Physics of the electron beam source: beam size, shape and lifetime - - PowerPoint PPT Presentation

Physics of the electron beam source: beam size, shape and lifetime and the relationship to the x‐ray radiation properties

Boaz Nash Accelerator Source Division, ESRF

Outline

• Schematic History of synchrotron

radiation

• Radiation brightness and beam size
• Physics of one electron in the storage

ring

• Electron beam emittance, size and

shape

• Beam lifetime

Schematic History of Synchrotron Radiation

• Accelerators were built for nuclear and particle
• physics. (~1930 onward)
• Synchrotron radiation was discovered (observed

1947) and seen as a problem: it limits acceleration!

• Schwinger, Ivanenko and Pomeranchuk, others

already described synchrotron radiation (1944 and earlier)

• radiation is useful!
• Machines built to optimize its production
• Generations… 1st, 2nd, 3rd ,4th, (5th?)…

The big picture: photons come from electrons

• electrons:gun‐>linac‐> booster ‐> storage ring
• xrays: source‐>front end ‐> beamline ‐>

experiment

xray properties determined by electron beam properties linac booster Storage ring beamline sample e- photons Focus on this! therefore this! gun x-ray source light source schematic (ESRF)

Brightness/brilliance of x‐ray beam

B = F(ω) 4π 2ΣxΣx 'ΣzΣz '

Where F is the photon flux in the central cone of a given harmonic for a given frequency bandwidth (e.g. .1%). The photons are created in the bending magnets and undulators. More about undulators, radiation and brightness:

• - J. Chavanne next month.

(photon beam sizes/divergence at source)

Photon beam size and divergence is determined by a combination of electron beam and single electron emission

2 2 2

, , photon x elec x

x

σ σ + = Σ

2 2 2 '

', ', photon x elec x

x

σ σ + = Σ

2 2 2

, , photon z elec z

z

σ σ + = Σ

2 2 2 '

', ', photon z elec z

z

σ σ + = Σ

These are at source. A distance D away, beam size become:

2 2 , ' 2 ,

D

x x

Σ + Σ

The relationship between electron beam and x‐ray beam may be more complex.

L

x

2

'

λ σ =

An example: The standard formula for the source divergence due to undulator radiation is given by Wavelength of radiation Length of undulator However, there are important corrections to this formula due to the electron beam energy spread, particularly at higher harmonics. As we look at smaller and smaller electron beam sizes, vertical and horizontal, we ought to revisit many questions regarding the interaction between electron beam and x-ray beam. Like any relationship… there are two parties involved. So, let’s consider the electron beam. (See Tanaka et. al., Journal of Synch. Rad. 16, 380-386 (2009) and talk by J. Chavanne next month)

ESRF electron beam synopsis.

What is going on behind this? Some basic questions: The current is 196.87 mA. What does this really mean? What is uniform multibunch. What is lifetime and emittance? More difficult questions: What determines the value of the emittances? What determines the value of the lifetime? How much control do we have over these parameters?

Start with the basics.

Try Wikipedia? No luck. How to answer these questions?

What is an electron?

Spin ½ elementary particle

Coulombs 10 6 . 1 charge

19 −

× − = = e kg me

31

10 11 . 9 mass

−

× = =

) ( B v E e F r r r r × + =

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛

• =

τ τ

μ μ

d dp d dp c m r P

e

2 2 0

For circular motion:

2 4 4

2 ρ π β

γ

E C c P =

3 5 3 2

) /( 10 846 . 8 ) ( 3 4 GeV m c m r C

e −

× = = π

γ

Bending radius

What do free electrons do?

They move and get pushed around by electric and magnetic fields. (Lorentz force law) They radiate when they turn or accelerate. (relativistic Larmour equation)

How to store a high energy electron

• Accelerate to high energy (E=6.04 GeV for ESRF)

in linac and booster, then inject into ring.

• Use dipole magnets to create circular trajectory.
• Use quadrupoles to confine the beam

transversely.

• Use sextupoles to fix chromatic aberration

caused by the quadrupoles.

• Use an RF cavity to replenish energy and

confine longitudinally.

Components needed to store electrons

dipole quadrupole sextupole RF cavity

Electron closed orbit

The dipole magnets have constant vertical magnetic fields that bend the electron into a big circular trajectory. There is an orbit that closes on itself that is called the ideal orbit. (controlled by dipoles+correctors) For recent work on orbit correction, see:

• E. Plouviez et. al. “Fast Orbit Correction for the

ESRF Storage Ring”, ipac ‘11 (MOPO002 ) dipole

s

x y

e-

ρ

GeV/c] [ 3.3357 Tm] [ p B = ρ

B=.86 T p/c=6.04 GeV

m 4 . 23 = ρ

use perpendicular coordinates, x, y and coresponding angles x’=px/P0, y’=py/P0 Transverse phase space: (x,x’,y,y’) Bending magnet

