Ring Accelerator Physics
for Non Accelerator Specialist
Hitoshi TANAKA
Beam Dynamics TeamXFEL Project Head Office
Cheiron School 2008@SPring-8

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Contents
1.
2.
3.
4.
5.
6.
7.

Introduction
Light source properties vs electron beam performance
Stochasticity of photoemission
Distribution of circulating electron beam
Current limiting factors
Beam stability
New idea towards higher beam performances

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Introduction (1)Even any member in an experiment-related division orBL scientists, who are not the accelerator specialists,should understand the relation between the SR
properties and electron beam performances,
to keep the present experimental condition,
to improve the condition as much as possible,
andto challenge new experiments by using
the advanced SR.
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Introduction (2)FilterChemicaltreatment

Pumping

SR light
SampleWhat determines
the brilliance ?
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Introduction (3)Mutual understanding between accelerator
physicists and experimental division staff
through the communication surely makes
the SR source performance better for users.

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Light source properties vs electron
beam performance(1)Main devices to supply SR to users are“undulators”, which are installed in magnet-freestraight sections in 3rd-generation SR sources.

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2. Light source properties vs electron
beam performance(2)In an undulator, radiation from a single electron
at each undulation interferes with each other.
Spectral narrowing
by resonant condition
for a single electron is
λu(1+K2/2+γ2θ2)λ1 =2γ2where K and γ are a
deflecting parameterand relative energy.

Deflecting angle ψ=γ-1K

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2. Light source properties vs electron
beam performance(3)Q: Why does the radiation concentrate within the γ cone ?
A: The electron can run after the emitted light with thealmost light speed only in this limited angle.
retarded timeR(t’)
t = t’+ cpresent timeΔtTime squeezing factor κ(t’)=Δt’κ(t’) small 䊲㻃large acceleration 䊲
strong radiationrr#t" (t' ) == 1 $ n (t ' ) % & (t' )
#t'

!

)
1& 1
2
" (t' ) # ( 2 + % +*2'$

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2. Light source properties vs electron
beam performance(4)Q: Why does the radiation concentrate within the γ cone ?
A: The electron runs after the emitted light with thealmost light speed only in this limited angle.
)
1& 1
2" (t' ) # ( 2 + % +
*
2'$

!

κ(t’)

Intense radiation
is emitted in thisnarrow angle

θ <1/γ 䊲㻃κ~1/γ2

Width
~2/γ2

θ
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2. Light source properties vs electron
beam performance(5)If electron beam were point-like without spatial
divergence and , all electrons could have noangular divergence, radiations were coherent !
Real radiation properties are obtained by convoluting
radiations from all N electrons
In the real world N
electrons are
distributed in aphase space, neverdegenerate on thesame point.

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2. Light source properties vs electron
beam performance(6)In order to obtain the same resonance condition
over most of circulating electrons distributed,
sufficiently small beam emittance is required.
Interferencezone

Large emittance

Small emittance~2/γ

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2. Light source properties vs electron
beam performance(7)In order to obtain the same resonance condition
over most of circulating electrons distributed,
sufficiently small beam energy spread is required.
λu(1+K2/2+γ2θ2)
λ1 =2(γ+Δγ)2

Nu:undulator period number

Spectral broadening by theenergy spread is much less thanintrinsic broadening ~1/Nu.

ΛnΔλ1 ~ 2λ1 <Δγ2>

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1/Nu=0.05

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2. Light source properties vs electron
beam performance(8)B∝22Ib⋅Nu ⋅γ ⋅Fn(K, δ)

σxσx’σyσy’,

B: Brilliance (phs/sec/mm2/mrad2/100mA)
Ib: Beam current (mA)Nu: Undulator period number
σx,σy: Horizontal and vertical beam sizes (m)σx’,σy’: Horizontal and vertical angular divergence (rad)δ: Beam energy spread
K :Deflection parameter
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2. Light source properties vs electron
beam performance(9)σx= σp2+βxεx+ηx2δ2 , σx’= σp’2+γxεx+η'x2δ2 , Photon
2+β ε +η 2δ2, σ = σ 2+γ ε +η’ 2δ2
σ
p’
y y
y
σy= p y y y,y’

1+αx,y2-dβx,ydηx,y,, γx,y=.
αx,y=η'x,y=
βds
dsx,yElectronemittance
Electronenergyspreadεx,εy: Horizontal and vertical emittance (m•rad)
βx,y : Horizontal and vertical betatoron functions at ID
ηx,y : Horizontal and vertical dispersion functions at ID
σp,σp’: Spatial and angular divergence of photon beam
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3. Stochasticity of photoemission(1)The photo-emission process is not continuous, in otherwords, quantized process. So, the emission position, thenumber of the emission photons, and the emission photon
energy have fluctuations.
This stochasticity (random fluctuation) causes finite
spread of the circulating electron beam in the 6D
phase space. The density distribution is generally
Gaussian due to the central limit theorem.

