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Physics of Electron Storage Rings
According to Larmor’s theorem, the instantaneous radiated power from an
accelerated electron is
The energy radiated from the particle with nominal energy E0 in one revolution is
where the proper-time is dτ = dt/γ, and pμ = (p0, p) is the 4-momentum vector.
The radiation power arising from circular motion is
where R is the average radius. For an isomagnetic ring with constant field strength in
all dipoles, the energy loss per revolution and the average radiation power become
where T0 = 2πR/βc is the orbital revolution period. Because the power of
synchrotron radiation is proportional to E4/ρ2, and the beam is compensated on
average by the longitudinal electric field, the longitudinal motion is damped. This
natural damping produces high brightness electron beams, whose applications
include e+e− colliders for nuclear and particle physics, and electron storage rings
for generating synchrotron light and free electron lasers for research in condensed
matter physics, biology, medicine and material applications.
Electron positron colliders and the discovery of leptons
• 1895: Jean Perrin found that the cathode ray can be deflected by electric field;
1897: Joseph John Thomson was able to determine e/m; 1909: R. A. Millikan
determined the charge e using the oil drop experiment.
• 1926: P.A.M. Dirac predicted the existence of positron. 1932: Carl W.
Anderson observed positron in cosmic ray.
• 1934: muons were observed from the cosmic ray experiment; thought to be the
Yukawa particle proposed in 1932. The actual Yukawa particle (pion) was
discovered in 1947 by C.F. Powell.
• 1934: W. Pauli proposed to solve the beta-decay puzzle by the existence of a
neutral particle, called neutrino. F. Reines and C.L. Cowan discovered the
neutrino in 1953. Lederman, Schwartz and Steinberger (1962) discovered that
neutrinos come in two kinds - one associated with electrons, the other with
muons.
• Discovery of J/ψ particle in 1974 - the "November Revolution".
• Nicola Cabibbo (1963) introduced the Cabibbo angle (θc) to preserve the
universality of the weak interaction. Makoto Kobayashi and Toshihide Maskawa
(1973) suggested the existence of six types of quark in order to explain the
mysterious violation of CP symmetry.
• In 1977, Martin Perl and his group at the SPEAR electron-positron collider at
the Stanford Linear Accelerator Center (SLAC) discovered a third electrically
charged much heavier lepton, τ (tau) (1777GeV/c2). The tau-neutrino was
discovered in 2000 at Fermilab.
• In 1977, the upsilon (9.5 GeV) was discovered at the Fermilab fixed target
experiment. Unlike the J/psi, the upsilon was not discovered via the electronpositron annihilation route, as in 1977 no electron-positron collider had enough
energy. Nearest in energy was the DORIS machine at DESY, Hamburg, and
following the upsilon news the DORIS beam energy was turbocharged in a crash
programm. By the following summer, the PLUTO and DASP detectors at
DORIS had seen their first upsilons.
• In 1979, Cornell's CESR electron-positron collider plays a key role in Bphysics. In 1999, PEP-II at SLAC and KEKB at KEK , high luminosity B
factories, play the central role in B-physics. In 2010, PEP-II was shutdown, and
super-B in Italy was approved.
• Since 2005, ILC has played a potential future high energy collider organization.
• Since 2000, Muon collider concept has been considered seriously, and an
intensive research program has been carried out in many national labs.
According to Larmor's theorem, the instantaneous radiated power from an
accelerated electron is
 2
2
2r0  dpi dpi  2r0   dp 
e2υ 2
1  dE  
P=
=
=








6πε0 c3 3mc  dτ dτ  3mc   dτ  c 2  dτ  
dτ=dt/γ is the proper time. The radiation power from circular motion in a dipole
becomes
 2
P=
2 r0  d p 
2r 2 2  2 2r 2
β 4c
E4
2
Cγ 2

 = 0 γ ω ρ | p | = 0 γ | F | =
3mc  dτ 
3mc
3mc
2π
ρ
8.846 105 m/(GeV)3 electron

4π r0
=4.840 1014 m/(GeV)3 muon
Cγ =
2 3
3 (mc ) 
18
3
7.783 10 m/(GeV) proton
P =
U0
=
T
3
4
C γ β E0
1
=
2
ρ
ρ
cC γ β 3 E 04
U 0 =  Pdt = Cγ β 3 E04 R
2 πRρ
Modern synchrotron radiation theory was formulated by many physicists;
in particular, its foundation was laid by J. Schwinger. Some of his many
important results are summarized below: [J. Schwinger, Phys. Rev. 70, 798
(1946); 75, 1912 (1949); Proc. Nat. Acad. Sci. 40, 132 (1954).]