Quadrupole focusing

) ( ' ' = + x s k x

x

) ( ) (

, ,

s k C s k

z x z x

= +

) ( ' ' = + z s k z

z

The fact that we use quadrupole magnetic fields for focusing implies:

> ⇒ < < ⇒ >

z x z x

k k k k

Cristofilos (1949) and Courant-Snyder (1952)discovered that a combination of focusing and defocusing quadrupoles leads to a net focusing effect. (principle of strong focusing).

quad

(Hill’s equation)

Harmonic oscillator in phase space Twiss Parameters

' x

x

x xβ

ε

x xγ

ε

slope:

β α −

2 2

' ' 2 x xx x β α γ ε + + =

tune is defined by number of oscillations about closed

• rbit over 1 turn

measuring the position

• ver time, it will oscillate

This is at one position in the ring. invariant with position! turn 1 turn 2 turn 3

Transverse dynamics

1

s

2

s

3

s

' x

x

turn 1 turn 2 turn 3 tune = phase advance per turn

Tune measurement

Shake the beam at different frequencies and measure the response. Intensity

• f response

Tune Resonance Diagram

we want to avoid tunes near resonances, i.e. n nux + m nuy = k for some integers, n, m, k.

Energy dependence of transverse motion

• electrons have an average energy of 6.04 GeV,

but will have a spread about this value (energy spread)

• The transverse oscillation vary with energy,

since the bending (dipoles) and focusing (quadrupoles) effect does so.

• Orbit shift with energy ‐> dispersion.
• Tune shift with energy ‐> chromaticity.
• This tune shift would be unacceptable.
• It is fixed with the sextupoles.

σδE /E = 10−3

Lattice

The layout of dipoles, quadrupoles and sextupoles is called a lattice. The lattice is chosen to try to satisfy many constraints, and to optimize important parameters. The strengths of the quadrupole and sextupole magnets are controllable. This gives a certain amount of flexibility in setting the beta functions and dispersion and improving the non-linear dynamics with the sextupoles even with the magnets fixed in place. ESRF Lattice is a double bend achromat, but with distributed dispersion.

• 0.050

0.000 0.050 0.100 0.150 0.200 0.250 0.300 0.350 0.400

• 10.00

0.00 10.00 20.00 30.00 40.00 50.00 60.00

• 5.000

0.000 5.000 10.000 15.000 20.000 25.000 30.000

ESRF Optical Functions

BetaX BetaZ Dispersion

recall: beta describes variation of beam envelope around ring. Dispersion is the off energy orbit function. This is one half of a super period which is repeated mirror symetrically. There are 16 super periods in the ring to create the full circumference of 844.39 meters.

(high beta straight) (low beta straight)

longitudinal motion

• energy loss from radiation would cause

particles to be lost.

• RF cavity gives this energy back.
• RF cavity also causes longitudinal focusing!

synchrotron oscillations and synchrotron tune. nus=6e-3

• r 1 oscillation every 166 turns.

ct dp/p RF buckets t V_RF f_RF = h f_0 h=992

RF

Non‐linearities from sextupoles

• sextupole fixes chromaticity, but introduces

cubic term in potential.

+ =

dynamic aperture

Stable

unstable

S

Stability over many turns: how to predict/control?

for ESRF, 10 hour lifetime = 13 billion turns! Age of earth = 4.5 billion years = 4.5 billion turns around the sun. Our solar system seems quite stable… but in fact… its chaotic and may be unstable in the long run! Jacques Laskar and his colleague Mickaël Gastineau in 2009 took a more thorough approach [to studying the solar system evolution] by directly simulating 2500 possible futures. Each of the 2500 cases has slightly different initial conditions: Mercury's position varies by about 1 metre between one simulation and the next.[13] In 20 cases, Mercury goes into a dangerous orbit and often ends up colliding with Venus or plunging into the sun. Moving in such a warped orbit, Mercury's gravity is more likely to shake other planets out of their settled paths: in one simulated case its perturbations send Mars heading towards Earth.[14]

http://en.wikipedia.org/wiki/Stability_of_the_Solar_System (30/11/2011) [13] New Scientist, “Solar system's planets could spin out of control “, 10 June 2009 [14] J. Laskar1 & M. Gastineau, “Existence of collisional trajectories

• f Mercury, Mars and Venus with the Earth”, Nature 459, 817-819 (2009)

Suppose we model our ring extremely well, and can track the electron around 1 turn to great accuracy. Tracking for 10^9 turns is not feasible! This was a great worry for the early accelerators such as SSC. Many mathematicians worked on this problem. Not fully solved! Fortunately, electrons are easier than protons: radiation damping! Really only need to compute stability over thousands of turns! practical implications of non-linear dynamics in ESRF: injection efficiency – on energy dynamic aperture Touschek lifetime – momentum acceptance

In the end, we do manage to store electrons…

Storing single electrons! (Sept. 19 MDT) (K. Scheidt)

Users of SR may not be so happy with single electron mode… In fact, we store many electrons

) (z f r

There are 992 RF buckets. Each of these buckets may have many electrons. In uniform filling, all buckets have electrons. Other filling patterns have not all buckets full, (e.g. 2/3 full, 16 bunches, 4 bunches…) Consider 200 mA, uniform filling. 200/992 = 0.2 mA per bunch. N=I T0/e. T0=2.82 microseconds. e=1.6e-19.

