Ring Accelerator Physics
for Non Accelerator Specialists
XFEL Research & Development Division
RIKEN SPring-8 Center
Hitoshi Tanaka

2014/09

Cheiron School 2014@SPring-81Outline
1. Light source properties vs electron beam
performance
2. Stochasticity of photoemission
3. Distribution of circulating electron beam
4. Approach to coherent X-rays	

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1. 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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1. 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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1. 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 timepresent timeR(t’)
t = t’+ cΔtTime squeezing factor κ(t’)=Δt’κ(t’) small → large acceleration →strong radiationΔtκ(t' ) == 1 − n(t' )⋅ β(t' )Δt'2014/09

€

)
1& 1
2
κ(t' ) ≈ ( 2 + θ +2'γ*Cheiron School 2014@SPring-8

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1. 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
κ(t' ) ≈ ( 2 + θ +
2'γ
*

κ(t’)

θ <1/γ → κ~1/γ2

Intense radiation
is emitted in thisnarrow angle

Width
~2/γ2

θ
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1. 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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1. 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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1. 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.
2014/09ΛnΔλ1 ~ 2λ1 <Δγ/γ>

Cheiron School 2014@SPring-8

1/Nu=0.05

91. Light source properties vs electron
beam performance(8)

B∝

2
2	

Ib⋅Nu ⋅γ ⋅Fn(K, δ)

σxσx’σyσy’ (~εx ⋅ εy),

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 parameter2014/09	

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1. Light source properties vs electron
beam performance(9)2+β ε +η 2δ2 , σ = σ 2+γ ε +η' 2δ2 , Photonσσx= p x x x
p’
x x
x
x’
Electronemittance2+γ ε +η’ 2δ2222σ
σy= σp +βyεy +ηy δ , σy’= p’ y y y ,Electron
1+αx,y2-dβx,ydηx,yenergy
,
,
.
γx,y=
αx,y=
η'x,y=
βx,y
ds
ds
spreadεx,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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1. Light source properties vs electron beam
performance
2. Stochasticity of photoemission
3. Distribution of circulating electron beam
4. Extra: Approach to coherent X-rays	

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2. Stochasticity of photoemission(1)The photo-emission process is not continuous but
quantized. So, the emission position, the number of theemission photons, and the emission photon energy havefluctuations.
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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2. Stochasticity of photoemission(2)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
re c
re c
2
E
2
2
2
.
E F⊥ =
P= 3
2
3
2
2
3
(m0c )
3 (m0c ) ρ<Averaged radiation energy per turn> Averaged properties
T
L
d" ,
are smooth !
U=∫Pdt = ∫Pc
88.5E4(GeV)
for the case with the constant ρ.
U(keV)=
ρ(m)2014/09

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2. Stochasticity of photoemission(3)In the quantized radiation process, the integrated parameter,
energy loss per unit time also fluctuates. Magnitude of thefluctuation can be defined by a mean square.
<Mean square of energy loss fluctuation per unit time>55
3 hc γ3 .
Uu ,
uc =
c
24 3
2 ρ
T
uc: critical photon energy ~ 3.2<u><Nu2> =

<Averaged photo-emission rate><N>(phs/s) ~ 3.2P/ uc
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2. Stochasticity of photoemission(4)~2keV~30keVρ=~40muc represents the photon energy at the peak brilliance of
the BM radiation.
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2. Stochasticity of photoemission(5)Let’s estimate the parameters!
<Case-1 E=1 GeV, ρ=5m>
88.5×14U= 1.7×108
P(keV/s)=U(keV)== 17.7,
T=2π×5/c
53 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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2. Stochasticity of photoemission(6)SPring-8 parameter is the next example.
<Case-2 E=8 GeV, ρ=40m>
88.5×84
U3= 1.1×1010
P(keV/s)=U(keV)== 9.1×10 ,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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1. Light source properties vs electron beam
performance
2. Stochasticity of photoemission
3. Distribution of circulating electron beam
4. Extra: Approach to coherent X-rays	

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3. 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 byεthe product of areas of the three orthogonal 2D spaces sz.
z=x,y,sTransverse motion
Longitudinal motion

εs6=εsxεsyεss
These are invariants ofthe motion for aconservative system.
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3. 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・cos(ωt+φ0)
dx=x’= -A・ω sin(ωt +φ0)
dt
1 dx
ω	

dt = -A・sin(ωt+φ0)

1 x’
ω
A
x

Action=A, Angle=φ= ωt +φ0 ,Invariant= x 2+(x’/ω)2=A2= circle area = εsxπ		2014/09

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ωt21

3. 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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3. 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)2014/09

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3. 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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3. Distribution of circulating electronbeam(6)
Remember the invariant of a harmonic oscillator.
dεzd< A2>= dt
dt

z = x,y,s.