[1] The angular distribution of synchrotron radiation is sharply peaked in the
direction of the electron's velocity vector within an angular width of 1/γ, where γ
is the relativistic energy factor. The radiation is plane polarized on the plane of
the electron's orbit, and elliptically polarized outside this plane.
[2] The radiation spans a continuous spectrum. The power spectrum produced by
a high energy electron extends to a critical frequency ωc=3γ3ωρ/2, where ωρ=c/ρ
is the cyclotron frequency for electron moving at the speed of light. [D.H.
Tomboulin and P.L. Hartman experimentally verified that electrons at high
energy (70 MeV then) could emit extreme ultraviolet (XUV) photons; Phys. Rev.
102, 1423 (1956).]
[3] Quantum mechanical correction becomes important only when the critical
energy of the radiated photon, ћω=(3/2)ћcγ3/ρ is comparable to the electron beam
energy, E=γmc2. This occurs when the electron energy reaches mc2(mcρ/ћ)1/2 ~
106 GeV. The beamstrahlung parameter, defined as =(2/3) ћωc/E is a measure of
the importance of quantum mechanical effects.
[Photons/(s-mm2-mrad2-0.1%bandwidth)]
Fields of a Moving Charged Particle
The total power radiated by the particle is
The first term, which is proportional to 1/R2, is a static field pointing away from the
charge at time t. This field can be transformed into an electrostatic electric field by
performing a Lorentz transformation into a frame in which the charge is at rest. The
total energy from this term is zero. The second term, related to the acceleration of
the charged particle, is the radiation field, which is proportional to 1/R. Both E and
B radiation fields are transverse to n-vector and are proportional to 1/R.
Frequency and Angular Distribution
The radiation from circular motion is at least a factor
of 2γ2 larger than that from longitudinal acceleration.
The radiation emitted by an extremely
relativistic particle subject to arbitrary
acceleration arises mainly from the
instantaneous motion of the particle along a
circular path. The radiation is beamed in a
narrow cone in the forward direction of the
velocity vector. The short pulse of radiation
resembles a searchlight sweeping across the
observer. The Figure shows the coordinate
system of a particle moving along a circular
orbit, where the trajectory is
which peaks around y = ω/ωc =1 or ω = ωc. Thus the radiation due to the bending
magnets has a smooth spectral distribution with a broad maximum at the critical
frequency ωc. The critical photon energy is
where E is the electron beam energy and B is the magnetic flux density.
The moments of energy distribution become
II Radiation Damping and Excitation
The instantaneous power radiated by a relativistic electron at energy E is
At a fixed bending radius, the quantum fluctuation
varies as the seventh power of the energy.
where B is the magnetic field strength, ρ is the local radius of curvature, and
Cγ = 8.85 × 10−5 m/(GeV)3 is given by Eq. (4.5). The total energy radiated in
one revolution becomes
The average radiation power for an isomagnetic ring is
where T0 = βc/2πR is the revolution period, and R is the average radius of a
storage ring. An electron at 50 GeV in the LEP at CERN (ρ = 3.096 km) will lose
0.18 GeV per turn. The energy loss per revolution at 100 GeV is 2.9 GeV, i.e. 3%
of its total energy. The energy of circulating electrons is compensated by rf
cavities with longitudinal electric field.
Since higher energy electrons lose more
energy than lower energy electrons and
the average beam energy is compensated
by longitudinal electric field, there is
radiation damping (cooling) in the
longitudinal phase space.
