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
light source schematic
(ESRF)
linac
gun
Focus on this!
x-ray
Storage ring source
booster
therefore this!
beamline
photons
e-
sample
Brightness/brilliance of x‐ray beam
F(ω )
B=
2
4π Σ x Σ x ' Σ z Σ z '
(photon beam sizes/divergence at source)
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:
--Æ J. Chavanne next month.
Photon beam size and divergence
is determined by a combination of electron beam and single electron emission
Σ = σ x ,elec + σ x , photon
2
x
Σ
2
x'
2
2
= σ x ',elec + σ x ', photon
2
2
Σ = σ z ,elec + σ z , photon
2
z
2
2
Σ = σ z ',elec + σ z ', photon
2
z'
2
2
2
2
2
These are at source. A distance D away, beam size become: Σ x ,0 + Σ x ',0 D
The relationship between electron beam and x‐ray beam may be more complex.
An example:
The standard formula for the source divergence due
to undulator radiation is given by
σ x' =
λ
Wavelength of
radiation
2L
Length of
undulator
However, there are important corrections to this formula
due to the electron beam energy spread, particularly at higher
harmonics. (See Tanaka et. al., Journal of Synch. Rad. 16, 380-386 (2009) and talk by J. Chavanne next month)
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.
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?
How to answer these questions?
Try Wikipedia? No luck.
Start with the basics.
What is an electron?
Spin ½ elementary particle
charge = e = −1.6 ×10 −19 Coulombs
mass = me = 9.11× 10 −31 kg
What do free electrons do?
They move and get pushed around by electric and magnetic fields.
r
r r r
F = e( E + v × B )
(Lorentz force law)
They radiate when they turn or accelerate.
For circular motion:
P=
2r0 ⎛ dpμ dpμ
⎜
•
2me c ⎜⎝ dτ dτ
P=
β 4c E 4
Cγ 2
2π
ρ
⎞
⎟⎟
⎠
Cγ =
(relativistic Larmour equation)
r0
4π
= 8.846 ×10 −5 m /(GeV ) 3
3 (me c 2 ) 3
Bending radius
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
dipole
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.
eBending magnet
y
x
s
ρ
Bρ [Tm] = 3.3357 p[GeV/c]
B=.86 T
use perpendicular coordinates, x, y
p/c=6.04 GeV
and coresponding angles x’=px/P0, y’=py/P0
ρ = 23.4 m
Transverse phase space: (x,x’,y,y’)
For recent work on orbit correction, see:
(controlled by dipoles+correctors) E. Plouviez et. al. “Fast Orbit Correction for the
ESRF Storage Ring”, ipac ‘11 (MOPO002 )
Quadrupole focusing
quad
x' '+ k x ( s ) x = 0
z ' '+ k z ( s ) z = 0
(Hill’s equation)
k x, z (s + C ) = k x, z (s)
The fact that we use quadrupole magnetic fields for focusing implies:
kx > 0 ⇒ kz < 0
kx < 0 ⇒ kz > 0
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).
Harmonic oscillator in phase space
Twiss Parameters
measuring the position
over time, it will oscillate
ε = γx + 2αxx'+ βx'
2
x'
2
invariant with position!
ε xγ x
ε xβx
turn 1
x
turn 2
α
slope: − β
turn 3
tune is defined by
number of oscillations about closed
orbit over 1 turn
This is at one position in the ring.
Transverse dynamics
turn 1
turn 2
turn 3
s3
s2
s1
x'
x
tune = phase advance per turn
Tune measurement
Shake 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 = 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) σ δ E / E = 10 −3
• 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.
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.
ESRF Optical Functions
BetaX
BetaZ
Dispersion
0.400
60.00
0.350
50.00
0.300
(high beta straight)
40.00
(low beta straight)
0.250
30.00
0.200
20.00
0.150
0.100
10.00
0.050
0.00
-5.000
0.000
-10.00
5.000
10.000
15.000
20.000
25.000
0.000
30.000
-0.050
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.
RF
longitudinal motion
• energy loss from radiation would cause particles to be lost.
• RF cavity gives this energy back.
