Preamble

Analysis of the beam orbit motion in the HEPS storage ring Xiyang Huang, Haisheng Xu, Fang Yan, Zhe Duan, Zihao Wang, Gang Xu, Na Wang, Yuemei Peng, Xiaohao Cui, Nan Li, Yaliang Zhao, Xiaohan Lu, Saike Tian, Hongfei Ji, Cai Meng, Yuanyuan Guo, Jiuqing Wang, Yi jiao, 1, 2, Yuanyuan Wei, and Daheng Ji

1 Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China 2 University of the Chinese Academy of Sciences, Beijing 100049, China

The High Energy Photon Source (HEPS) is a high-performance and high-energy synchrotron radiation light source with a beam energy of 6 GeV and an ultra-low emittance of 34 . In order to enable users to fully harness the potential of this high-brightness beam, it is essential to maintain the stability of the beam orbit with precision to a fraction of the beam size, which corresponds to a sub-micron level of orbit stability. The primary sources of orbit motion in accelerators are ambient ground vibrations and electrical noise from the power supply of the magnets. The extremely strong focusing required to achieve low beam emittance will amplify the impact of such noise sources on the beam orbit compared to existing accelerators. Hence, predicting the expected beam orbit motion becomes crucial for validating the design approaches of the components. In this paper, we will describe the calculation of the anticipated beam orbit motion in HEPS, taking into account the effects of the measured frequency-dependent ground motion coherence, structural resonances of magnet supports and power supply noise. By comparing with the measured power spectral density (PSD) of actual beam motion, we verify the model and calculation accuracy and check for extra noise sources affecting the light source. Finally, by predicting the beam orbit with fast orbit feedback (FOFB) using the measured hardware response, we confirm it meets the orbit stability requirements.

Keywords

light source, orbit motion, ground vibration, electrical noise

INTRODUCTION

The High Energy Photon Source (HEPS) is a fourth- generation synchrotron facility located in Beijing, China[ Operating at 6 GeV, HEPS features an ultralow- emittance storage ring designed with 48 modified hybrid 7 bend achromats (7BA), achieving a natural emittance of 34.8 . This configuration enables HEPS to produce re- markably high synchrotron radiation brightness, peaking at in the hard X-ray regime. The stability of the electron beam orbit is one of the most important performance criteria at synchrotron light sources, as they support ultrafast and time-resolved experi- ments across diverse beamlines. For example, the hard X- ray nanoprobe (HXN) is a high-resolution imaging tool de- signed for nanoscale material characterization, offering nm spatial resolution and 1nm motion precision at the sam- ple, which refers to tens nrad angular stability[ ]. The soft

inelastic X-ray scattering (SIX) studies electronic excitation 18

with ultrahigh energy resolution, which is determined by the gratings and exit slit with a critical vertical aperture. During the measurement, efforts focus on improving noise sources to

minimize disturbances and maintain sub- µ m stability at the 22

exit slit[

Variations in beam position can lead to synchronization is- 24

sues, compromising experimental accuracy. To maintain or- bit stability, the root mean square (rms) position and angular Supported by the High Energy Photon Source project and the National Natural Science Foundation of China (No. 12205315).

motion of the electron beam must be minimized to less than 27

of the beam size and its divergence in both the horizontal and vertical planes. Specifically, for HEPS, in the frequency range of 0.1 Hz to 1 kHz, the allowed orbit fluctuation toler- ances are set at for the horizontal plane and 0.3 the vertical plane at the light source point in low-beta straight section [ ]. In this context, developing predictive models for orbit motion is essential. Such models will ensure that the design and operational protocols of the storage ring are adequately aligned with user needs, thereby facilitating the high-quality experimental outcomes that HEPS aims to de- liver. As research and methodologies evolve, the capabilities

of HEPS are poised to significantly enhance scientific explo- 39

ration across a range of fields. Within the frequency range of our concern, the electron

beam motion is primarily influenced by two significant fac- 42

tors: the movement of focusing elements as a result of ground vibrations, and the fluctuation in the guiding field of the mag- nets due to electrical noise from power supplies[ ]. To

minimize the relative motion among the individual magnets, 46

they are typically mounted on rigid girders. This strategy effectively reduces random relative motion between nearby magnets. However, it is essential to acknowledge that gird- ers are not entirely rigid structures, they have resonant modes due to deformation[ ]. This means that the analysis of beam motion must incorporate these resonant modes. Then the motion of the magnets driven by ground vibrations can be decomposed into two distinct components: the motion of the girders driven by ground vibration, and the motion of the magnets on the girders that corresponds to the resonant modes of the girders. In the HEPS storage ring, the beam orbit will be affect by the noise from quadrupoles and correc- tors, which are all equipped with individual power supplies.

The magnetic field is directly proportional to the current, any

fluctuations in the current can lead to magnetic field varia- tions. Such instabilities can cause beam particles to deviate from their designated paths, resulting in unwanted orbit dis- tortions. This issue is particularly critical in regions of strong focusing, where even minor changes in the magnetic field can

produce magnified effects on the beam trajectory. In the case 66

of HEPS, the gradients of quadrupoles can reach up to 80 T/m, indicating that contributions from power supply noise to orbit perturbations are substantial and cannot be ignored.

