Proceedings of the 20th Annual Meeting of Particle Accelerator Society of Japan
August 29 - September 1, 2023, Funabashi

PASJ2023 WEOB14

POTENTIAL-WELL BUNCH LENGTHENING IN ELECTRON STORAGE
RINGS
Demin Zhou∗ , Gaku Mitsuka, Takuya Ishibashi,
KEK Accelerator Laboratory, 1-1 Oho, Tsukuba 305-0801, Japan
Karl Bane, SLAC National Accelerator Laboratory, Menlo Park, California 94025, USA
Abstract

motion are modified to:

The cubic equation derived by B. Zotter has been popularly used for electron storage rings to describe the scaling
law of potential-well bunch lengthening. This equation has
also often been used to calculate the effective impedance
when the bunch lengthening is measured or simulated. This
paper discusses the validity of Zotter’s equation and presents
an alternative but self-consistent equation for potential-wellbunch lengthening. Its applications to predicting bunch
lengthening and extracting effective impedance from bunch
length measurements are also addressed.

INTRODUCTION
Individual electrons oscillate around a fixed point in thelongitudinal phase space in electron storage rings. The RF
system creates a potential well that confines the beam ina bucket. When the oscillations are of small amplitude,
the motion of the electrons is linear. Using the longitudinalcoordinate 𝑧 and momentum deviation 𝛿, the linear equationsof motion are [1]𝑑𝑧= −𝜂𝛿,
𝑑𝑠𝑒𝑉 𝜔 cos 𝜙s𝑑𝛿= − rf rf
𝑧.
𝐸𝐶𝑐
𝑑𝑠

(1)

The following quantities are used: the electron charge 𝑒, thespeed of light 𝑐, the slip factor 𝜂, the RF voltage 𝑉rf , the RF
frequency 𝜔rf , the electron energy 𝐸, the circumference ofthe ring 𝐶 and the synchronous phase 𝜙s . We assume thatthe reference particle is moving at the speed of light, 𝑐. In thepresence of radiation damping and quantum excitation, thebeam reaches a stable equilibrium distribution in the longitudinal phase space, which is Gaussian [2]. The momentum
spread 𝜎𝛿0 in an electron storage ring at equilibrium is determined by the beam energy and the bending radii of thedipole magnets but is not affected by the RF parameters 𝑉rf
and 𝜔rf (see Eq. (7.94) of [1]). Radiation effects cause thebeam to bunch, with an rms length of 𝜎z0 = −𝑐𝜂𝜎𝛿0 /𝜔z0 .
From Eqs. (1), the synchrotron frequency 𝜔z0 is defined by
𝑒𝑐𝑉rf 𝜂𝜔rf cos 𝜙s𝜔z0 ≡ sgn[𝜂]√−
,
𝐸𝐶

(2)

where sgn[] is the sign function. We define 𝜔z0 to have theopposite sign of 𝜂 (this is the choice of the SAD code [3])so that 𝜎z0 is always positive.
The longitudinal wakefields deform the potential wellcreated by the RF system. As a result, the equations of𝑑𝛿 𝜔2z0= 2 𝑧 − 𝐹(𝑧, 𝑠).
𝑑𝑠
𝜂𝑐

𝑑𝑧
= −𝜂𝛿,𝑑𝑠(3)

The wakefield term 𝐹(𝑧, 𝑠) is calculated from the convolutionof charge density and longitudinal wake function∞

