Potential-well bunch lengthening
in electron storage rings
Demin Zhou, Gaku Mitsuka, Takuya Ishibashi (KEK)
Karl Bane (SLAC)

Acknowledgements
G. Bassi, A. Blednykh, A. Chao, Y. Cai, L. Carver, K. Hirata, R. Lindberg, M. Migliorati, T.
Nakamura, K. Ohmi, K. Oide, B. Podobedov, Y. Shobuda, V. Smaluk, and M. Tobiyama

20th Annual Meeting of Particle Accelerator Society of Japan, Nihon Univ., Aug. 30, 2023

Motivation• Discrepancy between impedance calculations and beam-based measurements- For several decades, theories and numerical tools for impedance calculations have been “well established”.
- Techniques for experimental observations of impedance e ects have also matured in parallel.
- However, discrepancies remain in each accelerator project, to varying degrees [1].

Zotter’s equation [2]:
m=1

ZcI
∥
b
3x −x−Im
2 (E/e)κηω0σz0σδ0(n)=0eff

We must be cautious:
The model can be a source of discrepancies
if its assumptions are violated.
ff[1] V. Smaluk, NIMA 888, 22 (2018). [2] B. W. Zotter, CERN-SPS-81-14-DI (1981).

2

Theories for longitudinal single-bunch impedance effects• Haissinski equation [1]- Exact solution of Vlasov-Fokker-Planck equation below microwave instability threshold.
- Bottom-up predictions of potential-well lengthening: Impedance calculations→Simulations→Experiments.
Iσ∞
z2I ∞
bz0
− 2 − σ ∫z dz′∫−∞ W∥(z′−z′′)λ0(z′′)dz′′
I=
2σz0z02 (E/e)λ0(z) = Aecησδ0• Zotter’s equation [2]- Simple scaling law of bunch lengthening, widely used to extrapolate e ective impedance from bunch length
measurements.

x = σz /σz0 is the bunch lengthening factor.
κ = 2π was used in [2].
κ = 4 π is more consistent to Haissinski equation and experiments [3].

m=1ZcI∥b
3x −x−Im2 (E/e)κηω0σz0σδ0(n)=0eff• Connecting Haissinski equation (HE) and Zotter’s equation (ZE)
- HE is self-consistent, knowing Z∥(k) means “knowing everything”, except for Z∥ /n.
- ZE is easy to use, but it has certain applicability conditions.
- In this talk, we examine ZE and present a new equation for potential-well bunch lengthening.

ff￼￼￼￼￼￼￼￼

[1] J. Haissinski, Nuovo Cimento 18, 1 (1973). [2] B. W. Zotter, CERN-SPS-81-14-DI (1981). [3] V. Smaluk, NIMA 888, 22 (2018).

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A revisit of Zotter’s equation• How Zotter derived the cubic equation?
- Equations of motion + Incoherent tune shift (from e ective impedance) + Energy spread condition [1]2∞∞2ωz0cInd zikz
˜F(z,s)=IW(z−z′)λ(z′,s)dz′=dkZ(k)λ(k,s)e=−z+ηF(z,s)
n∥
∥22∫∫
2π −∞
dsc−∞
2ωz2 = ωz0(1 − ξZ1)

σzωz = σz0ωz0 = − cησδ0Z1(σz) = −• An alternative way to derive the cubic equation

2πcω03σz33ImZ∥

m=1(n)

Incorrect factor
leading to κ =eff

Z∥

m(n)

=e∞

Z∥(ωmp)∑p=−∞∞phm(ωmp)∑ hm(ωmp)p=−∞

Effective impedance

2π- Equations of motion + Incoherent tune shift (from from linearized wake force) + Energy spread condition- The standard formulae of e ective impedance is unnecessary2ωz0d z=−z+ηF(z,s)
ds 2c22￼￼ffLinearization

222ωzωz0 c ηInLeff=−c2c24 πσz32

￼2ωzd z=−z
ds 2c2F(z, s) ≈ F1(s)zCorrect factor∞∂F(z, s) icIn
F1 =dzλ(z, s)=dkkZ∥(k)λ(k, s)λ*(k, s)∫−∞∂z2π ∫−∞→

Average Linear part ofEffective inductancewake force

cIb3x −x−L
=0eff
2 (E/e)κησz0σδ0Modi ed Zotter’s equationfffffi[1] B. W. Zotter, CERN-SPS-81-14-DI (1981).