The transmission pathways through which these two sources of noise affect the beam orbit are clearly illustrated in Fig. , highlighting the interplay between ground vibrations and power supply fluctuations on beam stability. Understanding these pathways is essential for developing effective strategies to mitigate their impact and enhance the orbit stability at the HEPS.

Furthermore, it is essential to consider several other poten- tial sources that may affect beam stability[ ]. Be- yond the previously mentioned influences, additional sources

can impact the beam orbit stability. One significant source 80

arises from the vibrations of the conductive vacuum chamber within the quadrupoles[ ]. The vibrations of this chamber can effectively mirror the motion of the quadrupole itself due to the presence of eddy currents induced within the vacuum chamber. These eddy currents seek to stabilize the magnetic field in the moving frame of the chamber, which can result in additional complexities for orbit stability. Such vibrations can originate from several factors, including the flow of cool- ing water around the vacuum chamber. Although the this type of vibration has not yet been quantified, it is anticipated that any potential issue can be readily mitigated and effectively damped, achieved by inserting shims between the poles of magnets and vacuum chambers.

The structure of this paper is as follows: The forthcom- ing section will introduce the general methodology employed

in analyzing orbit motion. Sec. III initiates the examina- 96

tion of vibration-induced orbit motion by investigating the coherence measurement of floor motion. This parameter is essential for understanding how the amplification factor de- pends on the frequency of ground vibrations. Following the coherence analysis, the focus shifts to the characterization of beam orbit motion, particularly as it relates to the reso- nances in girders. Shifting focus to electrical motion, the dis- cussion begins with the presentation of findings from power supply noise measurements in Sec. . Utilizing these data, along with the associated amplification factors, we estimate the beam orbit motion induced by power supply noise from focusing magnets and correctors. With an assessment of the beam motion originating from both vibration and electrical sources in hand, we synthesize these results to formulate a predictive spectrum of beam motion. Through previous cal- culations, the total orbit motion can be obtained in Sec.

Sec. VI , we present the initial beam commissioning results of 113

the HEPS storage ring and the actual power spectral density (PSD) spectrum of beam orbit fluctuations. By comparing these with the computational results, we aim to identify exter- nal noise sources that have not been previously considered in the model. The final section is a summary and includes some discussion of the results. This structured approach guarantees a comprehensive and systematic evaluation of the factors in- fluencing beam motion. The insights derived from this study lay a robust foundation for future design improvements and experimental strategies aimed at reducing the adverse effects of these disturbances.

METHODOLOGY IN THE ORBIT MOTION ANALYSIS The PSD of a stationary random signal is defined ����� �����

S x ( f ) = lim T →∞ 1 T

) exp

= � ∞

where ω = 2 πf is the angular frequency, R x ( τ ) = 129

is the auto-correlation function of . With known in frequency region , measured or pro- vided in the data sheet, can be generated by cos [2

x ( t ) =

where a k = �

2 S

ϕ k is a random phase angle distributed uniformly over the 135

range . As defined in Eq. ( ), the PSD of a stochas- tic process is a statistical measure that characterizes its fre-

quency content. Consequently, generating x ( t ) over a finite 138

time interval from a given PSD is not unique due to the ran- 139

dom nature of the process. Thus, the simulation should be repeated multiple times using different realizations to ensure reliable conclusions.

One of the main properties of the PSD is that its integral gives the motion variance:

σ 2 x = � ∞

When integrating over a finite frequency range, the result rep- 146

resents the motion variance within that frequency band. If the motion arises from multiple independent factors, the PSD of the overall motion is the sum of the PSDs of each individual contribution, provided they are uncorrelated. This approach decomposes the complex motion into its constituent parts, en- abling a detailed analysis of the electron beam’s behavior and characteristics across the frequency range of interest.

The orbit motion due to a specific noise source can be an- alytically calculated using the PSD of the source and the am- plification or attenuation factor along the propagation path, as illustrated in Fig. . Additionally, orbit feedback is imple- mented to correct fluctuations by adjusting the beam position based on real-time measurements. Finally, the total orbit mo- tion resulting from various noise effects can be expressed in

coherence Ground vibration Girder resonant vibration resonant Electrical noise Attenuated by Vacuum chamber the form:

σ 2 tot = �

where denotes th noise source, are the amplification factor and correction factor for each source. Additionally, based on the integration time and spatial resolution require- ments of the beamline experiments, the frequency range we are concerned with is from 0.1 Hz to 1 kHz.

ORBIT MOTION INDUCED BY GROUND VIBRATION Ground vibration

Ground motion poses a significant challenge for particle 171

accelerators, including beam oscillations that degrade beam quality in synchrotron facilities and cause beam-beam off- sets in colliders[ ]. At HEPS, ground vibrations have been measured for six years, staring before construction.

The measurements were conducted using CMG seismome- ters equipped with GPS, which recorded the wave velocity of ground vibration wave. The seismometers had a timing preci- sion of s and a sampling rate of 500 Hz. Fig. displays the PSD of ground vibrations measured in the storage ring tunnel in 2023. At frequencies that are exceptionally low, specifically below 1 Hz, primary contributors include atmo- spheric phenomena, the movement of ocean tides, and fluc- tuations in temperature, among others. Above 1 Hz, human activities, such as machinery operation and vehicular traffic, dominate.