𝐹(𝑧, 𝑠) = 𝐼𝑛 ∫−∞𝑊∥ (𝑧 − 𝑧′ )𝜆(𝑧′ , 𝑠)𝑑𝑧′ ,(4)with the scaling factor 𝐼𝑛 = 𝑁𝑒2 /(𝐸𝐶) = 𝐼𝑏 /(𝑐(𝐸/𝑒)), theline density distribution 𝜆(𝑧, 𝑠), and the wake function 𝑊∥ (𝑧).
Here, 𝑁 is the bunch population, and 𝐼𝑏 = 𝑁𝑒/(𝐶/𝑐) is thebunch current. The wake function and the corresponding
impedance are Fourier transforms of each other: 𝑊∥ (𝑧) =∞∞𝑐∫ 𝑍 (𝑘)𝑒𝑖𝑘𝑧 𝑑𝑘 and 𝑍∥ (𝑘) = 1𝑐 ∫−∞ 𝑊∥ (𝑧)𝑒−𝑖𝑘𝑧 𝑑𝑧 with
2𝜋 −∞ ∥𝑘 ≡ 𝜔/𝑐. Using the impedance, 𝐹(𝑧, 𝑠) can be rewritten as
𝐹(𝑧, 𝑠) =𝑐𝐼𝑛 ∞̃ 𝑠)𝑒𝑖𝑘𝑧 ,∫ 𝑑𝑘𝑍∥ (𝑘)𝜆(𝑘,
2𝜋 −∞

(5)

̃ 𝑠) = ∫∞ 𝜆(𝑧, 𝑠)𝑒−𝑖𝑘𝑧 𝑑𝑧. Sincewith the beam spectrum 𝜆(𝑘,−∞𝑊∥ (𝑧) and 𝜆(𝑧, 𝑠) are real, the reality condition applies:̃𝜆̃ ∗ (𝑘, 𝑠) = 𝜆(−𝑘,𝑠) and 𝑍∥∗ (𝑘) = 𝑍∥ (−𝑘).

THE CUBIC EQUATION FOR BUNCHLENGTHENING
In [4], Zotter solved the equations of motion for a single
particle as described by Eqs. (3). The leading-order perturbation from the longitudinal wakefields was found to shift
the incoherent synchrotron frequency. For electron storagerings, Eq. (5.10) of [4] is of interest to us:
𝜔2z = 𝜔2z0 (1 − 𝜉𝑍1 ) ,(6)with 𝜉 = 𝜂𝜔20 𝐼𝑏 /(2𝜋𝜔2z0 (𝐸/𝑒)) and𝑍1 (𝜎z ) = −𝑍∥ 𝑚=1
Im
.
(
)
𝑛 eff
𝜔30 𝜎3z

√2𝜋𝑐3

(7)

Here 𝜔z is the incoherent synchrotron frequency after bunch
lengthening, 𝜎z is the rms length of the lengthened bunch,𝑚=1
𝜔0 = 2𝜋𝑐/𝐶 is the revolution frequency, and (𝑍∥ /𝑛)eff
is the effective impedance for azimuthal mode 𝑚 = 1. The
effective impedance is a quantity used in collective instability
theories [5]. It is defined as follows:∗ dmzhou@post.kek.jp

∞

𝑍 (𝜔

𝑝=−∞

- 117 -)

∑ ∥ 𝑝𝑚𝑝 ℎ𝑚 (𝜔𝑚𝑝 )
𝑍∥ 𝑚
𝑝=−∞
( ) =
,
∞
𝑛 eff
∑ ℎ (𝜔 )
𝑚

𝑚𝑝

(8)

Proceedings of the 20th Annual Meeting of Particle Accelerator Society of Japan
August 29 - September 1, 2023, Funabashi

PASJ2023 WEOB14

where 𝑛 = 𝜔/𝜔0 , 𝜔𝑚𝑝 = 𝑝𝜔0 + 𝑚𝜔z the mode frequencies.
The power density of the 𝑚-th mode is defined asℎ𝑚 (𝜔) = 𝜆̃ 𝑚 (𝜔) ⋅ 𝜆̃ ∗𝑚 (𝜔),(9)with 𝜆̃ 𝑚 (𝜔) the Fourier transform of the line density of the𝑚-th azimuthal mode. It was suggested in [4] that, whenusing Eq. (8) to calculate 𝑍1 , 𝜔𝑚𝑝 should be evaluated with𝜔z = 0.
Usually, the ring length is much larger than the bunch
length, suggesting that the sampling frequency step 𝜔0 in
Eq. (8) is much smaller than the typical frequency of thebeam spectrum, that is, 𝑐/𝜎z0 . With this condition, the summations over 𝑝 in Eq. (8) can be fairly replaced by integrals,
yielding∞𝜔∫−∞ 𝑍∥ (𝜔) 𝜔0 ℎ𝑚 (𝜔)𝑑𝜔
𝑍∥ 𝑚
( ) =
.
∞
𝑛 eff
∫−∞ ℎ𝑚 (𝜔)𝑑𝜔(10)