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3. Stochasticity of photoemission(2)1stContinuous fluid

2nd

3rd

4th

Photoemission process

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3. Stochasticity of photoemission(3)A relativistic electron accelerated in a magnetic field willradiate electromagnetic energy at a rate which is
proportional to the square of the accelerating force.
classical electron radius<Averaged radiation power>
4
rec
rec
2
E
2
2
2
.
E F⊥ =
P= 32
2
3(m0c2)33 (m0c ) ρ<Averaged radiation energy per turn> Averaged properties
TLdl ,are smooth !
U=䌶Pdt = 䌶㻳 c
88.5E4(GeV) for the case with the constant ρ.
U(keV)=
ρ(m)Cheiron School 2008@SPring-8

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3. Stochasticity of photoemission(4)As a first approximation, we can assume that a photoemission process is continuous. However, allelectromagnetic radiation occurs in quanta of discreteenergy. This quantization of the energy loss is characterized
by instantaneous emission, random emission timing, andrandom quantized energy, etc.
<Mean square of energy loss fluctuation per unit time><Nu2> =

55
24 3

Uu ,
c
T

uc =

uc: critical photon energy ~ 3.2<u>3 hc γ3 .
2 ρ<N>(phs/s) ~ 3.2P/ uc

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3. Stochasticity of photoemission(5)~2keV~30keVρ=~40muc represents the photon energy at the peak brilliance of
the BM radiation.

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3. Stochasticity of photoemission(6)Let’s estimate the parameters!
<Case-1 E=1 GeV, ρ=5m>
88.5×14
U
8
U(keV)== 17.7, P(keV/s)=T=2π×5/c= 1.7×10
5
3 6.85×10-19×2.998×108×19573= 0.44uc (keV)=25
55 1.7×108 × 0.44=0.99×108<Nu2> (keV2/s) =24 3<N>(photons/s) ~ 3.2×1.7×108 / 0.44 = 1.2×109
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3. Stochasticity of photoemission(7)SPring-8 parameter is the next example.
<Case-2 E=8 GeV, ρ=40m>
88.5×84
U3= 1.1×1010
U(keV)== 9.1×10 , P(keV/s)=
40
T=2π×40/c
3 6.85×10-19×2.998×108×156563= 29.6
uc (keV)=240
55 1.1×1010 × 29.6=4.31×1011<Nu2> (keV2/s) =24 3<N>(photons/s) ~ 3.2×1.1×1010 / 29.6 = 1.2×109
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4. Distribution of circulating electronbeam(1)6D-phase space volume of a single electron, εs6 comprisesof canonical variables (x, px(x’), y, py(y’), t, ps(ΔE/E)).
In an ideal case, 6D-phase space volume can be written by
the product of areas of the three orthogonal 2D spaces

εsz.

z=x,y,sTransverse motion
Longitudinal motion

εs6=εsxεsyεss
These are invariants ofthe motion for aconservative system.
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4. Distribution of circulating electronbeam(2)
You can understand the relation between the 2D
phase space and the emittance by using a simple
harmonic oscillator.
x = A䝿cos(ωt+φ0)
dx =x’= -A䝿ω sin(ωt +φ )0ds
1 dx = -A䝿sin(ωt+φ )
0
ω ds

1 x’
ω
A
x

Action=A, Angle=φ= ωt +φ0 ,Invariant= x 2+(x’/ω)2=A2= circle area = εsxπCheiron School 2008@SPring-8

ωt23

Distribution of circulating electronbeam(3)
We need potential wells to stabilize threeorthogonal oscillation modes to keep electron
beam in a storage ring.
Quadrupole magnets generate theadequate potential wells for two
transversal oscillation modes, which
are called betatron oscillations in thehorizontal and vertical planes.
RF acceleration electric field
generates the adequate potential wellfor longitudinal oscillation mode,
which is called a synchrotron
oscillation.
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Distribution of circulating electronbeam(4)
SR light properties reflects the 3×2D phasespace distribution of circulating electrons.
We use the following three ensemble-averaged
emittances to express beam distribution in threeorthogonal phase spaces.