According to Eq. ( ), the measurement in the storage tun- nel gives approximately 23 nm rms ground vibration noise in the frequency range of 1-100 Hz, and the induced frequency- independent orbit motion are 1.4 m in the horizontal plane and 0.8 m in the vertical plane, which compares favorably with the orbit stability requirements.

However, when the range is extended to 0.1-100 Hz, the induced orbit motion exceeds 6 m in both planes. Even with perfect orbit feed- back, the beam orbit cannot be controlled at a level of 0.3 in the vertical plane, let alone the fact that our BPMs are also vibrating with the floor at the same amplitude.

Amplified by Attenuated by Feedback Orbit motion lattice Fortunately, the long wavelengths of low-frequency ground vibrations cause nearby accelerator components, such as magnets and vacuum chambers, to move in phase, effectively

reducing relative displacements. This synchronization miti- 202

gates the impact on orbit motion, as the wavelength at these frequencies may span the entire storage ring, resulting in neg- ligible net motion. Understanding the coherence of ground motion is crucial for analyzing low-frequency beam orbit mo- tion. Notably, diffusive ground movements lack spatial co- herence, rendering the concept of coherence length irrelevant

and having minimal impact on frequencies above 0.1 Hz. 209

Girder motion The characterization of ground vibration coherence con- stitutes a fundamental procedure in the site assessment for

accelerator facilities[ 24 , 26 , 30 ]. The magnitude-squared co- 213

herence between two signals is mathematically defined as

C xy ( f ) = | P xy ( f ) | 2

where represent PSD of individual signals and denotes their cross-spectral density[ ]. Prior to con-

ducting ground vibration coherence measurements within the tunnel, it was imperative to validate the consistency of data acquisition between two identical devices operating simulta- neously at the same location, ensuring a high degree of co- herence. Seismometric measurements were performed across three orthogonal directions, with each measurement compris- ing 900 seconds of data sampled at 1024 Hz. Fig. illustrates the coherence spectrum of horizontal and vertical ground mo- tion captured by two adjacent Guralp 6TD seismometers. It is observable that the two sensors exhibit excellent coherence in the frequency range of 1.7-80 Hz. The data demonstrate exceptional coherence within the frequency range of 1.7-80 Hz. While theoretical predictions suggest sustained coher- ence at frequencies below 1.7 Hz, empirical observations re- veal a notable decline in coherence within this regime. The observed incoherence at frequencies range of DC-1.7Hz can-

not be readily attributed to any physical mechanism other than 235

diffusion. However, the influence of diffusion is negligible in

this context, as its effect diminishes with decreasing separa- 237

tion between measurement points. Consequently, this reduc- tion in coherence is primarily ascribed to measurement noise.

Following the validation of device coherence within the specified frequency band, systematic ground vibration coher- ence measurements were conducted. The experimental pro-

tocol involved maintaining one seismometer at a fixed posi- 243

tion while progressively relocating the second device along the storage ring tunnel, with a maximum separation of 110 meters constrained by cable length. The results are demon- strated in Fig. , where the horizontal and vertical axes repre- sent frequency and the distance between the two seismome- ters, respectively, and the color indicates the level of coher- ence. The measurement results reveal that the vertical align- ment shows high coherence at frequencies below 1 Hz, even at distances up to 110 meters, as indicated by a coherence level above 0.9, highlighted by the yellow shading in the right figure. In contrast, the horizontal plane measurements reveal

a diminishing coherence with increasing distances. The rela- 255

tionship between coherence length and frequency was estab- lished through analysis of the dataset presented in Fig. identifying the frequency threshold at which coherence ex- ceeds 0.9 for each separation distance, we derived empirical models for both planes:

11 f 0 . 66 , for f > 1 Hz

L x =

11 f 2 . 1 , for f ≤ 1 Hz (6) 261

20 f 0 . 73 , for f > 1 Hz

L y =

20 f 2 . 6 , for f ≤ 1 Hz (7) 263

The model parameters indicate coherence lengths of 11 me- ters and 20 meters at 1 Hz for the horizontal and vertical planes, respectively. Notably, the coherence length exhibits

significant frequency dependence below 1 Hz, suggesting that 267

low-frequency vibrations are unlikely to be amplified by the lattice structure..

The relationship between ground vibration coherence and the interplay of distance and frequency has been system- atically established, providing a foundation for determin- ing frequency-dependent orbit amplification factors. These factors quantitatively characterize the amplification of beam orbit motion in response to ground vibrations across vary- ing frequencies. The orbit amplification factors are derived through static closed-orbit simulations utilizing the Accelera- tor Toolbox (AT)[ ]. The comprehensive calculation process is outlined as follows:

1. A two-dimensional grid is constructed to encompass the

entire storage ring. Each grid point is assigned random displacements that follow a Gaussian distribution with a predefined amplitude, ensuring a statistically representa- tive model of ground vibrations.

2. A low-pass filter is implemented to smooth the random

displacements in both transverse planes. The cutoff fre- quency of the filter is carefully selected to correspond to

the coherence length being modeled, thereby maintaining 289

the physical relevance of the displacement patterns.

3. The amplitude of filtered displacements is adjusted to

match the initial predefined amplitude. Subsequently, the 292

average displacement is subtracted to eliminate any po- tential bias, ensuring the displacement field remains zero- mean.