For a Gaussian bunch, the power densities ℎ𝑚 (𝜔) for 𝑚mode is given by
2 2 2ℎ𝑚 (𝜔) = (𝜔𝜎z /𝑐)2𝑚 𝑒−𝜔 𝜎z /𝑐 .

(12)

Combining Eqs. (6) and (12), we arrive at Zotter’s cubic
equation for potential-well bunch lengthening:𝑥3 − 𝑥 +𝑍∥ 𝑚=1
𝑐𝐼𝑏
Im
= 0,
(
)
𝑛 eff
𝜅𝜂𝜔0 𝜎z0 𝜎2𝛿0 (𝐸/𝑒)(14)

We assume the incoherent synchrotron frequency with wakefield perturbation is given by
𝜔2z
𝑑 2𝑧
=
−
𝑧.
𝑑𝑠2
𝑐2

(15)

∞

−∞

=

𝑑𝑧𝜆(𝑧, 𝑠)𝜕𝐹(𝑧, 𝑠)𝜕𝑧

𝑖𝑐𝐼𝑛 ∞
̃ 𝑠)𝜆̃ ∗ (𝑘, 𝑠).
∫ 𝑑𝑘𝑘𝑍∥ (𝑘)𝜆(𝑘,
2𝜋 −∞

(16)

From the reality condition of the impedance and beam spectrum, it follows that only the imaginary part of 𝑍∥ (𝑘) contributes to 𝐹1 . Taking a Gaussian bunch and pure inductive
impedance 𝑍∥ (𝑘) = −𝑖𝑘𝑐𝐿, one can obtain𝐹1 =𝑐𝐼𝑛 √𝜋𝑐𝐿.
2𝜋 2𝜎3z(17)

Alternatively, for an absolute impedance, the effective inductance can be defined as follows [8]:∞̃ 𝑠)𝜆̃ ∗ (𝑘, 𝑠)
1 ∫−∞ 𝑑𝑘𝑘𝑍∥ (𝑘)𝜆(𝑘,𝐿eff =.
∞𝑐 ∫ 𝑑𝑘𝑘 2 𝜆(𝑘,̃ 𝑠)𝜆̃ ∗ (𝑘, 𝑠)

(18)

−∞

For a Gaussian bunch, it becomes
𝐿eff =

2𝑖𝜎3z ∞
̃ 𝑠)𝜆̃ ∗ (𝑘, 𝑠),∫ 𝑑𝑘𝑘𝑍∥ (𝑘)𝜆(𝑘,√𝜋𝑐 −∞(19)

which is equivalent to
𝐿eff = −𝑍∥ 𝑚=1
1
Im ( )
𝜔0
𝑛 eff

(20)

with the effective impedance calculated by Eqs. (10) and (11).
The concept of effective inductance was first introduced by
K. Bane in [8].
If we replace 𝐹(𝑧, 𝑠) with 𝐹1 𝑧 in Eq. (14) and compare
the resulting equation with Eq. (15), we can obtain(13)

where 𝑥 = 𝜎z /𝜎z0 and 𝜅 = √2𝜋. Note that from Eq. (6) toEq. (13), 𝜉𝑍1 (𝜎z ) is replaced by 𝜉𝑍1 (𝜎z0 )/𝑥 3 .
There are alternative ways of deriving the cubic equation
similar to Eq. (13) (for example, see Refs. [5–7]). Here,
we present a simple way that does not need the effective
impedance defined by Eq. (8). From Eqs. (3), we can obtain the second-order differential equation of longitudinalcoordinate as
𝜔2𝑑 2𝑧= − 2z0 𝑧 + 𝜂𝐹(𝑧, 𝑠)2𝑑𝑠𝑐