II.1 Damping of Synchrotron Motion
radiation
cavity
Electrons lose energy in a cone with an
angle about 1/γ of their instantaneous
velocity vector, and gain energy through
rf cavities in the longitudinal direction.
This mechanism provides transverse
phase-space damping.
The damping (e-folding) time is
generally equal to the time it takes for
the beam to lose all of its energy.
The damping re-partition
To evaluate the damping rate, we need to evaluate W. Since the radiation energy
loss per revolution is
cdt/ds = (1+x/ρ),
(cτs, 0): the longitudinal phase-space coordinates
of a synchronous particle.
(c(τ +τs),ΔE): of a particle with energy deviation
ΔE from the synchronous energy.
The path length difference between these two
particles is ΔC = αcC ΔE/E , where αc is the
momentum compaction factor, C is the accelerator
circumference, and the difference in arrival time is
During one revolution, the electron loses energy
U(E) by radiation, and gains energy eV (τ ) from
the rf system. Thus the net energy change is
where K(s) = B1/Bρ is the quadrupole gradient function with B1 = ∂B/∂x. The
damping partition number D is a property of lattice configuration. For an
isomagnetic ring,
1.
For an isomagnetic ring with separate
function magnets, where K(s) = 0 in dipoles
The damping time constant, which is the inverse of αE, is nearly equal to the
time it takes for the electron to radiate away its total energy.
2.
For an isomagnetic combined function accelerator, we find
II.2 Damping of Betatron Motion
A relativistic electron emits synchrotron radiation primarily along its direction of
motion within an angle 1/γ. The momentum change resulting from recoil of
synchrotron radiation is exactly opposite to the direction of particle motion. Figure
4.6 illustrates betatron motion with synchrotron radiation, where vertical betatron
coordinate z is plotted as a function of longitudinal coordinate s. The betatron
phase-space coordinates are
When an electron loses an amount of energy u by radiation, the momentum vector
P changes by δP , such that δP is parallel and opposite to P with |cδP | = u. Since
the radiation loss changes neither slope nor position of the trajectory, the betatron
amplitude is unchanged except for a small increment in effective focusing force.
Now the energy gain from rf accelerating force is on the average parallel to the
designed orbit, i.e.
where A is betatron amplitude,  is betatron phase, and β is betatron function.
The corresponding change of amplitude A in one revolution becomes
Schematic drawing of the damping of
vertical betatron motion due to
synchrotron radiation. The energy loss
through synchrotron radiation along the
particle trajectory with an opening
angle of 1/γ. Energy is replenished in
the rf cavity along the longitudinal
direction. This process damps the
vertical betatron oscillation to a very
small value.
Horizontal betatron motion
where <..> averages over betatron oscillations in one revolution, and U0 is
synchrotron radiation energy per revolution. Since the betatron motion is sinusoidal,
we obtain <(βz′)2> = A2/2,
The damping rate applies also to the horizontal betatron motion.
The fractional betatron amplitude increment in one turn becomes
The horizontal motion of an electron is complicated by the off-momentum
closed orbit. The horizontal displacement from the reference orbit is
Including the phase space damping, the net horizontal amplitude change:
In summary, radiation damping coefficients for the three degrees of freedom in a
bunch are
Robinson theorem
We neglected all terms linear in xβ′,
because their average over the
betatron phase is zero. The time
average over the betatron phase gives
<xβ> = 0 and <xβ2>=A2/2.
II.3 Damping Rate Adjustment
A. Increase U to increase damping rate (damping wiggler)
Robinson wiggler
If the gradient and dipole field of each magnet satisfy Kρ < 0, as shown in Fig. 4.9,
the damping partition of Eq. (4.100) can be made negative.
The damping time is shortened by a factor of (1+Uwiggler/U0)−1.
B. Change D to repartition the partition number
Many early synchrotrons, such as 8 GeV synchrotron (DESY) in Hamburg, 28 GeV
CERN-PS, 33 GeV AGS, etc., used combined function isomagnetic magnets, where
D ≈ 2. Thus the energy oscillations are strongly damped (JE ≈ 4) and the horizontal
oscillations become anti-damped (Jx ≈ −1).