4. The resulting grid displacements are mapped onto the cor-

responding magnets. Linear regression is then applied to smooth the displacements of magnets sharing the same girder. It is important to note that while this smoothing process is not strictly necessary, however, its omission may lead to excessive deviations in certain results, potentially compromising the accuracy of the amplification factor cal- culation.

5. The closed-orbit distortion is calculated, and the orbit am-

plification factor is determined through systematic analysis of the resulting beam motion.

Next, we will provide a detailed explanation of the numeri- cal simulation. As indicated in the step , the simulation pro-

cess initiates with the generation of a two-dimensional grid, 309

where points are uniformly spaced at intervals of 0.5 me- 310

ters to encompass the entire ground area of the storage ring.

Each grid point is assigned two random displacement val- ues in horizontal and vertical planes, denoted as To reduce the inherent randomness of the displacements, a

two-dimensional Gaussian smoothing technique is employed. 315

This process is governed by a transfer function that maintains a constant value of 1 for frequencies below a lower thresh- , decreases linearly to 0 for frequencies above an upper threshold , and exhibits a linear transition between these cedure yields ground displacements characterized by coher- ence lengths of 20 meters and 60 meters, as illustrated in Fig. . From the figure, it is evident that the ground displacement

profiles exhibit significant variations depending on the filter 324

length applied. Specifically, a larger filter length results in a smoother distribution of ground displacements, highlighting the impact of the coherence length on the spatial characteris- tics of the displacement field.

The determination of magnet displacements is achieved through the extraction of ground displacement data at the entry and exit points of the magnetic components. This ex- traction employs bilinear interpolation, a computational tech-

nique that estimates values at target locations based on the 333

four nearest points within the square grid. Subsequently, a linear regression analysis is performed across all elements

of a single girder, aligning them to an optimal linear tra- 336

jectory. The resulting displacements are then meticulously adjusted to ensure accuracy. provides illustrative examples of the derived magnet displacements for coherence lengths of 1 meter, 60 meters, and 600 me- ters. The results demonstrate a clear trend: as the coherence length increases, the variation in magnet displacements be- comes progressively smoother. Notably, when the coherence

length (or filter length) reaches a sufficiently large magnitude, 344

the entire accelerator system behaves as a unified entity, ef- 345

fectively eliminating relative displacements between its con- stituent components.

The procedure outlined above was employed to generate 200 datasets of ground displacements, each associated with a distinct filter length.

For every dataset, the closed orbit was computed, and the standard deviation of the orbit at the positions of insertion devices was evaluated relative to the ground position. These standard deviations were subse- quently averaged across the 200 datasets to obtain the mean value, also denoted as . The amplification factors were then calculated by normalizing with respect to the stan- dard deviation of the ground motion:

A c ( L i ) = ⟨ σ i ⟩ β 0 σ g

The results of this analysis are presented in the left panel of Fig. . For shorter coherence lengths, the simulated ampli- fication factors exhibit excellent agreement with the analyti-

) with coherence length of 20 m (left) and 60 m (right), the black dots indicate the storage ring elements. with different coherence length. cally derived values, which are based on the assumption that all displacements are statistically independent. This align- ment underscores the validity of the simulation approach and provides confidence in the robustness of the results for scenar- ios characterized by smaller coherence lengths. By utilizing the relationship between the coherence length and frequency as described in Eqs. ( ) and ( ), we can derive the amplifi- cation factors at different frequencies , the results are plotted in the right panel of Fig.

Subsequently, the PSD of non-resonant orbit motion the source point, induced by ground vibrations, can be calcu- lated using Eq. ( and illustrated in Fig. . The non-resonant orbit motion obtained by integrating over the frequency range from 0.1 Hz to 1000 Hz, as defined by Eq. ( ), yielding values of 0.56 and 0.20 . It should also be noted that the ground vi- bration PSD in the 100–1000 Hz range depicted in the figure is extrapolated from the PSD below 100 Hz, with the ampli- fication factor for this frequency range being identical to that at 100 Hz.

At the conclusion of this section, it is imperative to ad- dress the uncertainties associated with the measurement and calculation of coherence length. As previously discussed, a coherence threshold of 0.9 was selected, which yields a no- tably shorter calculated coherence length compared to scenar- ios where a lower threshold is applied (e.g., a threshold of 0.8 would result in exponential fits yielding coherence lengths of approximately 100 meters in both planes). In practical terms, adopting a lower coherence threshold would shift the curve in the right panel of Fig. to the right, thereby reducing the amplification factor below 10 Hz and resulting in a smaller calculated orbit motion. However, the influence of this co- herence threshold-induced uncertainty on the calculations be- comes negligible once orbit feedback is implemented, as the feedback system is particularly effective in mitigating low- frequency disturbances.

C. Vibration on girder 399

As described in the preceding section, ground motion in- duces solid-body-like displacements of the girders. In addi-

tion to these rigid-body motions, such vibrations can also ex- cite resonant deformation modes of the girders. In the case

of uniform girder motion, the effects of the focusing and de- 404

focusing quadrupoles tend to partially cancel each other out due to their opposing influences[ ]. However, it is possi- ble for a deformation mode to exist where the displacements of the focusing and defocusing quadrupoles on the girder act

in a cumulative manner, thereby exerting a more significant 409

impact on the beam orbit motion. Therefore, it is crucial to consider the influence of these girder deformation modes in the analysis.