• > N=3.5 billion electrons in each bunch!

each bunch has a phase space distribution

Radiation Damping effect

• radiated power in dipoles:
• So higher energy radiates more, lower energy
• less. Causes damping.
• For ESRF, we have

Tx = 6.97 ms , Tz = 6.97 ms, Ts = 3.48 ms

) 10 * 85 . 8 ( 2

3 5 2 4 − −

− = = GeV meter C E cC P

γ γ γ

ρ π

Revolution time is 2.82 microseconds So damping occurs in 2500 turns transversely and 1250 turns longitudinally.

What would this predict?

• Damping effect causes all electrons to spiral

towards the ideal orbit.

• Beam sizes should be zero!

ct dE/E ct dE/E damping in all directions… Robinson’s theorem relates the different damping rates.

What limits the beam size?

• Is it the repulsion of the electrons?
• In fact, for highly relativistic beams, the

coulomb repulsion is largely canceled by the magnetic attraction

• Another effect dominates the beam size well

beyond the Coulomb repulsion.

• What could this other effect be?

Look more closely at the radiation process

The radiation power spectrum coming out of a dipole is given by: (first computed by Schwinger (1948) )

) ( ) (

c c

S P ω ω ω ω

γ

= ℘

ρ γ ω

3

2 3 c

c =

(critical frequency)

ξ ξ ξ π ξ

ξ

∫

∞

= d K S ) ( 8 3 9 ) (

3 / 5

(follow Sands, Slac report 121 (1970) ) (corresponds to 18.8 KeV)

In fact, this radiation will be emitted from the electron as photons

We relate the power spectrum to the distribution

• f the number of emitted photons per unit time

as follows:

h h / ) / ( ) ( du u du u un ℘ =

ω h = u One may compute the average # of photons emitted per radian:

137 3 2 5 γ =

For 6.04 GeV electrons, this results in 756 photons per turn. Or, for ESRF, an average of about 1 photon emitted per meter! The total emission rate is given by:

c

u P

γ

8 3 15 = Ν

What is the effect of this “graininess” of photon emission on the electron beam?

The energy given to a photon will come from the electron. The fluctuations about the average energy loss will cause a random walk in the electrons, or “diffusion”. To compute The diffusion coefficient, one needs the following quantity:

∫

∞

= Ν =

2 2

) ( du u n u u d The result of this calculation is:

3 5

3 24 55 ρ γ α ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = mc d h

Equilibrium electron beam

• The diffusion effect from the quantum

fluctuations will be balanced by the classical damping effect. Thus, longitudinally, there is an equilibrium energy spread and bunch length.

• The energy dynamics are translated into

transverse dynamics via dispersion. In the uncoupled case, the horizontal equilibrium emittance is given by:

x x x x x

T H ds b d / 2 ) 3 48 / 55 ( 2

3 5 ∫

= = ρ αγ ε

2 2

' ' 2 βη αηη γη + + = Η z

vertical motion

• no intrinsic vertical diffusion (no vertical disp.)
• equilibrium vertical emittance determined by

errors.

• this is why the beam is flat.

Compute electron beam sizes

Compute beam sizes and divergences:

2 2 2 δ

σ η β ε σ

x

x x x

+ =

2 2 2 '

'

δ

σ η β ε σ

x x x x

+ =

2 2 2 δ

σ η β ε σ

z

z z z

+ =

2 2 2 '

'

δ

σ η β ε σ

z z z z

+ =

However, vertical dispersion is very small! Example- center of high beta straight section:

' , 13 . , 95 . 2 , 6 . 37 = = = = η η β β m m m

x z x

3

10 06 . 1 , 3 , 4

−

× = = =

δ

σ ε ε pm nm

z x

rad m

x x

μ σ μ σ 3 . 10 , 6 . 411

' =

= rad m

z z

μ σ μ σ . 1 , 97 . 2

' =

=

Recall:

Equilibrium Gaussian distribution for arbitrarily coupled lattice

• In the general case with x‐y coupling and/or x‐z coupling, the

equilibrium distribution is still a Gaussian.

• Find the invariants quadratic (actions) g_a (a=1,2,3) from the
• ne turn map matrix M.
• Project the diffusion and damping onto the eigenvectors of

the one‐turn map matrix to find d_a, b_a. Equilibrium emittances given by d_a/2 b_a. ***

• Other general numerical algorithms exist as well.