∞4A new equation derived from Hassinski equation• rms bunch length σz2De nition: σz =∞∫−∞2(z − zc) λ0(z)dz =∞∫−∞22z λ0(z)dz − zc

dλ0(z)z
1+−F(z)λ(z)=0002dz
ησδ2[ σz0]→Trick: Multiply by z and then integrate this equation over z“Effective impedance” for bunch lengthening:∞
2πeffZ∥ =dz(z − zc)λ0(z) ∥(z)c ∫−∞Stretching force average over charge density

Wake potential:∥(z) =∞∫−∞W∥(z − z′)λ0(z′)dz′cIeff
2x −1−Z∥ (x) = 0
2πσz01) Exact prediction by Haissinski equation
2) A generalized version of Zotter’s equation

Normalized current:

I=

Ibσz0

2 (E/e)cησδ0x = σz /σz0d
˜ (k) + zcλ*˜ (k)dkZ∥(k)λ˜0(k) i λ*00[]
dk−∞eff
Z∥ = −∫∞

Both real and imaginary parts of impedance contributeto bunch lengthening if the bunch is deformed￼￼￼𝕎𝕎- σz is sensitive to imaginary part of impedance- If real part of impedance is large, it also contributes to bunch lengtheningfi5

A new equation derived from Hassinski equation• Relation between the “quadratic” equation and Zotter’s equationcI eff2x −1−Z∥ (x) = 0
2πσz0effZ∥ = −∫d˜ (k) + zcλ*˜ (k)dkZ∥(k)λ˜0(k) i λ*00[]dk−∞∞- First, we assume a Gaussian bunch to approximate the Hassinski distribution.

zc =Iσz0c∞

π ∫0dkRe[Z∥(k)]e−k 2σz2eff2Z∥ = − 2σz∫∞0dkkIm[Z∥(k)]e−k 2σz2eff
and zc depends on the imaginary and real parts of impedance, respectively.
∥1) Zeff
2) For a full understanding of broad-band impedance, we need to measure both zc (real part) and Z (imaginary part).
∥- Second, we approximate the ring impedance by Z∥(k) = − ikcLeff. Immediately, we obtain Zotter’s equation.

1eff
Z∥ ∝x→

cIb
3x −x−L=0eff
2 (E/e)κησz0σδ0Z∥
1Leff = − Imω0(n)m=1=eff2σz0πceff
Z∥eff,
andshould be evaluated at nominal bunch length σz0Leff, (Z∥ /n)m=1Zeff∥- We conclude that Zotter’s equation is one special case of self-consistent “quadratic” equation. ZE is only a good
approximation for electron storage rings where the inductance is the dominant impedance.

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Potential-well bunch lengthening for various impedance sourceseff• A table of zc and Z∥ with Gaussian bunch approximation [1][1] D. Zhou, G. Mitsuka and T. Ishibashi,
arXiv:2307.01286 (2023).

- Bunch shortening for positive momentum compaction: Pure capacitance, Free-space steady-state CSR and CWR

7Example 1: SuperKEKB LER

cI effx −1−Z∥ (x) = 0
2πσz02

Haissinski distribution

Effective impedancexZ∥eff (x) = Constant

T. Ishibashi et al., ICFA Newsletter #85, 2023

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Example 2: SuperKEKB HER

cI effx −1−Z∥ (x) = 0
2πσz02

Haissinski distribution

Effective impedancexZ∥eff (x) = Constant

T. Ishibashi et al., ICFA Newsletter #85, 2023

9

Example 3: SLC damping ring with original vacuum chamber [1]Haissinski distribution

Effective impedancexZ∥eff (x) = Constant

SLC is a special case:* With original vacuum chamber,
inductive character is dominant.
* Zotter’s equation does not apply
because of large resistive impedance[1] R. Warnock and K. Bane, Phys. Rev. Accel. Beams 21, 124401 (2018).

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Example 4: SLC damping ring with improved vacuum chamber [1]Haissinski distribution

Effective impedancexZ∥eff (x) = Constant

SLC is a special case:* With improved vacuum chamber,resistive character is dominant.
* Zotter’s equation does not apply
because of relative large resistive
impedance[1] R. Warnock and K. Bane, Phys. Rev. Accel. Beams 21, 124401 (2018).

11Summary∂ψ dz ∂ψ dδ ∂ψ2 ∂2 ∂ψ++=δψ + σδ0
VFP equation:∂s ds ∂zds ∂δctd ∂δ [∂δ ]“Quadratic” equation:cI effx −1−Z∥ (x) = 02πσz02Zc equation:zc(I) = Iσz0κ∥Zm equation:

zm = Iσz0∥(zm)

Haissinski equation:

λ0(z) = Aez2I ∞− 2 − σ ∫z dz′
2σz0z0∥(z′)

m=1Zotter’s equation:ZcI∥b
3x −x+Im
2 (E/e)κηω0σz0σδ0(n)=0eff

Vlasov-Fokker-Planck equation
Haissinski equation“Quadratic” equation

￼𝕎

￼

𝕎

Cubic equation(Zotter)Zc equation

Zm equation

Subset diagram

12

Summary• Zotter’s equation is valid with assumptions- The longitudinal total impedance of the ring can be well approximated by a pure inductance.
- The impact from the real part of the total impedance is negligible.
- The potential well remains well quadratic so that the lengthened bunch is close to Gaussian.
- Applicable cases: NSLS-II, Diamond (light sources with dominating inductive impedances from small discontinuities(bellows/ anges) and tapers (insertion devices)) [1]; SuperKEKB LER (colliders with dominating impedances fromsmall-gap collimators).
Non-applicable cases: damping rings (no insertion devices or collimators, SLC damping ring with smoothedchamber is a good example), Future Circular Colliders (FCCs, resistive-wall impedance dominates because of large
ring circumference).