𝐹1 = ∫(11)

For the case of potential-well bunch lengthening, 𝜎z can
𝑚=1
be replaced by 𝜎z0 so that (𝑍∥ /𝑛)eff is a constant machineparameter independent of the bunch current. Note that theeffective impedance depends on the beam properties, but
the original impedance 𝑍∥ (𝜔) only depends on the beam’s
surroundings [5].
For electron storage rings, the energy spread is constant
below the microwave instability threshold, resulting in𝜎z 𝜔z = 𝜎z0 𝜔z0 = −𝑐𝜂𝜎𝛿0 .

To find the explicit form of 𝜔z , we can approximate 𝐹(𝑧, 𝑠)by taking the leading term that is linear to 𝑧. To do so, wetake the derivative over 𝑧 on the right side of Eq. (5) andthen average over the line density:

𝜔2z0 𝑐2 𝜂𝐼𝑛 𝐿eff
𝜔2z
=
−
.
𝑐2
𝑐2
4√𝜋𝜎3z(21)

Using the relation given in Eq. (12), we arrive at a cubic
equation of bunch lengthening factor 𝑥:𝑥3 − 𝑥 −𝑐𝐼𝑏
𝐿eff = 0,
𝜅𝜂𝜎z0 𝜎2𝛿0 (𝐸/𝑒)(22)with 𝜅 = 4√𝜋. It can be seen that Eqs. (13) and (22) are
identical except for the constant 𝜅 in the denominator. We
conclude that the factor of √2𝜋/(𝜔0 𝜏)3 for 𝑍1 in Eq. (5.9)
of [4] is incorrect and should be replaced by √𝜋/(2(𝜔0 𝜏)3 ).
This conclusion will be justified in the following section.
The cubic equations (13) and (22) have been widely usedin electron storage rings. During the design and construction
phases, an impedance database can be created and used to
calculate the effective impedance. This can then be used to
predict the potential-well bunch lengthening using Eq. (22).

- 118 -

Proceedings of the 20th Annual Meeting of Particle Accelerator Society of Japan
August 29 - September 1, 2023, Funabashi

PASJ2023 WEOB14

During beam commissioning phases, beam-based measurements are typically performed to extrapolate the effective
impedance or effective inductance, which is used to verify
the accuracy of the impedance database. A comprehensive
review can be found in [9]. The author of [9] suggests thatEq. (22) is a model more consistent with the prediction of the
Haissinski equation [10] and with the measurements in theelectron storage rings. However, it is important to emphasizethat this consistency is conditional on the following factors:(1) The total longitudinal impedance of the ring can be wellapproximated by a pure inductance, 𝑍∥ (𝑘) = −𝑖𝑘𝑐𝐿, with
the impact of its real part negligible. (2) The potential well,which is distorted by wakefields, remains well quadratic sothat the lengthened bunch is close to Gaussian. These factors
allow us to replace 𝜂𝐹 in Eq. (14) with 𝜂𝐹1 𝑧, where 𝐹1 isgiven by Eq. (17). This leads to the cubic equation (22).
Typically, important sources of inductive impedance include
small 3D discontinuities, such as obstacles or cavities withlongitudinal dimensions comparable to or smaller than therms bunch length (𝜎z0 ), on beam pipes; small angle transitions, such as collimators in colliders or tapers sandwichingsmall-gap insertion devices in light sources, in flat chambers.
Small 3D discontinuities from the flanges and bellows arecommon in all storage rings.
The cubic equation (22) may not apply well in some cases.
One example is damping rings, where small-angle transitions
are usually unnecessary. A good example is the SLC damping ring with the improved chamber [11], where the resistive
part dominates the total impedance [12]. Another example
is future circular colliders (FCCs), where the resistive-wall
impedance is dominant because of their large circumferences.