The HEPS will incorporate focusing magnets mounted on six girders, in addition to the longitudinal gradient dipoles

spanning the plinths. All girders are supported by concrete 415

plinths, and in our design, the vibration at the top of the plinths is assumed to be identical to that on the floor. Fig. illustrates the panoramic layout of the magnet arrangement within one cell, which has a total length of approximately 22 meters. The beam direction runs from right to left, with the girders arranged as follows: DQ-I, MP-I, FODO-I, FODO- II, MP-II, and DQ-II. The girders at both ends of the cell are of the DQ type, each housing one focusing quadrupole (QF), one defocusing quadrupole (QD), and an independent fast corrector. The two central modules, which contain bend- ing magnets with defocusing gradient (BD), are referred to

as FODO. The remaining two modules, which include sex- 427

tupoles, octupoles, and anti-bending magnets with focusing gradient (ABF), are designated as MP. The vibration mode

analysis of was conducted utilizing the ANSYS Mechani- 430

cal software, specifically the Release 19.1 edition[ order to accurately predict the modal response of the mod- ules, dynamic stiffness testing was completed on the support mode of the FD-type girder in the horizontal plane.

During accelerator operation, the primary cause of the girder’s vibration is the ground motion. For frequencies ap- proaching resonance, the amplitude can be character- ized by the classical resonance curve:

z g ( f ) = R ( f ) z d , (10) 440

where is the drive amplitude, and

R ( f ) = f 2 0 �

where is the resonant frequency, is the quality factor

of the girder, related to the damping ratio ζ : Q = 1 / 2 ζ . 444

The damping ratio is determined by striking the girder with

a hammer and subsequently monitoring the resulting vibra- 446

tions. Table shows the first two deformation modes and the measurement quality factors of three girder types.

To calculate the orbit motion due to girder deformation, for quality factor , the full-width half-maximum of the reso-

FD-II, MP-II, DQ-II. nance curve equals to . A key feature, which will prove

significant later, is that at low frequencies where f is far from 453

, Eq. ( ) approaches 1. This indicates the absence of am-

plification, meaning the girder simply follows the ground mo- 455

tion. In our analysis, we consider the orbit motion due to one single mode in all girder assemblies. The orbit displacement at the source point (center of the straight section) due to a

2 N

σ 2 m =

odd,I odd,II even,I

where N = 48 is the number of period, and λ 2 m accounts for 482

the contribution of m -type girder in both odd and even units. 483

The normalized amplification factor th mode can be single girder assembly displacement in a resonant mode

z m = κ m,i d cos( ϕ − πν z ) , (12) 460

where denotes either represent the normalized amplification factor and girder displacement for th mode, respectively; distinguishes between the four girder types( type I/II of odd cells and type I/II of even cells), is the phase advance between the girder and the source point. Since the girder motion induced by a resonant mode is oscillatory, one can also use rms values for in Eq. ) . The motion of every girder assembly in the same mode is independent because the coherence length of ground mo- tion at frequencies above 65 Hz is less than 0.69 m, which is smaller than the distant between two individual girders, as was shown in Eq. ( ). Consequently, the rms displacements generated by each girder can be combined in quadrature. Fur- thermore, because girder motion is driven by ground move- ment and the frequency spectrum of ground motion is largely consistent across all points around the ring, the amplitude of

girder motion is approximately uniform for all girders at a 477

given frequency. The total rms motion resulting from one mode of the same girder type around the whole ring can be calculated as derived using Eq. (

Λ m = σ m √ β 0 d rms = √

even,II odd,I odd,II even,I even,II

all resonant modes below 150 Hz. for each mode. To calculate the contribution of a single mode to orbit mo- tion, we begin by multiplying the PSD of ground motion by the square of the resonance curve derived from Eq. ( ). This resulting PSD is then further multiplied by the square of the mode’s amplification factor, yielding the PSD of the orbit mo- tion attributed to this specific mode. The final step involves integrating this orbit motion PSD to obtain the rms orbit mo- tion for the mode:

G r ( f ) = �

Notably, the integration limits must be set close to the res- onance frequency to avoid artificially inflating the PSD at lower frequencies. As previously noted, Eq. ( ) approaches 1 when . Thus, summing the effects of resonant modes over a broad frequency range could erroneously am- plify the low-frequency PSD to times that of the ground motion PSD. To address this, the integration limits were care- fully chosen as , which encompass approxi- mately of the total integral under the squared resonance curve for a given quality factor . The integrated rms orbit motion for each mode is plotted in Fig.

Total orbit motion induced by ground vibration Based on the contents of Sec.

III B III C , Fig. presents the expected rms orbit motion induced by ground vibration, encompassing both non-resonant and resonant vi- brations (as shown in the zoomed-in subplot). Using Eq. ( the orbit motion induced by all resonant modes are 0.08 and 0.05 , while the total motion induced by ground vibration can be calculated as Eq. (

σ vib = �

The total motion is 0.57 and 0.21 in the two planes respectively, as shown in Fig.