-• A simple equation for potential-well bunch lengthening is derived from Haissinski equation,
useful for correlating impedance computations with simulations and beam-basedmeasurements.

fl[1] V. Smaluk, NIMA 888, 22 (2018).

13Backup

14• Center of mass zc- Starting from the di erential equation instead→dλ0(z)z
1+−F(z)λ(z)=0002dz
ησδ2[ σz0]
Trick: Integrate this equation over zzc =∞∫−∞∥(z) =∞∫−∞W∥(z − z′)λ0(z′)dz′∞

Loss factor:

λ0(z) = Ae∞z2I ∞− 2 − σ ∫z dz′∫−∞ W∥(z′−z′′)λ0(z′′)dz′′2σz0z0zc(I) = Iσz0κ∥zλ0(z)dz

Wake potential:∞cκ∥ =dzλ0(z) ∥(z) =Re[Z∥(k)]h(k)dk∫−∞
π ∫0- zc is sensitive to real part of impedance, simply proportional to loss factor κ∥.
- The most trivial way of measuring loss factor might be collecting data of power assumptions: RF power, temperature of￼

￼

￼

￼

￼

￼

￼

￼

￼
￼

￼

𝕎

cooling water, …

ff𝕎

Equations derived from Hassinski equation

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Equations derived from Hassinski equation• Peak position of bunch pro le zmdλ0(z)=0dzzm = Iσz0∥(zm)- zm is sensitive to real part of impedancedλ0(z)
z1+−F(z)λ(z)=00022dzσησ[ z0]δ
Trick: Take this term as zero- From measurement viewpoint, zc and zm have di erent meanings.
- Mostly, acc. physicists believe measuring zc is trivial. They usually share simulation data of zc to experimental experts.
- Practically, it is not trivial to measure zc with good accuracy using synchrotron radiation light or using BPM signal.
- However, measuring zm using BPM signal is possible (realized at SuperKEKB)

𝕎

Realistic BPM signal [1]fffi[1] T. Ieiri et al., NIMA 606 (2009) 248–256.

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Equations derived from Hassinski equation• Inverse problem of Haissinski equation- Wake potential extracted from simulated or measured bunch pro leλ0(z) = Ae∞z2I ∞− 2 − σ ∫z dz′∫−∞ W∥(z′−z′′)λ0(z′′)dz′′
2σz0z0σz0d ln λ0(z)z(z)=+∥2 ]
I [ dzσz0- Impedance extracted from wake potential [1]∞𝕎σz0d ln λ0(z)z−ikzZ∥(k) =+ 2 e dz
σz0 ]Ic 2λ˜0(k) ∫−∞ [ dz17fi￼￼￼￼￼￼fi￼￼

[1] A. Chao, Lectures on Accelerator Physics (World Scienti c, 2020)

Some practical examples• Bunch shortening by free-space CSR/CWR- Signi cant bunch shortening/lengthening for positive/negative momentum compaction in simulations for EIC ring-electron cooler [1]Practically, chamber shielding suppresses low-frequency CSR/CWR, such e ects have not be observed inmeasurements

Positive αc

Negative αc18fffi[1] A. Blednykh et al., PRAB 26, 051002 (2023).

• Using BPM to measure beam phase- It’s not trivial to detect the center of mass using BPM signals- Detecting the peak position of the bunch pro le using BPM signals (zero-cross point) was developed at SuperKEKB

Simulated center of mass did nott the measured data well

BPM signal measured at KEKB in 2008 [1]Simulated peak position did t
the measured data well,especially at low bunch currents[1] T. Ieiri et al., NIMA 606 (2009) 248–256.
fi19fifi

Some practical examples

Some practical examples• SuperKEKB LER- Pseudo-Green function wakes constructed and used inputs of simulations

Pseudo-Green function wakes with 0.5mm Gaussian bunch

Fourier transform of short-bunch wakes

Haissinski solutions

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Some practical examples• SuperKEKB LER- Pseudo-Green function wakes constructed and used inputs of simulations

The slopes at zero current have clear
meanings with given impedance andnominal bunch

E ective impedance shows machine
properties

ff21

Some practical examples• SuperKEKB LER- Pseudo-Green function wakes constructed and used inputs of simulations

Wake potential with di erent bunch
pro les

Real part of impedance extracted fromHaissinski solutions

Imaginary part of impedance extracted
from Haissinski solutions

fffi22