A SELF-CONSISTENT EQUATION FOR
BUNCH LENGTHENING
For electron storage rings, the circulating beam can bemodeled as a continuous distribution 𝜓(𝑧, 𝛿, 𝑠), with its evolution governed by the Vlasov-Fokker-Planck (VFP) equation [13]. Specifically, when considering the synchrotron
motion, the VFP equation is [14]the so-called Haissinski equation [10]2𝑑𝜆0 (𝑧)𝑧
1+[ 2 −𝐹0 (𝑧)] 𝜆0 (𝑧) = 0.
𝑑𝑧
𝜎z0 𝜂𝜎2𝛿0(24)Here 𝐹0 (𝑧, 𝑠) is from Eq. (4) with 𝜆(𝑧, 𝑠) replaced by theequilibrium distribution 𝜆0 (𝑧). The solution of Eq. (24) is

z0

,

(25)

with the new scaling parameter 𝐼 = 𝐼𝑛 𝜎z0 /(𝜂𝜎2𝛿0 ) and thewake potential of the bunch𝒲∥ (𝑧) = ∫∞−∞𝑊∥ (𝑧 − 𝑧′ )𝜆0 (𝑧′ )𝑑𝑧′ ,(26)With the synchrotron tune defined by 𝜈z0 = 𝜔z0 /𝜔0 , there
is 𝐼 = −𝑁𝑒2 /(2𝜋𝜈z0 𝐸𝜎𝛿0 ) [12, 15]. The stability of theHassinski equation is beyond the scope of this paper, andthe reader is referred to [16] and references therein.
Integrating over 𝑧 on both sides of Eq. (24) and recognizing that the center of mass of the bunch is𝑧𝑐 = ∫∞−∞𝑧𝜆0 (𝑧)𝑑𝑧,(27)

we obtain 𝑧𝑐 = 𝐼𝜎z0 𝜅∥ , with the well-known loss factor∞

𝑑𝑧𝜆0 (𝑧)𝒲∥ (𝑧).

(28)

𝑐 ∞
∫ Re[𝑍∥ (𝑘)]ℎ(𝑘)𝑑𝑘
𝜋 0

(29)

𝜅∥ (𝐼) = ∫−∞In terms of impedance, it is𝜅∥ =with the spectral power density ℎ(𝑘) = 𝜆̃ 0 (𝑘)𝜆̃ ∗0 (𝑘).
Given the bunch profile 𝜆0 (𝑧), by definition, the rms
bunch length is calculated by
∞

∞

−∞

−∞

𝜎2z = ∫ (𝑧 − 𝑧𝑐 )2 𝜆0 (𝑧)𝑑𝑧 = ∫𝑧2 𝜆0 (𝑧)𝑑𝑧 − 𝑧𝑐2 . (30)

We show how to calculate 𝜎z from Eq. (24). We can obtain
three terms by multiplying 𝑧 on both sides of this equationand performing integration over 𝑧. The first term is a constant22-1 (consider that 𝜆0 (𝑧) decays exponentially as 𝑒−𝑧 /(2𝜎z0)
when 𝑧 → ±∞, according to the Haissinski equation). The
second term equals 𝜎2z + 𝑧𝑐2 . The third term is an integrationthat contains the wake function. Combining the three terms,
we can arrive at an equation that describes the potential-wellbunch lengthening𝜕𝜓 𝑑𝑧 𝜕𝜓 𝑑𝛿 𝜕𝜓
2 𝜕
𝜕𝜓
+
+
=
[𝛿𝜓 + 𝜎2𝛿0
] , (23)𝜕𝑠𝑐𝑡𝑑 𝜕𝛿
𝜕𝛿
𝑑𝑠 𝜕𝑧
𝑑𝑠 𝜕𝛿
with 𝑡𝑑 the longitudinal damping time. The equations of motion are given by Eqs. (3), with the line density distribution∞
𝜆(𝑧, 𝑠) = ∫−∞ 𝜓(𝑧, 𝛿, 𝑠)𝑑𝛿.
The VFP equation has an 𝑠-independent stationary solution 𝜓0 (𝑧, 𝛿) in the form of 𝜓0 (𝑧, 𝛿) = 𝜓̂ 0 (𝛿)𝜆0 (𝑧) below
the microwave instability threshold. The momentum distribution 𝜓̂ 0 (𝛿) is Gaussian with rms spread 𝜎𝛿0 , and thespatial distribution satisfies∞− 𝑧 2 − 𝜎𝐼 ∫𝑧 𝑑𝑧′ 𝒲∥ (𝑧′ )𝜆0 (𝑧) = 𝐴𝑒 2𝜎z0