ORBIT MOTION INDUCED BY ELECTRICAL NOISE

In addition to vibrations, another significant noise source 525

contributing to orbit disturbances is the noise from the mag- net power supplies. Given that there are over 2000 electro- magnets in the HEPS storage ring, the impact of electrical

noise could be more substantial than initially anticipated. All 529

storage ring magnets, with the exception of sextupoles and octupoles, are equipped with independent power supplies,

which fall into three categories: unipolar, slow bipolar, and 532

fast bipolar power supplies[ 35 , 36 ]. Unipolar power supplies 533

are used to power quadrupoles, BD, and ABF. Slow and fast bipolar power supplies are used to power slow correctors (SC) and fast correctors (FC), respectively.

The rms current ripple of a power supply, denoted as , can be calculated by integrating the PSD of power supply noise, , over a specified frequency range:

∆ I 2 rms = � f 2

where typically represents the maximum noise level in parts per million (ppm) relative to the maximum output current:

n = ∆ I rms I max . (18) 543

Measurements of the power supply current ripple were per- formed using a CoCo-80X oscilloscope operating in Dynamic Signal Analysis (DSA) mode. The schematic of the FC noise measurement setup is presented in Fig. , and the measured noise spectra of all types of power supplies provided as illus- trative examples, see in Fig. 15A [FIGURE:15]/0.65V DS2000IDSA 500:1 I/V Resistor CoCo80X FC current ripple measurement using DSA mode with CoCo-80X oscilloscope.

The measurement results revealed numerous additional

spectral lines, predominantly identifiable as harmonics of 50 551

Hz. These lines are suspected to be artifacts originating from the measurement process itself, rather than genuine compo- nents of the power supply output. Furthermore, the impact of electrical noise on the beam orbit is attenuated by the vacuum chamber. Based on the experimental results presented in Fig. , this attenuation effect for quadrupoles and slow correctors can be approximately characterized by the fitting curve given in Eq. ( for FC for other magnets As indicated by the measurement results in Fig. , negli- gible attenuation is observed for fast correctors at frequencies below 1 kHz. These correctors are featuring with laminated core and are positioned within the Inconel vacuum chamber of 0.5 mm thickness. Therefore, when calculating the orbit motion induced by power supply noise in fast correctors, the suppressive effect of the vacuum chamber will be neglected.

To account for the effect of different vacuum chambers, we define the reduced power supply noise as follows:

˜ n = ∆˜ I rms I max =

Next, we will directly use the Eq. ( ) to calculate the ampli- fication factor.

Inconel vacuum chambers, respectively. Amplification factors of electrical noise Similar in form to the ground vibration noise, the lattice- related normalized amplification factor for the reduced power supply noise can be expressed as

A ps = ⟨ z ⟩ √ β 0 ˜ n, (21) 577

where the representing either , denotes the orbit dis- tortion at source point generated by a kick angle

z = √ ββ 0 2 sin πν z cos ( ϕ − πν z ) θ. (22) 580

Based on this definition of Eq. ( 18 ), for quadrupoles, θ rms 581

can be written as

θ rms = ˜ nKLd rms I max I K , (23) 583

where are the nominal strength and correspond- ing operational current of the quadrupole, is the effective length, is the rms displacement of the quadrupole. For correctors, however:

θ rms = ˜ nθ max . (24) 588

) remains valid when the power supply noise is expressed as a rms value.

As previously mentioned, quadrupoles and correctors in HEPS each have independent power supplies. For a given family represents the total effect of all magnets in that family, assuming each indi- vidual magnet generates an rms orbit kick of . By cat- egorizing the magnets into distinct families, we can derive Eq. ( ) as follows: 4 sin

σ 2 j, rms =

= β 0 4 sin 2 πν z Φ j θ 2 j, rms ,

Normalized amplification factors of different magnet types.

where Φ j = � β l cos 2 ( ϕ l − πν z ) is computed across all N j 598

Combining Eqs. ( 21 - 25 ), the amplification factor of all 600

quadrupoles and correctors on the orbit can be expressed as the sum over all quadrupole and corrector families: 4 sin , for corrector

A 2 j =

4 sin , for quadrupole In addition to regular quadrupoles, the BD and ABF mag- nets are designed as quadrupoles and installed with a offset.

In the horizontal plane, the orbit displacement is dominated by a design offset of several millimeters—specifically, dur- ing operation, the offset for BD is set to 14 mm, while that for ABF is set to 2.5 mm. In the vertical plane, however, the orbit displacement remains characterized by the rms orbit er- ror, similar to regular quadrupoles. Table summarizes the normalized amplification factors for different magnet types.

Orbit motion induced by electrical noise As discussed earlier, the orbit motion induced by elec- trical noise for each magnet type can be calculated by multi- plying the measured power supply noise PSD by the attenua- tion curve described in Eq. ( ) and then by the correspond- ing amplification factors listed in Table , as shown in Eq.

G ps = �

where represents different magnet families. One should note that the fast correctors are subject to different attenua- tion effect. The PSD of total orbit motion induced by all the power supply noise is shown in Fig.

The integrated orbit motion, calculated using Eq. (

z ps = �

and the results are summarized in Table . It is evident that in the horizontal plane, the dominant contributions originates from the transverse gradient dipoles and SC, whereas in the vertical plane, SC and FC have relatively greater influence.