• RF cavity also causes longitudinal focusing!
V_RF
t
f_RF = h f_0
h=992
synchrotron oscillations and synchrotron tune. nus=6e-3
or 1 oscillation every 166 turns.
dp/p
ct
RF buckets
S
Non‐linearities from sextupoles
• sextupole fixes chromaticity, but introduces cubic term in potential.
unstable
+
=
Stable
dynamic aperture
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
of 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
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
r
f (z )
Radiation Damping effect
• radiated power in dipoles:
cCγ E 4
Pγ =
2π ρ 2
(Cγ = 8.85 *10 −5 meter − GeV −3 )
• 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
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.
damping in all
• Beam sizes should be zero!
directions…
Robinson’s theorem
relates the different
damping rates.
ct
ct
dE/E
dE/E
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
(follow Sands, Slac report 121 (1970) )
The radiation power spectrum coming out of a dipole is given by:
(first computed by Schwinger (1948) )
℘(ω ) =
Pγ
ωc
S(
ω
)
ωc
3 cγ 3
ωc =
2 ρ
(critical frequency)
(corresponds to 18.8 KeV)
9 3 ∞
S (ξ ) =
ξ ∫ K 5 / 3 (ξ )dξ
ξ
8π
In fact, this radiation will be emitted from the electron as photons
We relate the power spectrum to the distribution
of the number of emitted photons per unit time
as follows:
un(u )du = ℘(u / h )du / h
u = hω
The total emission rate is given by:
Ν=
15 3 Pγ
8 uc
One may compute the average # of photons emitted per radian:
=
5
γ
2 3 137
For 6.04 GeV electrons, this results in 756 photons per turn.
Or, for ESRF, an average of about 1 photon emitted per meter!
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:
∞
d = Ν u 2 = ∫ u 2 n(u )du
0
The result of this calculation is:
55 ⎛ h ⎞ γ 5
α⎜ ⎟ 3
d=
24 3 ⎝ mc ⎠ ρ
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:
( 55 / 48
ε
x
dx
=
=
2bx
3 ) αγ
2 / Tx
5
∫ ds
H
ρ
x
3
Η = γη 2 + 2αηη ' z + βη '2
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
Recall:
σ x2 = ε x β x + η 2σ δ2
x
σ
2
x'
εx
+ η ' 2x σ δ2
βx
=
σ z2 = ε z β z + η 2σ δ2
z
σ
2
z'
=
Compute beam sizes and divergences:
εz
+ η ' 2z σ δ2
βz
However, vertical
dispersion is very small!
ε x = 4nm, ε z = 3 pm, σ δ = 1.06 ×10 −3
Example- center of high beta straight section:
β x = 37.6m, β z = 2.95m,η x = 0.13m,η ' = 0
σ x = 411.6μm, σ x ' = 10.3μrad
σ z = 2.97 μm, σ z ' = 1.0μrad
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.
ga =
da
2ba
***See B. Nash et. al. Phys. Rev. ST Accel. Beams, 9, 032801,
(2006)
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.
y
electrons
X-rays
y
x
x
Beam lifetime
1
1 dI
=
τ I dt
1
τ
=
1
τT
+
1
τV
• beam lifetime has two components:
• Touschek lifetime and vacuum lifetime
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.
Typical values:
MB: 50 hrs
16 bunch: 15 hrs
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:
τT ≈
σ s εzδ 3
acc
Ib
(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 σ s
70
60
50
sig_L (ps)
40
8MV
6MV
30
Recent measurements
With streak camera
20
10
0
0
1
2
3
I_b (mA)
4
5
6
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.
Vertical emittance ε z
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.
Touschek Lifetime vs. RF Voltage
~2.6% for MB,
2.5% for 16 bunch, and
2.2% for 4 bunch.
12
10
tau (hrs)
Recent measurements for energy
acceptance give
14
8
6
multibunch
4
16 bunch
2
4 bunch
0
5
7
9
11
V_rf (MV) (scaled with k=0.905)
Finding new sextupole settings may improve this.
Measurement of energy
acceptance. See B. Nash et.
al. IPAC ‘11 for further details.
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)