<εs6> = < εsx > < εsy > < εss >=εx εy εs,εx : horizontal emittance (m rad)εy : vertical emittance (m rad)εs : longitudinal emittance (m rad)Cheiron School 2008@SPring-8

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4. Distribution of circulating electronbeam(5)
Width of the Gaussian distribution of N
circulating electrons is determined by the

dynamical equilibrium between theradiation excitation and damping.

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4. Distribution of circulating electronbeam(6)
Remember the invariant of a harmonic oscillator.
dε zd< A2>= dt
dt

z = x,y,s.

Equilibrium condition:2
2
d< A2> lim <(A+ΔA) - A > lim <AΔA> + lim <ΔA2> = 0= Δt䊲㻓
=Δt䊲㻓 2
Δt䊲㻓
Δt
Δt
dt
Δt
Damping term
Averaged energy dissipation

Excitation term
Quantum effect

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4. Distribution of circulating electronbeam(7)
Energy spread σΔΕ
2>
<
A
<AΔA>
,
= -2
Δt䊲㻓 2
τε
Δt

lim

E×T ,
U

τε =

lim <ΔA2>
55
2> =
=
<Nu
Δt䊲㻓
Δt
24 3

Uu .
c
T

Since σε is not the emittance, phase average factor2><
A2
should be considered, σΔΕ = 2 ,

σΔΕ=

55
96 3

E×uc , σΔΕ/Ε=

55
96 3

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uc
E

.
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4. Distribution of circulating electronbeam(8)τ=(z-z )/c0zBunch length στ

στ has two components;
reference electron z=z0
dilation factorστ1= time spread by energy spread
α ΔΕ/Ε
στ2= time spread by stochastic photoemissionωangular frequency
of energy oscillationστ1 >>στ2
10-4
-3
α
10
σ
στ = ω ΔΕ/Ε104

= ~10 psec
@ SPring-8

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τ=t-t0ωt

Phase space of synchrotron(energy) oscillation29

4. Distribution of circulating electronbeam(9)
Horizontal emittance εxExcitation: due to the discrete energy jump + energy
dispersion
Damping: due to the decrease of transverse momentum
by the photoemission + acceleration along the running
direction
Excitation
Damping

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4. Distribution of circulating electronbeam(10)For the typical magnetic lattice structure (Chasman
Green: CG) based storage ring, the horizontal
minimum emittance is written by

εx min @general ~=1@achromat
x
min
2
Cqγ2
θb3
8 15 JxεCq: Quantumn constant 3.832×10-13 (m)θb: Deflection angle of a single bending magnet (rad)Jx: Horizontal damping partition number ~ 1
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4. Distribution of circulating electronbeam(11)
Chasman-Green (CG) lattice is the most popularmagnet cell structure for a low emittance SR souce,
where a achromatic arc is composed of a pair ofbending magnets.
2 bends: CG or CBA
3 bends: TBA
4 bends: QBA

εx min∝θb3

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θb smallerwith same
cell No.
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4. Distribution of circulating electronbeam(12)
Vertical emittance εyεy has two components;
εy1= vertical emittance by stochastic photoemission
εy2= vertical emittance by HV coupling
Usually, εy1 << εy2

1-GeV storage ring
γ=1000/0.511∼2000

1/γ∼5×10-4
The Angular divergence
In the vertical plane is ~ 1/γ

Principally, εy =1/1000 εx
is possible

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4. Distribution of circulating electronbeam(13)
Vertical emittance is determined by magnetic error
components mixing the horizontal and vertical betatronoscillation. The main sources are vertical misalignmentsof sextupole magnets and rotational errors of quadrupole
magnets.
The effect of these error fields can be corrected to
0.1 % level by the combination of beam response
analysis and skew quadrupole corrector magnets.
One-dimensional diffraction limited X-ray beam is
now available
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5. Current limiting factors(1)B ∝ IbWhat limits the stored current ?

(1) The circulating electrons trapped in the RF acceleration
potential forms the bunch.
(2) The electron-electron scattering scarcely causes theemittance growth and instability in high energy region.
(3) The electron bunch circulates at the inside of the vacuum
chamber, which has lots of non-smooth structures suchas bellows, valves, cavities.
(4) The indirect interaction of the electrons via the externalboundary determines the current upper limit.
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5. Current limiting factors(2)Wakefield by the circulating electron beam
Example: accelerating structure
Acc.structure

exciting electron
bunchwakefield

The electron bunchleaves a part of beamenergy as a electromagnetic (wakefield)
field which perturbs the
successive bunches.

