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

linacbooster

Storage ring

beamlinesample

e-

photons

Focus on this!

therefore this!

gun

x-raysource

light source schematic(ESRF)

Brightness/brilliance of x‐ray beam

B =F(ω )

4π 2Σ xΣ x 'ΣzΣ z '

Where F is the photon flux in the central cone of a given harmonicfor 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 divergenceis determined by a combination of electron 

beam and single electron emission

222,, photonxelecxx σσ +=Σ222

' ',', photonxelecxx σσ +=Σ222,, photonzeleczz σσ +=Σ

222' ',', photonzeleczz σσ +=Σ

These are at source. A distance D away, beam size become: 220,'

20, Dxx Σ+Σ

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

complex.

Lx 2'λσ =

An example:

The standard formula for the source divergence dueto undulator radiation is given by

Wavelength ofradiation

Length of undulatorHowever, there are important corrections to this formula

due to the electron beam energy spread, particularly at higherharmonics.

As we look at smaller and smaller electron beam sizes, vertical and horizontal, we ought to revisit many questions regarding the interaction between electronbeam 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 theemittances? What determines thevalue of the lifetime? How muchcontrol do we have over theseparameters?

Start with the basics.Try Wikipedia? No luck.How to answer these questions?

What is an electron?Spin ½ elementary particle

Coulombs106.1charge 19−×−== ekgme

311011.9mass −×==

)( BvEeFrrrr

×+=

⎟⎟⎠

⎞⎜⎜⎝

⎛•=

ττμμ

ddp

ddp

cmrPe2

2 0

For circular motion:2

44

2 ρπβ

γECcP = 35

320 )/(10846.8

)(34 GeVm

cmrCe

−×==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 orbitThe dipole magnets have constant vertical magnetic fields thatbend 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

xy

e-

ρ GeV/c][3.3357Tm][ pB =ρ

B=.86 Tp/c=6.04 GeV

m4.23=ρ

use perpendicular coordinates, x, yand coresponding angles x’=px/P0, y’=py/P0Transverse phase space: (x,x’,y,y’)

Bending magnet

Quadrupole focusing

0)('' =+ xskx x

)()( ,, skCsk zxzx =+

0)('' =+ zskz z

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

0000

>⇒<<⇒>

zx

zx

kkkk

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

quad

(Hill’s equation)

Harmonic oscillator in phase spaceTwiss Parameters

'x

xxxβε

xxγε

slope: βα

−

22 ''2 xxxx βαγε ++=

tune is defined bynumber of oscillations about closedorbit over 1 turn

measuring the positionover time, it will oscillate

This is at one position in the ring.

invariant with position!

turn 1turn 2

turn 3

Transverse dynamics

1s2s

3s

'x

x

turn 1turn 2

turn 3

tune = phase advance per turn

Tune measurementShake the beam at different frequencies and measure the response.

Intensity of response

Tune Resonance Diagram

we want to avoid tunes near resonances, i.e. n nux + m nuy = kfor 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 magnetsare 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-3or 1 oscillation every 166 turns.

ct

dp/p

RF buckets

t

V_RF

f_RF = h f_0h=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 andmay 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 trajectoriesof Mercury, Mars and Venus with the Earth”, Nature 459, 817-819 (2009)

Suppose we model our ring extremely well, and can track the electronaround 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 apertureTouschek 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

)(zf r

There are 992 RF buckets. Each of these buckets may have many electrons.In uniform filling, all buckets have electrons. Other filling patternshave 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 hasa 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

352

4−− −== GeVmeterCEcC

P γγ

γ ρπ

Revolution time is 2.82 microsecondsSo 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 alldirections…Robinson’s theoremrelates the differentdamping 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 radiationprocess

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

)()(cc

SP

ωω

ωω γ=℘

ργω

3

23 c

c =(critical frequency)

ξξξπ

ξξ∫∞

= dKS )(8

39)( 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 distributionof the number of emitted photons per unit timeas follows:

hh /)/()( duuduuun ℘=

ωh=u

One may compute the average # of photons emitted per radian:

137325 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:cu

Pγ8

315=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 causea random walk in the electrons, or “diffusion”. To computeThe diffusion coefficient, one needs the following quantity:

∫∞

=Ν=0

22 )( duunuud

The result of this calculation is:

3

5

32455

ργα ⎟

⎠⎞

⎜⎝⎛=

mcd 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

xx T

Hds

bd

/2

)348/55(

2

35 ∫

==ρ

αγ

ε22 ''2 βηαηηγη ++=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:

222δσηβεσ

xxxx +=

222' ' δση

βεσ x

x

xx +=

222δσηβεσ

zzzz +=

222' ' δση

βεσ z

z

zz += However, vertical

dispersion is very small!

Example- center of high beta straight section:

0',13.0,95.2,6.37 ==== ηηββ mmm xzx

31006.1,3,4 −×=== δσεε pmnm zx

radm xx μσμσ 3.10,6.411 ' ==

radm zz μσμσ 0.1,97.2 ' ==

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 one 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

aa b

dg2

=

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

yelectrons X-rays

Beam lifetime

• beam lifetime has two components:

• Touschek lifetime and vacuum lifetime

Typical values:MB: 50 hrs16 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.

1τ=

1I

dIdt

1τ=

1τT

+1τV

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

zsT I

acc

3δεστ ≈

(weak dependence on horizontalemittance)

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 sσ

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.

0

10

20

30

40

50

60

70

0 1 2 3 4 5 6

sig_L (ps)

I_b (mA)

8MV

6MV

Recent measurementsWith streak camera

Vertical emittance zεCan be controlled to some extent. Equilibrium value determined byErrors 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 increasesthe equilibrium emittance.

Multibunch mode: run at minimum vertical emittance. Still allows 50 hrlifetime 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, and2.2% for 4 bunch.

Finding new sextupole settings may improve this. Measurement of energy acceptance. See B. Nash et. al. IPAC ‘11 for further details.

0

2

4

6

8

10

12

14

5 7 9 11tau (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 SandsSLAC-R-121(available to download from SLAC pubs website)

“Accelerator Physics”, by S.Y. Lee, Ch. 4,World Scientific, (1999)