The Robinson wiggler has been successfully employed in the CERN PS to obtain
Jx ≈ 2, which enhances damping of horizontal emittance and reduces damping in
energy oscillation. The resulting line density of beam bunches is likewise reduced
to prevent collective instabilities.
Note that a large compaction factor is necessary for achieving de-bunching
for the electron beams in a single path!
Using a pair of quadrupoles as in the DBA
II.4 Radiation Excitation and Equilibrium Energy Spread
The time during which a quantum is emitted is about
Emission of individual quanta are statistically independent because the energy of
each photon [keV] is a very small fraction of electron energy.
Discontinuous quantized photon emission disturbs electron orbits. The cumulative
effect of many such small disturbances introduces diffusion similar to random
noise. The amplitude of oscillation will grow until the rates of quantum excitation
and radiation damping are on the average balanced. The damping process depends
only on the average rate of energy loss, whereas the quantum excitation fluctuates
about its average rate.
A. Effects of quantum excitation
B. Equilibrium rms energy spread
C. Adjustment of rms momentum spread
II.5 Radial Bunch Width and Distribution Function
Emission of discrete quanta in synchrotron radiation also excites random
betatron motion. The emission of a quantum of energy u results in a change of
betatron coordinates, i.e.
The resulting change in the Courant-Snyder invariant is
D. Beam distribution function in momentum
The normalized phase-space coordinates are (ΔE, θ = Eωsτ/αc).
Central Limit Theorem: If the probability P(u) of each quantum emission is
statistically independent, and the probability function falls off rapidly as |u| → ∞,
then the probability distribution function for the emission of n photons is a
Gaussian,
II.6 Vertical Beam Width
The transverse kick is then equal to θγu/c. The transverse angular kicks on
phase-space coordinates become
The emittance of Eq. (4.164) is called the natural emittance. Since the H-function is
proportional to Lθ2 ∼ ρθ3, where θ is the dipole angle of a half cell, the natural
emittance of an electron storage ring is proportional to γ2θ3. The normalized
emittance is proportional to γ3θ3. Unless the orbital angle of each dipole is
inversely proportional to γ, the normalized natural emittance of an electron storage
ring increases with energy.
The horizontal distribution function
The total radial beam width has contributions from both betatron and energy
oscillations. The rms beam width is Gaussian quadrature
Emittances in the presence of linear coupling
II.7 Radiation Integrals
III Emittance in Electron Storage Rings
Since H ∼ Lθ2 = ρθ3, the <H> and the resulting natural emittance obey the
scaling laws:
8
5 3
where the scaling factor Flattice depends on the design of the storage ring lattices,
and θ is the total dipole bending angle in a bend-section. The resulting
normalized emittance is
A. FODO cell lattice
B. Double-bend achromat (Chasman-Green lattice)
Achromatic: d0=0, d0′=0,
The H-function at the end of the dipole is
This criterion can be used to evaluate the goodness of DBA lattice match.
C. Minimum emittance (ME): Minimum <H>-function lattice
SESAME full period optical functions
for (Qx=7.23 - Qz=6.19), ε=26 nm.
 nat  Cq
 2 3
12 15 J x
 12 nm
1
Define the H-function at the end of TME lattice:
3 15
We find
3
4
1
4
Effective emittance
x  x ( s )  D( s )
A smaller emittance in a NON_ACHROOMATIC lattice does not
necessary produce a smaller effective emittance!!
H/HTME of one superperiod two DB
cells for a small emittance DBA
lattice red dashed line and a small
emittance DB lattice black. Here,
HTME=ρθ3/3√15 is the maximum
H-function for the theoretical
minimum emittance TME lattice.
The dotted and dot-dashed lines
are the averaged H-function in
the dipoles. If the design reaches
ME condition, they should be 0.25
and 0.75 respectively!