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5. Current limiting factors(3)Average current limit by coupled bunch instability(1) The circulating electron bunches have a wide frequency
range from revolution to higher frequencies up to 10 GHz
corresponding to the bunch length.
(2) These bunches excite harmful wakefields in high-Q
structure or at narrow gap ID with a quite large
impedance at lower frequencies.
(3) The excited wakefield causes the coherent instability.

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5. Current limiting factors(4)Built-up wakefield
Image of the coupled bunch instability

High Q structure (RF cavity)

Narrow gap undulator

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5. Current limiting factors(5)Peak current limit by single bunch instability(1) The wakefield excited in non-smooth structures, of which
size is much smaller than the bunch length, works on thebunch as an inductive force.
(2) This wakefield generates the energy chirp in the bunchand deforms the bunch distribution.

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5. Current limiting factors(6)Towards the high stored current
Passive(1) Suppression of harmful impedance(2) Landau damping enhancement
Active(3) Introduction of some feedback system
At SPring-8,
average current ~120 mA (target 200 mA)
Single bunch current ~5mA (Max. 15mA)
High current vs low emittance, which is better ?
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6. Beam stability(1)Beam stability is quite important for effective
use of sharp (brilliant) SR
Perturbation sources• Compressors & Pumps• Magnet current ripple• Vibration by cars, trains, ..
• Temperature drift• Earth tide
Sometimes, river also becomes a• Earthquake
serious perturbation source….
• etc…
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6. Beam stability(2)To supply stable brilliant SR,
we have continued to improvethe beam stability from multiangles.

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6. Beam stability(2)Beam energy stabilization by RF frequency FB

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6. Beam stability(3)Beam orbit stabilization by the global COD FBCheiron School 2008@SPring-8

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6. Beam stability(4)Beam orbit stabilization by the source suppression

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6. Beam stability(5)Making IDs transparent for users

baredynamic term
corrected
dynamic + static
term corrected
BG
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6. Beam stability(6)Introducing top-up operation
(a)

(b)

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6. Beam stability(7)Beam stability of a few µm was achieved

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7. New idea towards higher beam
performances(1)• EUV-FEL and XFEL
• 4-dimensional phase space
manipulation for an intense
micro-beam

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Towards XFEL(a) Oscillator(b) Single Pass (SASE) FEL(a) Mirror development for EUV~X-ray region(b) High quality electron beam for a high FEL gain
and long undulators with high field quality
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Laser amplification by SASE
Undulator

E-beamSASE

Micro-bunching

)
g
/L

=

)
z
(
P

e

P0

(z
p
x

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Interaction with a builtup laser fieldGain~105

Log(Gain)

Energy modulation by
spontaneousemission51

4-dimensional phase space manipulation
Motivation
In the SPring-8 storage ring, spatial coherence of
X-ray SR is limited by horizontal electron beamemittance, being much larger than photon beamemittance. On the other hand, vertical one issmaller compared with photon beam emittance.
The spatial coherence is potentially improved byadjusting the miss-match between two kinds ofemittance,i.e.,
optimizingtheelectrondistribution in the 4-dimensional transverse
phase space.
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Spatially coherent X-ray
from unmatched electron beam
Narrow coherent area
S N S N S N

N S N S N

S N

S N

S

S N

Slit

Week micro beam

Spatially coherent X-ray
from matched electron beam
Wide coherent area
S N S N S N

N S N S N

S N

S N

S

S N

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Intense micro beam
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How to realize
Magnet-free long straight section of 30m+ 4-dimensional phase space transformer
60
4D Phase-space
Transformer

y'
x

2.4

40

2
!x

30

1.6

!y

1.2
20
0.8
10

0.4

Dispersion Function "x [m]Betatron Function !x, y [m]

50

x'

2.8

x'

y

y'
x

y

ID
0

0
0

20

y

40
60
Pathlength s [m]y80:Electron beam :Photon beam

y

y

y

y

x

x

x

x

x

x

Uncoupled Mode

4D-Transformer

Circular or
Elliptic Mode

Rotation Stopper

4D-Inverse
Transformer

Uncoupled Mode

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