𝑥2 − 1 −𝑐𝐼
𝑍 eff (𝑥) = 0,
2𝜋𝜎z0 ∥

(31)

where 𝑍∥eff is formulated by𝑍∥eff =2𝜋 ∞∫ 𝑑𝑧(𝑧 − 𝑧𝑐 )𝜆0 (𝑧)𝒲∥ (𝑧).
𝑐 −∞

(32)

In terms of impedance, it is equivalent to∞𝑍∥eff = − ∫−∞𝑑𝑑𝑘𝑍∥ (𝑘)𝜆̃ 0 (𝑘) [𝑖 𝜆̃ ∗0 (𝑘) + 𝑧𝑐 𝜆̃ ∗0 (𝑘)] . (33)𝑑𝑘Here, we define 𝑍∥eff as an effective impedance, which isalways real but not complex, to indicate bunch lengthening.
Equation (31) shows that the term 𝑍∥eff is a quadratic function of 𝑥, where 𝑥 is a function of the normalized current 𝐼- 119 -

Proceedings of the 20th Annual Meeting of Particle Accelerator Society of Japan
August 29 - September 1, 2023, Funabashi

PASJ2023 WEOB14𝑧𝑐 =𝐼𝜎z0 𝑐 ∞2 2∫ 𝑑𝑘Re[𝑍∥ (𝑘)]𝑒−𝑘 𝜎z ,0𝜋∞𝑍∥eff = −2𝜎2z ∫ 𝑑𝑘𝑘Im[𝑍∥ (𝑘)]𝑒−𝑘 𝜎z .
2

2

0

(34)
(35)

The above quantities can be formulated explicitly for somewell-defined impedances in the literature. For more details,
the reader is referred to [17], where it is shown that 𝑍∥eff (𝑥)has different scaling laws, depending on the properties of
specific impedances. In particular, when the ring impedanceis approximated by 𝑍∥ (𝑘) = −𝑖𝑘𝑐𝐿eff , we immediately obtainthe cubic equation (22) from Eqs. (31) and (35). This justifies
Eq. (22) and leads us to conclude that the original Zotter’sequation (13) should be recast with 𝜅 = 4√𝜋, as suggested
in [9]. Meanwhile, the conventional effective impedance
and inductance are connected to the newly defined effective
impedance 𝑍∥eff by
𝐿eff = −𝑍∥ 𝑚=1 2𝜎𝑧0 eff
1
Im ( )
=
𝑍 ,
𝜔0
𝑛 eff
√𝜋𝑐 ∥

(36)

and these quantities should be evaluated at zero bunch current.

BUNCH LENGTHENING IN SUPERKEKBAs a specific example, we consider the high-energy ring(HER) of SuperKEKB (the case of LER was discussed

3000

Pseudo-Green wake function without CSR

2500

W||(z) [V/pC]

2000
1500
1000
500
0
-500
-1000
-0.1

-0.08

-0.06

30

-0.04
z [m]

