

We find:

ε = 0 min ,R RL

ε′ = ≠min0

0.RR

By focusing a beam with a lens and measuring the width at a fixed distance z=L downstream from the lens, one candetermine the emittance.

R0

′0Rslope

R(L)radius

Z=L

Z=0

Z

Basic Optics II IREAPIREAP

ε μ= 67 .m1 cm

Z = 1.28 cm

Z = 3.83 cm

Z = 6.38 cm

Z = 8.88 cm

Z = 11.28 cm

0

50

100

150

200

250

0 5 10 15 20X (mm)

1.28 cm

3.83 cm

6.38 cm

8.88 cm

11.28 cm

2

3

4

5

6

7

8

9

10

0 2 4 6 8 10 12

X (mm)Y (mm)SPOT

Distance from Aperture (cm)

Beam free expansion with space charge: emittance can be determined as a fit parameter in envelope calculation.

ε′′ − − =3

2

( ) 0,( ) ( )KR z

R z R z

Additional beam optics checks possible with solenoid and quadrupole focusing.

2

4

6

8

10

12

0 20 40 60 80 100

X expY expX math adj.Y math adj.

Z (cm)

IREAPIREAPPepper-Pot Technique

IREAPIREAP

(c)

10keV, 100 mA, 700 experiment

εX = 67 μm, εY = 55 μm

(b)

0.166 mm/pixel

(a)

Pepper-Pot Technique

IREAPIREAP

Calculating emittance and plotting phase space:

Pepper-Pot Technique

IREAPIREAPBeam (σ) and Transfer (R) Matrices

Define the beam matrix for, say, x-plane:

Consider 2 planes z0 and z1, z1>z0. Particle coordinates transform as:

⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

β -= ε -ασ α γ

So beam ellipse can be written as′ ′2 2x - 2 xx + βx = εγ αT -1x σ x = 1, ⎡ ⎤⎣ ⎦≡ ′Tx x, x

-11 0 0 1x = Rx or x = R x R is the

Transfer matrix.

The condition βγ−α2=1 is written σ = εdet

Then, the beam matrix is transformed according to:T

1 0σ = Rσ R

IREAPIREAPQuadrupole Scan⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

2 22

2 21 1 1quad 2 quad 3 2

1 1x = A - 2AB + ︵C + AB ︶,ff=m I - 2m m I + ︵m +m m ︶,

1 .he heqeff e

qg ll Im cf κ γ β= =

2ACε = ,d

αβ

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

2A 1 A 1+= B+ , = , = .dC Cα β γ

IC2

Solenoid

IC1Q2

d

IREAPIREAPQuadrupole Scan: results for 6 mA beam

0.0

5.0

10.0

15.0

20.0

25.0

30.0

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0

USPAS08-QuadScan-II08-05-09 Exp

X^2

(mm

^2)

Quad Current (A)

y = m1*M0*M0-2*m1*m2*M0+(m3+...ErrorValue

0.215917.7808m1 0.0172810.8538m2

0.177850.27948m3 NA2.7872ChisqNA0.99846R

X 5.4 μmε =

1. Martin Reiser, Theory and Design of Charged-Particle Beams, 2nd Ed,

Secs. 3.8.2, 5.3.4 (see Table 5.1) and Chapter 6.

2 D. A. Edwards and M. J. Syphers, An Introduction to the Physics of High Energy Accelerators, Sec. 3.2.4 (see Table 3.1).

3 Claude Lejeune and Jean Aubert, Emittance and Brightness: Definitions

and Measurements, Advances in Electronics and Electron Physics, Supplement 13A, Academic Press (1980).

4 Andrew M. Sessler, Am. J. Phys. 60 (8), 760 (1992); J. D. Lawson, “The Emittance Concept”, in High Brightness Beams for Accelerator

Applications, AIP Conf. Proc. No. 253; M.J. Rhee, Phys. Fluids 29 (1986).

5. Michiko G. Minty and Frank Zimmermann, Beam Techniques, Beam

Control and Manipulation, USPAS, 1999.

6. S. G. Anderson, J. B. Rosenzweig, G. P. LeSage, and

J. K. Crane, “Space-charge effects in high brightness electron

beam emittance measurements”, PRST-AB, 5, 014201 (2002).

7. Manuals to computer codes like TRACE3D, TRANSPORT, etc. .

References

IREAPIREAP

Backup Slides

Detectors IREAPIREAP

Type Characteristics

Phosphor ScreenExample: P43 phosphor (green output). Fairly linear. Inexpensive but delicate. Slow time response (up to ms tails.) Used in combination with CCD or MCP/CCD.

Fast Phosphor Screen

Example: P22 phosphor (blue output). Fairly linear. Delicate and expensive. Response time in the ns range.Used in combination with CCD or MCP/CCD.

OTR ScreenAny mirrored surface. OTR maps beam in real time (fsresponse) with high spatial resolution. Good for any beam energy. Used in combination with PMT or CCD cameras. Low current beams require long int. times.

Standard CCDStandard, inexpensive (8-bit.) Can be intensified for extra sensitivity. Slow response.Cooled CCD higher sensitivity + 16-bit.

Gated Camera0.1 ns time response possible. Expensive.Streak cameras: pico-second time resolution.

24 mA, 10 keV, χ=0.9

Location S (m) εX (µm) εY (µm)

Aperture Plate 0 1.50±0.25 1.5±0.25

After 1/4 turn 3.8 1.50 1.80

After 1/2 turn 7.0 2.10 1.40

After 2/3 turn 9.0 1.60 2.50

IREAPIREAPEmittance (rms. Norm.) in DC Injection Exps.