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

2013/09

Cheiron School 2013@SPring-81Outline
1. Light source properties vs electron beamperformance
2. Stochasticity of photoemission
3. Distribution of circulating electron beam
4. Approach to coherent X-rays

2013/09

Cheiron School 2013@SPring-82

Towards survival
Even any member in an experiment-related division orBL scientists, who are not the accelerator specialists,should understand the relation between the SRproperties 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.

2013/09

Cheiron School 2013@SPring-83

Outline
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

2013/09

Cheiron School 2013@SPring-84

2. 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.

2013/09

Cheiron School 2013@SPring-85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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Cheiron School 2013@SPring-86

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 time
∆tTime squeezing factor κ(t’)=∆t’κ(t’) small → large acceleration →strong radiation

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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.

θ <1/γ → κ~1/γ2

κ(t’)

Intense radiation
is emitted in thisnarrow angle

Width~2/γ2θ
2013/09Cheiron School 2013@SPring-88

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 aredistributed in aphase space, neverdegenerate on thesame point.

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Cheiron School 2013@SPring-89

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+∆γ)2

Nu:undulator period number

Λn∆λ1 ~ 2λ1 <∆γ/γ>

1/Nu=0.05

Spectral broadening by theenergy spread is much less thanintrinsic broadening ~1/Nu.
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Cheiron School 2013@SPring-811

2. Light source properties vs electron
beam performance(8)Ib⋅Nu2⋅γ2 ⋅Fn(K, δ) ,B∝
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 parameter2013/09

Cheiron School 2013@SPring-812

2. Light source properties vs electron
beam performance(9)2+β ε +η 2δ2 , σ = σ 2+γ ε +η' 2δ2 , Photonσσx= p x x xp’x xxx’
Electronemittance2+γ ε +η’ 2δ2222σσy= σp +βyεy +ηy δ , σy’= p’ y y y ,Electron
21+αx,y-dβx,ydηx,yenergy, η'x,y=, γx,y=.
αx,y=βx,yds
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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Cheiron School 2013@SPring-813

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

2013/09

Cheiron School 2013@SPring-814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 generallyGaussian due to the central limit theorem.

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Cheiron School 2013@SPring-815

3. 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
rec
rec
2
E
222.
E F⊥ =P=2
32
2
3
3 (m0c )3 (m0c ) ρ<Averaged radiation energy per turn> Averaged properties
TLd ,
are smooth !
U=∫Pdt = ∫P c88.5E4(GeV)
for the case with the constant ρ.
U(keV)=
ρ(m)2013/09

Cheiron School 2013@SPring-816

3. 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 =c24 32 ρT
uc: critical photon energy ~ 3.2<u><Nu2> =

<Averaged photo-emission rate><N>(phs/s) ~ 3.2P/ uc
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Cheiron School 2013@SPring-817

3. Stochasticity of photoemission(4)~2keV~30keVρ=~40muc represents the photon energy at the peak brilliance ofthe BM radiation.
2013/09

Cheiron School 2013@SPring-818

3. Stochasticity of photoemission(5)Let’s estimate the parameters!
<Case-1 E=1 GeV, ρ=5m>
88.5×14
UU(keV)== 17.7, P(keV/s)=T=2π×5/c= 1.7×108
5
3 6.85×10-19×2.998×108×19573= 0.44uc (keV)=2555 1.7×108 × 0.44=0.99×108<Nu2> (keV2/s) =24 3<N>(photons/s) ~ 3.2×1.7×108 / 0.44 = 1.2×1092013/09

Cheiron School 2013@SPring-819

3. 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
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
2013/09

Cheiron School 2013@SPring-820

Outline
1. Introduction
2. Light source properties vs electron beam
performance3. Stochasticity of photoemission
4. Distribution of circulating electron beam
5. Extra: Approach to coherent X-rays

2013/09

Cheiron School 2013@SPring-821

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(∆E/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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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・cos(ωt+φ0)
dx=x’= -A・ω sin(ωt +φ0)dt
1 dx
ω dt = -A・sin(ωt+φ0)

1 x’
ω
Ax

Action=A, Angle=φ= ωt +φ0 ,Invariant= x 2+(x’/ω)2=A2= circle area = εsxπ2013/09

Cheiron School 2013@SPring-8

ωt23

4. 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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Cheiron School 2013@SPring-824

4. 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)2013/09

Cheiron School 2013@SPring-825

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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Cheiron School 2013@SPring-826

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

z = x,y,s.

Equilibrium condition:
2 - A2>2<(A+∆A)
lim <A∆A> lim <∆A2>
lim
d< A >+ ∆t→0 ∆t=0= ∆t→0= ∆t→0 2 ∆t
dt∆t
Damping term
Averaged energy dissipation

Excitation term
Quantum effect

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Cheiron School 2013@SPring-827

4. Distribution of circulating electronbeam(7)Energy spread σ∆Ε2><A<A∆A>,= -2∆t→0 2∆t
τεlim

τε =lim <∆A2>
55
2> ==<Nu∆t→0∆t
24 3

E×T ,UUu .
cTSince σ∆Ε is not the emittance, phase average factor
2><Ashould be considered, σ 2 = 2 ,∆Εσ∆Ε=
2013/0955
96 3

E×uc ,σ∆Ε/Ε=55
96 3

Cheiron School 2013@SPring-8ucE.
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4. Distribution of circulating electronbeam(8)τ=(z-z0)/cBunch length στzreference electron z=z0στ has two components;
dilation factorστ1= time spread by energy spreadα ∆Ε/Εστ2= time spread by stochastic photoemission
ωστ1 >>στ2angular frequency
of energy oscillation

10-4-3α10στ = ω σ∆Ε/Ε104
2013/09= ~10 psec
@ SPring-8

Cheiron School 2013@SPring-8

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
Damping2013/09

Cheiron School 2013@SPring-830

4. Distribution of circulating electronbeam(10)For the typical magnetic lattice structure (Chasman
Green: CG) based storage ring, the horizontal
minimum emittance is written byεx min @general ~=ε1
@achromatx min
2
Cqγ2
θb3
8 15 Jx

Cq: 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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Cheiron School 2013@SPring-831

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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Cheiron School 2013@SPring-8

θb smallerwith samecell No.
32

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
2013/09

Principally, εy =1/1000 εxis possible

Cheiron School 2013@SPring-833

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 isnow available
2013/09

Cheiron School 2013@SPring-834

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

2013/09

Cheiron School 2013@SPring-835

5. Approach to coherent X-rays (1)欧米

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5. Approach to coherent X-rays (2)3.4 nmrd

4th Gen.
3rd Gen.
2nd Gen.

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5. Approach to coherent X-rays (3)Equation of natural emittance:
Conventional reduction scheme:
1. Reduction of bending angle (θ)
by increasing the number ofbending magnets

Additional reduction schemes:
2.
3.
4.
5.

γ : Lorentz factorθ : Bending angle
ρ : Bending radius

H : H-function
J : Damping partition
x number

Reduction of stored energy (γ) with the help of advanced
undulator design
Optimization of dipole field (ρ) in a dipole and / or inside
unit cell)
Damping enhancement (<H/ρ3>/<1/ρ2>) by additionalradiation
Damping partition number (Jx) control
2013/09

Cheiron School 2013@SPring-838