-0.02

0

Real, without CSR
Imaginary, without CSR

25
20
15
Z||(k) [kΩ](i.e., 𝑥 = 𝑥(𝐼)). Since Eq. (31) is derived from the Haissinski equation without any approximations and the Haissinski
equation is the exact stationary solution of the VFP equation,
we conclude that it is a self-consistent equation for the potential well lengthening applicable to electron storage rings.
We emphasize that Eq. (31) is not valid at beam currents
above the microwave instability threshold, where the beamenergy spread will increase.
Equation (32) shows that the effective impedance 𝑍∥eff is ameasure of the average stretching force over the whole bunch.
From the impedance viewpoint, Eq. (33) states that both thereal and imaginary parts of the impedance contribute to thebunch lengthening when the density distribution is deformed,
although the imaginary part is usually the dominant source.
At 𝐼 = 0, the density distribution is Gaussian with 𝑧𝑐 = 0
and 𝑥(0) = 1. The bunch lengthening rate at 𝐼 = 0 is given𝑐by 𝑥0′ ≡ 𝑑𝑥/𝑑𝐼|𝐼=0 = 4𝜋𝜎𝑍 eff (1). Here, the effective
z0 ∥eff
impedance 𝑍∥ (1) is given by Eq. (33) with 𝑧𝑐 = 0 and2 2
𝜆̃ 0 (𝑘) = 𝑒−𝑘 𝜎z0/2 . It only depends on the imaginary part of𝑍∥ (𝑘), suggesting that the inductive part of the impedancesolely determines the lengthening rate of the bunch at zerocurrent. The relation between 𝑍∥eff and the normalized current 𝐼 is complicated. Here, we only give the slope at 𝐼 = 0
2𝜋𝜎
as 𝑑𝑍∥eff /𝑑𝐼|𝐼→0 = 𝑐 z0 [𝑥0′2 + 𝑥 ″ ] with 𝑥 ″ = 𝑑 2 𝑥/𝑑𝐼 2 to
be determined.
Further calculations can be carried out when a Gaussian
distribution with rms length 𝜎z and center of mass 𝑧𝑐 isused to approximate the Haissinski distribution Eq. (25).
From Eqs. (28) and (33), the center of mass and effective
impedance are explicitly written as follows:

10
5
0
-5

-10
-15
-20

0

1000

2000

3000
k [m-1]

4000

5000

6000

Figure 1: Pseudo-Green’s function wake and corresponding
impedance without CSR for SuperKEKB HER.

in [17]). Longitudinal wakes for various components havebeen calculated using a Gaussian driving bunch with length𝜎̂ 𝑧 =0.5 mm and summed to create the pseudo-Green function wake [18] as shown in Fig. 1. The corresponding
impedance shown in the same figure is calculated by Fouriertransforming the wake data (indeed, the chirp-Z transform
is used to improve the frequency revolution). The decayof impedance data at high frequencies is due to the chosen 0.5mm Gaussian bunch, which acts as a Gaussian win2 2
dow function of 𝑒−𝑘 𝜎̂ 𝑧 /2 . This is justified when the ratio of the nominal length of the bunch 𝜎z0 to the lengthof the driving bunch 𝜎̂ 𝑧 is very large (for our example,
10). The bunch should not be strongly deformed or microbunched, so the high-frequency impedances are not sampled. The beam parameters used to solve the Haissinski
equation are beam energy 𝐸=7.00729 GeV, ring circumference 𝐶=3016.315 m, slip factor 𝜂=4.543 × 10−4 , bunch
length at zero current 𝜎z0 =5.05 mm, momentum spread𝜎𝛿0 =6.3 × 10−4 , and synchrotron tune 𝜈z0 =-0.0272. The
scaling parameter at 𝑁 = 1011 is 𝐼 = 0.0212 pC/V. The
numerically obtained Haissinski solutions with bunch populations 𝑁 = (0.0471, 4.71, 9.42, 14.1)×1010 (corresponding
to 𝐼=(0.0001, 0.01, 0.02, 0.03) pC/V) are shown in Fig. 2.
The lengthening of the bunch and the centroid shift as a function of the normalized current are shown in Fig. 3. Using thebunch length data from Fig. 3, the effective impedance 𝑍∥eff iscalculated by Eq. (31), as shown in Fig. 4. In this figure, we
also plot the 𝑥𝑍∥eff data, which corresponds to Zotter’s cubic
equation. According to the cubic equation, the inductance
impedance dominates the total impedance, and 𝑥𝑍∥eff should- 120 -

Proceedings of the 20th Annual Meeting of Particle Accelerator Society of Japan
August 29 - September 1, 2023, Funabashi

PASJ2023 WEOB14

0.4

real part of the total impedance is negligible. The Haissinski equation, a self-consistent solution of the VFP equation below the microwave instability threshold, has beenused to derive a new equation to describe potential-wellbunch lengthening. The effective impedance used to describe bunch lengthening has simply been redefined. This
new equation is useful for comparing impedance calculations with beam-based measurements in electron storagerings.