Estimated orbit motion due to power supply noises of different magnet type, the quadrupole type include the regular quadrupoles and transverse gradient dipoles.

Magnet All calculations based on electrical noise were performed using an idealized model of the storage ring lattice that incor- porates the cross-talk effect [ ]. However, in practical sce- narios,coupling between horizontal and vertical orbital mo- tions is unavoidable. In HEPS, the coupling is set to quadrupoles and horizontal gradient dipoles to vertical orbit motion. This increment is sufficiently small to be negligible.

The total orbit motions induced by power supply noise are and 0.24 , respectively, as illstrated in Fig.

The overall PSD of the orbit motion can be obtained by summing the individual PSD attributed to ground vibrations and power supply noise, based on Eqs. ( ) and (

G tot ( f ) = G c ( f ) + G r ( f ) + G ps ( f ) . (29) 646

The total integrated rms orbit motion is calculated as:

σ tot =

). It can be observed that power supply noise dominates across all frequency range in the horizontal plane. In the vertical plane, ground vibration is dominant below 100 Hz, while power sup- ply noise prevails above 100 Hz. The total rms orbit motion within the frequency range from 0.1 Hz to 1 kHz is expected to be 0.94 in horizontal and 0.32 in vertical planes.

Notably, the anticipated overall rms motion in the vertical plane exceeds the orbit stability criteria in the absence of a feedback system.

VALIDATION AND DISCUSSION OF SIMULATION RESULTS AND ACTUAL BEAM MOTION In this section, we will verify the validity of our analyti- cal approach by comparing the simulation results calculated above with the actual beam motion in HEPS.

A. Commissioning of the HEPS storage ring and optics 664

correction

Commissioning of the HEPS storage ring started from 666

July 23, 2024, to August 6, the first beam storage was

achieved[ 38 ]. Initially, the electron beam was transmitted 668

through the high-energy transfer line (BR) and injected into the storage ring using self-developed software, achieving single-turn transport. A semi-automatic correction program

based on PYAPAS was employed to tuning the beam, over- 672

coming challenges like small apertures[ ]. During multi-

turn commissioning, RF cavities and sextupole magnets were 674

gradually activated, with parameters optimized to increase the number of revolutions. Despite issues including cumulative errors and strong nonlinearities, beam storage was accom- plished on August 6, reaching a current of 60 mA. This mile- stone represented a crucial advancement in the HEPS project.

Following the completion of beam accumulation, more

detailed beam commissioning work was conducted, suc- 681

cessively accomplishing closed-orbit correction, beam-based alignment (BBA) and optics correction, among other tasks.

Detailed measurements of beam parameters were conducted, and local vacuum anomalies were repaired. The first syn- chrotron radiation light was obtained on September 23. By the end of December 2024, the beam intensity had reached 30 mA, the lifetime was 1000 s, the closed-orbit deviation

was < 100 µm , beta-beating was < 5% , chromatic dispersion 689

deviation was < 5 mm, coupling was < 10% , and emittance 690

was < 100 pm. 691

Theoretical calculation and comparison with actual beam oscillation Since the hardware for our FOFB system has not yet been fully constructed, we can currently only analyze short-term beam orbit motion using turn-by-turn (TBT) data comprising 1 million turns. age orbit at the source points of all low-beta straight sections (recorded on January 7, 2025) and our simulation results. It can be observed that across most of the 0.1 Hz to 1 kHz fre- quency range, the calculated results are in good agreement with the actual beam orbit motion—particularly within the 70–86 Hz frequency band corresponding to support structure resonance. Below, we discuss three frequency bands where the agreement is less satisfactory.

First is the 50 Hz peak: in both horizontal and verti- cal planes, the measured peak amplitude exceeds the cal- culated result. By inverting the response matrix, we deter- mined that the 50 Hz signal source is distributed across the entire ring, with a particularly strong contribution near the R42 cell—indicating an additional noise source in this region.

Further precise measurements identified a water pump for the RF cavities near the R42 cell, which exhibits a vibration fre- quency of 49.6 Hz, as illustrated in Fig. . Using a simpli- fied vibration attenuation model, we evaluated its impact on the beam, finding it contributes an orbit fluctuation of approx- imately 0.2 rms around 50 Hz.

The second frequency band with notable discrepancy is around 300 Hz. This relatively broad peak arises from the synchrotron frequency, which couples longitudinal motion to transverse motion via residual dispersion[ ]. To verify this hypothesis, we conducted a beam experiment: adjusting the

total cavity voltage and monitoring the corresponding PSD 724

peak, results are shown in Fig. . Currently, three RF cavi-

ties are in use in the storage ring. Initially, with a total cavity 726

voltage of 3.84 MV, the peak appeared at 360 Hz (theoretical calculations predicted a longitudinal oscillation frequency of 380 Hz under this setting). Next, when we reduced the volt- age of one cavity by 0.1 MV while keeping the other two con- stant, the peak split into two and shifted to lower frequencies.

Conversely, reducing the voltage of all three cavities by 0.1 MV simultaneously caused the entire peak to shift to 320 Hz.

These results confirm that the peak around 300 Hz is closely linked to longitudinal motion. Fortunately, it requires little attention at present, as its contribution to transverse motion is less than in both planes.