***See B. Nash et. al. Phys. Rev. ST Accel. Beams, 9, 032801, (2006)

a a a

b d g 2 =

Effects/causes of coupling

• In the ring with no errors, horizontal and vertical motion is

decoupled.

• Coupling comes from rotated quadrupoles, off‐center

sextupoles and insertion devices.

• With coupling, there is vertical diffusion, and hence vertical

emittance.

• Eigen‐emittance given by ratio of generalized diffusion

coefficient to damping decrement, d_a/b_a.

• With coupling, the electron beam (and hence photon beam)

may be tilted.

x y x y electrons X-rays

Beam lifetime

• beam lifetime has two components:
• Touschek lifetime and vacuum lifetime

Typical values: MB: 50 hrs 16 bunch: 15 hrs How fast are electrons really being lost? For 200 mA and a 50 hour lifetime: dN/dt = N/50 hrs N= 3.5e12 electrons stored. So dN/dt = 1.9e7 electrons per second lost.

1 τ = 1 I dI dt

1 τ = 1 τT + 1 τV

Measuring the vacuum lifetime

• When the beam is very large, the particle

density is small and so the scattering is small.

• Blow up the beam with a white noise shaker

and measure the lifetime, this gives vacuum lifetime.

Tau_V = 387 hrs (quite large, not the dominant effect!)

Touschek lifetime

• Scattered particles change energy. If energy

exceeds acceptance, it is lost.

• Compute # of electrons per unit time that will

scatter out of the energy acceptance.

• result is:

b z s T

I

acc

3

δ ε σ τ ≈

(weak dependence on horizontal emittance)

Bunch Current

200 mA, uniform filling. 200/992 = 0.2 mA per bunch. 16 bunch, 90 mA -> 5.6 mA per bunch 4 bunch, 40 mA -> 10 mA per bunch

Bunch length

s

σ

Increases with single bunch current. Zero current value is 4.36 mm, 16 bunch bunch, I_b=6.5mA, bunch length is 14.6 mm.

10 20 30 40 50 60 70 1 2 3 4 5 6 sig_L (ps) I_b (mA)

8MV 6MV

Recent measurements With streak camera

Vertical emittance

z

ε

Can be controlled to some extent. Equilibrium value determined by Errors and coupling. These may be corrected to minimize errors. Recent work has allowed us to reduce to 3-4pm.*** The emittance may also be increased by applying a “white noise shaker”. This essentially provides an additional vertical diffusion term and increases the equilibrium emittance. Multibunch mode: run at minimum vertical emittance. Still allows 50 hr lifetime for 4pm. Few bunch modes: apply white noise to increase vertical emittance to ~50 pm.

**For details, see IPAC 11 paper:

• A. Franchi et. al. “Vertical Emittance Reduction and

Preservation at the ESRF Electron Storage Ring” (TUODA01) also:

• A. Franchi et. al. “Vertical emittance reduction and preservation in

electron storage rings via resonance driving terms correction”

• Phys. Rev. ST Accel. Beams 14, 034002 (2011)

Energy acceptance

• scattered electron with changed energy gets

transverse orbit shift due to dispersion. Eventually exceeds the dynamic aperture.

Recent measurements for energy acceptance give ~2.6% for MB, 2.5% for 16 bunch, and 2.2% for 4 bunch. Finding new sextupole settings may improve this. Measurement of energy

• acceptance. See B. Nash et.
• al. IPAC ‘11 for further details.

2 4 6 8 10 12 14 5 7 9 11 tau (hrs) V_rf (MV) (scaled with k=0.905)

Touschek Lifetime vs. RF Voltage

multibunch 16 bunch 4 bunch

Conclusions

• Electron beam distribution and current determine photon beam

distribution and intensity together with the bending magnet or undulator.

• Electron beam is mainly Gaussian, and distribution changes around ring,

emittances set the overall size.

• Emittances are determined by a combination of radiation damping and

quantum diffusion.

• Horizontal emittance is much larger than vertical due to dispersion in the

bending plane.

• Beam lifetime comes from vacuum scattering and intrabeam scattering

(Touschek). Touschek lifetime limits the ESRF as long as the vacuum is

• good. We may improve the lifetime by increasing the vertical emittance or

decreasing the current. Both will decrease brightness, however. Work on the non‐linear dynamics of the ring via the sextupole settings can improve the momentum acceptance, which increases the lifetime as well.

Acknowledgements

• All my colleagues in ASD.

References

“The Physics of Electron Storage Rings: An Introduction” By Matthew Sands SLAC-R-121 (available to download from SLAC pubs website) “Accelerator Physics”, by S.Y. Lee, Ch. 4,World Scientific, (1999)