8

N=4.71×10
N=4.71×1010
N=9.42×1010
N=1.41×1011

0.35
0.3

λ(q)

0.25
0.2
0.15
0.1
0.05
0
-10

-8

-6

-4

-2

0
2
q=z/σz0

4

6

8

10

Figure 2: Haissinski solution for SuperKEKB HER with
𝑁 = (0.0471, 4.71, 9.42, 14.1) × 1010 .
1.8

zc/σz0σz/σz0

1.6
1.4
1.2

ACKNOWLEDGEMENTS
The author D.Z. thanks many colleagues, including 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, M. Tobiyama, andL. Zhang for inspiring discussions on various aspects ofimpedance issues in electron storage rings.

1

REFERENCES

0.8
0.6

[1] A. Wolski, Beam dynamics in high energy particle accelerators, Imperial College Press, 2014.

0.4
0.2
0

0

0.005

0.01

0.015
I [pC/V]

0.02

0.025

[2] A. Chao, Lectures on accelerator physics, World Scientific,
2020.

0.03

Figure 3: Bunch centroid and rms bunch length as a functionof normalized current for SuperKEKB HER.
be a constant or have a weak dependence on the bunch current. Figure 4 shows that this is not the case for SuperKEKB
HER. Therefore, we conclude that Zotter’s equation does
not apply well to this ring. However, the same analysis forSuperKEKB LER showed that 𝑥𝑍∥eff is fairly constant [17],suggesting that a pure inductance is good enough to describe
the lengthening of the bundle in LER.

SUMMARY[6] K.-Y. Ng, Physics of intensity dependent beam instabilities,
World Scientific, 2006.
[7] A. Chao, Special Topics in Accelerator Physics, World Scientific, 2022.
[8] K. Bane, SLAC-PUB-5177 (1990).
[10] J. Haissinski, Neural Comput. 18B (1) (1973).
[11] K. Bane and C.-K. Ng, Impedance calculations for the improved SLC damping rings, in Proceedings of the 15th Particle
Accelerator Conference (1993).

[13] T. D. Frank, Linear and nonlinear fokker-planck equations,
Synergetics, 149 (2020).

Z||eff [Ω]

4000
3800

[14] J. M. Jowett, in AIP Conference Proceedings, Vol. 153 (1987).

3600[15] K. Oide and K. Yokoya, KEK-Preprint-90-10 (1990).

3400[16] Y. Cai, Physical Review Special Topics-Accelerators andBeams 14, 061002 (2011).

3200
3000

[5] A. Chao, Physics of collective beam instabilities in high energy accelerators, Wiley series in beam physics and accelerator technology (1993).

[12] R. Warnock and K. Bane, Phys. Rev. Accel. Beams 21, 124401(2018).

Quadratic equation
Cubic equation

4200[4] B. Zotter, CERN SPS/81-14 (1981).

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

Zotter’s equation is valid under the following conditions:
The longitudinal total impedance of the ring can be wellapproximated by a pure inductance, and the impact of the4400[3] SAD homepage, https://acc-physics.kek.jp/SAD/

0

0.005

0.01

0.015
I [pC/V]

0.02

0.025

0.03

Figure 4: Effective impedance as a function of normalized current calculated from simulated bunch lengths forSuperKEKB HER.

[17] D. Zhou, T. Ishibashi, and G. Mitsuka, arXiv:2307.01286(2023).
[18] T. Ishibashi et al., Impedance modelling and single-bunchcollective instability simulations for the SuperKEKB mainrings, to be published in Journal of Instrumentation.

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