Finally, discrepancies are observed in the low-frequency range: notable differences between calculated and actual beam motion appear in the 1–5 Hz band. Given that the floor vibration data used to calculate beam motion was measured earlier, we relocated some probes on the storage ring to mon- itor vibrations during light source operation, aiming to iden-

tify any significant low-frequency vibrations. As shown in 744

larger vibrations around 2.5 Hz compared to other locations.

Based on the maximum ground vibration amplitude, the PSD of the beam at 2.5 Hz can be estimated as:

S b ≈ β 0 ∗ A ( f ) 2 ∗ S g ( f ) | f =2 . 5 = 0 . 3 µm 2 /Hz. (31) 749

This is consistent with the highest frequency observed in the beam PSD. However, since there is a large irrigation canal near the R42 cell in this direction, in addition to the pump room, we have not yet been able to identify the source of the 2.5 Hz peak. Although low-frequency noise currently has a

significant impact on beam motion, it is anticipated that once 755

our FOFB system is implemented[ ], its impact will be less pronounced.

(right). in horizontal plane with pump on and less than 0.1 with pump off.

Prediction of orbit motion with FOFB Since the actual beam fluctuations exceed our calculations, we need to estimate the beam orbit after the FOFB system is implemented. The FOFB system of HEPS will include 192 FCs in each plane and is planed to operate at 22 kHz. The characteristic of FOFB system can be investigated by the sen- sitive equation[

where s = Ω+ jω is the complex frequency, G PI ( s ) 766

are the transfer functions of PID compensation and correction system in domain, is the total delay of FOFB system. can be obtained by measuring the amplitude-frequency response of the correction system, and is simply expressed as a first-order plus time-delay transfer

function[

G FC ( s ) = 33100 s + 33770 e − 0 . 00002 s . (33) 773

When we select a simple integral controller,

G PI ( s ) = 7000 /s, (34) 775

with Eqs. ( ) and ( ), the disturbance rejection gain at each frequency point can be calculated from Eq.

noted that G 0 ( f ) is in unit of dB, then the attenuation ratio 778

is derived as:

F ( f ) = 10 G 0 ( f ) / 20 . (35) 780

The orbit motion with FOFB control can be expressed as Eq. (

G f ( f ) = � f 2

it is reduced to 0.36 and 0.26 in both planes, respec- tively, shown in Fig. and Fig.

SUMMARY AND DISCUSSION This study provides a comprehensive analysis of electron beam orbit motion in the HEPS storage ring, focusing on the two primary sources of noises: ground vibrations and power supply noise. Through a combination of advanced computa- tional modeling, experimental measurements, and theoretical analysis, we have quantified the contributions of these factors to beam motion and validated the predictive model against experimental data.

Ground vibrations have been identified as a critical contrib- utor to beam orbit motion, affecting the beam through both

non-resonant and resonant mechanisms. Non-resonant mo- 797

tion, derived from frequency-dependent coherence measure- ments, results in rms displacements of 0.56 in the hor- izontal plane and 0.20 in the vertical plane within the 0.1 Hz–1 kHz range. Resonant deformations of magnet sup- port girders, characterized using ANSYS simulations and dy- namic testing, further contribute 0.08 and 0.05 to the horizontal and vertical motions, respectively. This leads to total vibration induced rms values of 0.57 and 0.21

Electrical noise from magnet power supplies also signifi- 806

cantly impacts orbit stability, with contributions varying by magnet type. Transverse gradient dipoles and SC dominate horizontal orbit motion, while SC and FC dominate the ver- tical orbit, resulting in total electrical noise induced rms dis- placements of 0.75 and 0.24 in both planes. When

combining both vibration and electrical noise sources, the to- 812

tal anticipated orbit motion without feedback reaches 0.94 and 0.32 The accuracy of the predictive model has been confirmed through comparisons between simulated and measured beam motion PSD spectra, particularly in the 70–86 Hz band corre- sponding to girder resonances. However, residual discrepan- cies observed at specific frequencies: the 50 Hz line exceeds the power supply noise contribution by a small margin, a fea- ture near 360 Hz arises from longitudinal–transverse coupling driven by synchrotron oscillations, and the pronounced low- frequency beam motion observed throughout the ring, espe- cially around 2.5 Hz, is likely excited by vibrations near the R42 cell. These observations underscore the need for targeted suppression of these additional noise sources. Finally, after the inclusion of FOFB, the predicted beam motion is 0.36 and 0.26 in both planes, well within the orbit stabil- ity specification for the HEPS storage ring.

This study demonstrates the feasibility of achieving the stringent orbit stability requirements for the HEPS storage ring and highlights the importance of integrating precise mea-

surement techniques with sophisticated simulation method- 833

ologies. The findings provide a robust framework for under- standing and mitigating beam motion in high-performance electron beam systems. Furthermore, the methodology de- veloped in this work can be extended to model and calculate orbit motion in future large scale circular accelerators, such as the 100 kilometer long Circular Electron-Positron Collider

(CEPC)[ 48 ]. This approach will enable the clear definition 840

of required parameter thresholds and their impacts on beam motion, advancing the field of accelerator physics and sup- porting the development of next-generation light sources and particle colliders.

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