The circumference of the storage ring is 780.3 m, with 15 long and short straight
sections at 11.14 and 7 m, respectively. The betatron tunes are 31.45 and 15.40,
respectively. The natural chromaticities are −70 and −33, respectively. The natural
emittances are 2.28 nm for the achromatic mode and 0.75 nm for the
nonachromatic modes. This lattice was obtained by removing 60 quadrupoles
from the lattice by S. Krinsky, J. Bengtsson, and S. L. Kramer, Proceedings of
EPAC, 2006, p. 3487, while keeping the same circumference.
Define the dispersion emittance as
E. Three-bend achromat
Problem: Dispersion function mis-match
EPAC92_43 (‘92)
The matching condition requires L2 = 31/3L1 for isomagnetic storage rings, or
ρ1 = √3ρ2 for storage rings with equal length dipoles.
The emittance of the matched minimum
TBA (QBA, or nBA) lattice is
Theoretical εMEDBA=1.85 nm
compared to 6.2 nm.
How about Three-Bend-Achromat
where θ1 is the bending angle of the outer dipoles, provided the middle dipole
is longer by a factor of 31/3 than the outer dipoles. The formula for the
attainable minimum emittance is identical to that for the MEDBA.
The Quadruple-bend achromat
L1
With L2=1.45L1, we find that the emittance should scale like [2/(1+31/3)]3=0.54.
L2=31/3L1
Emittance optimization in three- and multiple-bend achromats
PRE 54, 1740 (1996)
The necessary condition for minimizing the emittance of the three- and
multiple-bend achromat lattices is derived. For isomagnetic three- or multiplebend achromat lattices, the minimum emittance can only be attained if the
length of the inner dipoles is 31/3 longer than that of outer dipoles. For the
three- or multiple-bend achromat with equal length dipoles the minimum
emittance can also be achieved by increasing the magnetic field of middle
dipoles by a factor of √3 larger than that of outer dipoles. The minimum
emittance formula for the isomagnetic three-bend achromat with equal length
dipole has also been derived.
1. Matching is relatively easy (as
compared to TBA lattices)
2. The chromatic properties is as
good as the DB lattice!
3. The emittance obeys the scaling
law: γ2θ3, i.e. 30 cells (15 QBA
cells) will give about 1.24 nm
emittance without any damping
wiggler!
30
Optical functions ( m )
L1
L2
QBA cell
25
βz
20 βx
15
10
5
0
0
10×Dx
10
20
s(m)
30
40
Consider a lattice with 16 QBA cells, a circumference of 780.3 m. The
parameters used in this lattice are L1=1.8 m and L2=3.0 m.The lengths of the
straight sections are 10.0 and 5.77 m. The emittance of 1.02 nm at the 3 GeV
beam energy.
H/HTME of one super QBA cell with
The HTME=ρθ23/3√15 is the
maximum H-function for the
theoretical minimum emittance
TME lattice based on the middle
dipole. The dotted line is the average
H-function in dipoles, derived from
the emittance of 1.02 nm at 3 GeV.
The ratio is 0.25 for a theoretical
MEQBA!
Consider a QBA lattice with a circumference of 486 m and 12 QBA cells for the
TPS design. We choose L2/L1=1.5 and L1+L2=2.5 m, i.e., θ1=6°, θ2=9°, and
B0=1.048 T. The length of L1+L2 is chosen to lower the dipole radiation power loss,
optimize emittance reduction from damping wigglers and wavelength shifters, and
allow for low field, low power undulators in the dispersive straight sections. To
maximize the number of insertion devices, we have achieved 6×10.91m and
18×5.31m straight sections. The ratio of straight sections to the total circumference
is 33%. The lattice design is thus very compact. There are ten families of
quadrupoles with reflection symmetry in a superperiod.
The betatron tunes are νx=26.28 and
νy=12.25, and the natural
chromaticities are −64 and −30. The
emittance of this lattice is 3.0 nm rad.
The factor fh is
=0.68
This is about 2.5 times the MEQBA.
Beter optimization can reach emittance
of 2.5 nm.
M. H. Wang, et al.
RSI 78, 055109 2007
Dynamic aperture based on tracking calculations with no random errors.
The rms beam sizes at this location are σx=0.19 mm and σz=0.016 mm.