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

Excitation term
Quantum effect

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3. Distribution of circulating electronbeam(7)
Energy spread σΔΕ	

2>
<
A
<AΔA>
,
= -2
Δt→0 2
Δt
τε	

lim

τε	

 =

lim <ΔA2>
55
2> =
=
<Nu
Δt→0
Δt
24 3

E×T ,
U

Uu .
c
T

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

σΔΕ=
2014/0955
96 3

E×uc ,

σΔΕ/Ε=

55
96 3

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uc
E

.
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3. Distribution of circulating electronbeam(8)τ=(z-z0)/czBunch length στ	reference electron z=z0
στ has two components;	

dilation factorστ1= time spread by energy spread
α ΔΕ/Ε
στ2= time spread by stochastic photoemission	ω	angular frequency
of energy oscillationστ1 >>στ2	

10-4

α	

10-3
στ = ω	σΔΕ/Ε104
2014/09= ~10 psec
@ SPring-8

Cheiron School 2014@SPring-8

τ=t-t0ωt

Phase space of synchrotron(energy) oscillation27

3. Distribution of circulating electronbeam(9)
Horizontal emittance εx	

Excitation: 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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3. 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
2C qγ 2
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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3. 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.
30

3. Distribution of circulating electronbeam(12)
Vertical emittance εy	εy 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
2014/09

Principally, εy =1/1000 εx
is possible

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3. 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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1. Light source properties vs electron beam
performance
2. Stochasticity of photoemission
3. Distribution of circulating electron beam
4. Approach to coherent X-rays	

2014/09

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4. Approach to coherent X-rays (1)Energy dispersion
NA: ~0.1 m
at ID source points	
A：~0.0 m

10 keV	
2.4 nmrd, NA
6.7 nmrd, A

3rd
Gen.	

4th
Gen.	

2nd Gen.	

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4. Approach to coherent X-rays (2)Emittance (nmrad)	

欧米	

Time-axis is
SSRF	
crucial	

ESRF
Upgrade★	
Targeting timely	

PLS2	
TPS	
TLS	
PLS	

Year starting user operation	
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4. Approach to coherent X-rays (3)γ 2 H ρ3γ 2θ 3∝Equation of natural emittance:	
 ε nat = Cq
2
Jx
Jx 1 ρ
Conventional reduction scheme:	
 γ : Lorentz factorθ : Bending angle
1. Reduction of bending angle (θ)ρ : Bending radius
by increasing the number of
H : H-function
bending magnets
Jx :	
Damping partition
Additional reduction schemes:	
number 	
2. Reduction of stored energy (γ) with the help of advanced
undulator design
3. Optimization of dipole field (ρ) in a dipole and / or inside
unit cell)
4. Damping enhancement (<H/ρ3>/<1/ρ2>) by additionalradiation
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4. Approach to coherent X-rays (4)SPring-8-II Unit Lattice	

SPring-8-II Lattice	

Quadrupoles
2014/09

Bendings

Sextupoles

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4. Approach to coherent X-rays (5)Main Parameter	

New Optics

Present Optics	

Energy (GeV)	

6	

8	

Circumference (m)1435.4	

1435.9	

Unit cell structure

5 BMs	

2 BMs	

Ring structure

2 Injection Cells + 44 Unit Cells +42 Unit Cells +4 Straight Cells	
4 Straight Cells	

ID straight length (m)

4.68	

6.65	

Natural emittance (nmrad)0.15 (Achro, w/o und)~0.10 (Achro, w und)	

2.4 (NA)
6.7 (Achro)	

Coupling ratio (%)

10 	

0.2	

Stored current (mA)

100	

100	

Filling pattern

Multi-bunches	

Multi-/Several
bunches	<10	

10~100	

Beam lifetime (hr)2014/09

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4. Approach to coherent X-rays (6)Main Parameter	

New Optics

Present Optics	

ID straight	

β Function @ID (βx, βy) (m)(5.5, 3.0)	

(31.2, 5.0)	

Dispersion ηx @ID (m)0.0	

0.146	

Beam sizes @ID (σx, σy) (µm)(24.0, 5.6)	

(316, 4.9)	

Angular div. @ID (σx’, σy’) (µrad)(4.4, 1.9)	

(8.8, 1.0)	

Critical photon energy (keV)

13.9	

28.9	

β Function (βx, βy) (m)(1.8, 14)	(2.9, 28)	Dispersion ηx (m)0.00016	

0.039	

Critical photon energy (keV)

22.8	

28.9	

β Function (βx, βy) (m)(0.9, 1.9)	

(2.4, 31)	0.0016	

0.059	

Bending magnet BM1

Bending magnet BM2

Dispersion ηx (m)2014/09

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4. Approach to coherent X-rays (7)Spectrum of Standard Undulator Radiation	

by SPECTRA	

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