Figure 4.15: Compilation of the
natural emittances in unit of nm
achieved for some synchrotron
light sources vs the number of
pairs of dipoles. For example, a
12 cell 7BA corresponds to 42
pairs of dipoles. The lines are
the minimum emittance for
DBA lattice and the theoretical
minimum emittance (TME)
respectively. The minimum
emittance of nBA lattice is
III.2 Insertion Devices
The IDs are normally made of dipole magnets with alternating dipole fields so
that the orbit outside the device is un-altered. A simple planer undulator with
vertical sinusoidal fields, that satisfies Maxwell’s equation, is
where kw = 2π/λw is the wiggler wave number, λw is the wiggler period, and Bw
is the magnetic field at mid-plane. The corresponding horizontal and
longitudinal magnetic fields, and the vector potential are
Bx = 0, Bs = −Bw sinh kwz sin kws, and
The Hamiltonian of particle motion is
where θ1 is the bending angle of the outer dipoles. The middle dipole is longer by a
factor of 31/3 than the outer dipole.
z  
sin 2 k w s
 z 
The nonlinear magnetic field can be neglected if the vertical betatron motion is
small with kwz ≪ 1. The horizontal closed orbit becomes
z  
sin 2 k w s
 w2
z0
 z 
sin 2 k w s
1
ds

z
4 
 w2
Helical undulators
 w2
z0
1
sin 2 k w s
z
ds

 w2
4
For a periodic wiggler
It is apparent to see that the periodic motion of the electron in the wiggler are
transformed to the observer at a frequency boosted by the factor:
where Nw is the number of the wiggler period, the apparent angular frequency
ωL(0) at the forward direction is
Here ωL corresponds to the characteristic frequency of the device in the observer’s
Frame: ωLt = ξ − p sin ξ + q cos ξ, where
If Kw ≤ 1, the radiation from each undulator period can coherently add up to
give rise to a series of spectral lines given by
The resonance condition for constructive interference is achieved when the path
length difference between the photon and electron, during the time that the electron
travels one undulator period, is an integer multiple of the electromagnetic
wavelength.
The critical wavelength from a regular dipole is
The pulse length of a photon from a short electron bunch is
III.3 Effect of IDs on beam dynamics
Thus the frequency bandwidth is
are the energy losses in the storage ring dipoles and in an undulator or wiggler in
one revolution respectively. Here ρ0 and ρw are the bending radii of storage ring
dipoles and wiggler magnets, Lw is the length of the undulator or wiggler, and E is
the beam energy.
A. Effects of insertion devices on the beam emittances
B. Effects of IDs on momentum spread
If Bw ≤ 3πB0/8 for the planer undulator, the beam momentum spread will decrease.
On the other hand, the high field IDs can increase momentum spread, particularly
important for the low main dipole field design.
The emittance will decrease slightly when the undulators field is Bw ≤ (3πfh/8) B0,
and the natural emittance will increase for strong field wigglers
C. Effect of the ID induced dispersion functions and its effect on emittance
We consider a simple ideal vertical field wiggler (Fig. 4.20), where ρw = p/eBw is
the bending radius, θ = Θw = Lw/ρw is the bending angle of each dipole, and Lw is
the length of each wiggler dipole. Since the rectangular magnet wiggler is an
achromat (see Exercise 2.4.20), the wiggler, located in a zero dispersion straight
section, will not affect the dispersion function outside the wiggler.
D. Effect of IDs on the betatron tunes
The IDs with rectangular magnets is achromatic, and the edge defocusing in the
rectangular vertical field magnets cancels the dipole focusing gradient of 1/ρ2.
Thus there is no net focusing in the horizontal plane. The focal length of the
vertical betatron motion and the tune shift resulting from the rectangular wiggler
dipole are respectively
where Lw,total = 4NwLw is the total length of the undulator, and <βz> is the
average betatron amplitude function in the wiggler region.
The vertical sinusoidal field undulator generates average vertical focusing
strength and vertical betatron tune shift:
The dispersion functions in the insertion region induced by a sinusoidal vertical
wiggler field