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Local impedance measurements from closed orbit distortion due to wake fields at CESR
Nuclear Inst. and Methods in Physics Research, A 927 (2019) 250–256
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Local impedance measurements from closed orbit distortion due to wake fields at CESR S.T. Wang ∗, J.D. Perrin, S. Poprocki, W. Hartung 1 , D.L. Rubin Cornell Laboratory for Accelerator-based Sciences and Education, Cornell University, Ithaca, NY 14853, United States
ARTICLE Keywords: Storage rings Wake fields Beam dynamics
INFO
ABSTRACT Short range transverse wake fields are excited by charged particles passing through a vacuum chamber. The wake fields can act back on a bunch of particles and alter its trajectory. We have measured the distortion of the closed orbit due to transverse wake fields in the Cornell Electron–positron Storage Ring. A transverse monopole wake field is induced by an asymmetric impedance associated with the insertion of a single vertical scraper. A transverse dipole wake field is generated by displacing the bunch from the symmetry axis of the chamber. The orbit is offset vertically through the gap of a pair of symmetrically inserted scrapers and also in a narrow-gap undulator chamber. A fit to the distorted orbits yields the average kick of the transverse monopole wake of the asymmetric scrapers and the kick factor of the symmetric scrapers; the scraper results are in reasonable agreement with the values from numerical simulation of the wake fields. The kick factor extracted from the closed orbit distortion of the dipole wake induced by the narrow-gap undulator chamber is significantly greater than that predicted by the numerical simulation. Possible sources of the discrepancy are discussed.
1. Introduction Wake fields can distort the bunch shape, induce instability, and limit beam intensity, thus compromising the performance of particle accelerators. The strength of the wake fields increases linearly with beam current, making them an important consideration for the design of high-current colliders and light sources. In a synchrotron light source, transverse wake fields become an important source of beam emittance dilution at high currents. Insertion devices with narrow vertical gaps, and high impedance, are often used to generate high-brightness X-ray beams [1–3]. The wake fields in these devices can dilute the emittance and limit the total charge in a bunch or train of bunches. The interaction between moving charged particles and the induced wake fields in a vacuum chamber are characterized by the wake functions, which are determined by the geometry and electromagnetic properties of the chamber. Analytic formulas to calculate the impedance (Fourier transform of the wake function) are limited to vacuum components with simple geometries, such as a round pipe [4], small pillbox cavity [5], tapered collimator [6], and a small-gap undulator chamber [7]. For complex geometries, numerical simulations such as GdfidL [8] or T3P [9] are typically used. Beam-based impedance measurement techniques are used in various store rings, including measurements of current-dependent bunch lengthening for the longitudinal impedance and current-dependent ∗ 1
betatron-tune shift for the transverse impedance. These techniques characterize the global impedance of a storage ring. To measure the local impedance of a vacuum component, one approach is to measure the global impedance before and after installation of a vacuum chamber, as the difference will be due to the impedance of the new chamber [10]. However, this method requires two separate measurements at two different times, so that different machine conditions and instrumentation errors may complicate the analysis. Making using of the fact that the transverse wake field can change both the betatron phase advance and the closed orbit, two additional methods have been developed to characterize the local distribution of impedances: measuring the current-dependent phase advance [11–14] and the closed orbit distortion [15–19]. The Cornell Electron–positron Storage Ring (CESR), built on the Cornell University campus, stores counter-rotating beams of electrons and positrons. Positrons circulate in the clockwise direction and electrons in the counter-clockwise direction as shown in Fig. 1. An undulator chamber with a vertical gap of ∼5 mm and length of ∼3.5 m was installed in CESR to accommodate two undulators for X-ray users. Because of its narrow gap, the undulator chamber produces a significant transverse wake which can affect the beam. Current-dependent betatron tune shifts measured before and after the installation of the vacuum chamber give an estimate of the transverse impedance of the chamber [10].
Corresponding author. E-mail address: [email protected] (S.T. Wang). Present address: Facility for Rare Isotope Beams, Michigan State University, East Lansing, MI 48824, United States.
https://doi.org/10.1016/j.nima.2019.02.060 Received 26 November 2018; Received in revised form 4 February 2019; Accepted 20 February 2019 Available online 23 February 2019 0168-9002/© 2019 Elsevier B.V. All rights reserved.
S.T. Wang, J.D. Perrin, S. Poprocki et al.
Nuclear Inst. and Methods in Physics Research, A 927 (2019) 250–256
The average kick 𝛥𝑦′ on the bunch causes the distortion of the closed orbit of the bunch as Eq. (2) described. The average kick is defined in Eq. (1) for an average wake field 𝑊⟂ . For the asymmetric scraper configuration, 𝑊⟂ is dominated by the transverse monopole wake and 𝑊⟂ is ∞
𝑊⟂ =
∫−∞ 𝑓 (𝑧)𝑊⟂ (𝑧)𝑑𝑧 ∞
∫−∞ 𝑓 (𝑧)𝑑𝑧
,
(3)
where 𝑓 (𝑧) is the distribution function of the bunch particles in the longitudinal direction. Typically the longitudinal distribution is approximately a Gaussian. For the cases with the beam displaced between the symmetrically inserted scrapers or at the undulator chamber, the transverse monopole wake is zero because of the top-down symmetry and the transverse dipole wake term dominates. The average wake field is then (4)
𝑊⟂ = 𝑘⟂ 𝑦,
where 𝑦 is the displacement of the beam from the chamber center and 𝑘⟂ is the kick factor, which is defined by the following equation in the limit where the vertical beam size is much less than the displacement 𝑦:
Fig. 1. CESR layout showing the locations of undulators and scrapers.
A pair of adjustable vertical scrapers with a minimum gap of 7 mm were installed in CESR in order to intercept the injected particles with large vertical oscillations. Studies have been done to explore the effects of both the undulator chamber and the scrapers [20]. It was found that the vertical beam size increases significantly due to wake fields generated by an asymmetric scraper configuration or by displacing the beam off the chamber center at the undulator or the scrapers. In this paper, we extend our investigation to the closed orbit distortion caused by the transverse monopole wake from an asymmetric scraper configuration and the dipole wake induced by displacing the beam at the undulator chamber or the scrapers. Using the same analysis method described in [18,19], we obtain the average vertical kick of the asymmetric scraper configuration and the kick factors from the scrapers and the undulator chamber. The values from the scrapers agree reasonably well with model values extracted from numerical simulations using T3P. On the other hand, there is a large discrepancy between the kick factor of the narrow-gap undulator chamber measured from the closed orbit distortion and the numerical simulation. Possible explanations are discussed.
∞
𝑘⟂ =
∫−∞ 𝑓 (𝑧)𝑊𝑑 (𝑧)𝑑𝑧 ∞
∫−∞ 𝑓 (𝑧)𝑑𝑧
,
(5)
where 𝑊𝑑 (𝑧) is the dipole wake function. As discussed in Section 3, we used numerical simulations to compute the wake functions for the asymmetric and symmetric scraper configurations and for the undulator chamber. The average kick of the transverse monopole wake from an asymmetric scraper configuration and the kick factors of the symmetric scrapers and the undulator chambers are calculated from those wakes. The CESR beam position monitors (BPM) have the capability of measuring the turn-by-turn (TBT) and bunch-by-bunch beam positions with a resolution of 10 μm [21]. That capability has been an important and useful tool for low emittance tuning [22] and studying collective effects such as fast ion instability [23]. In CESR, there are 100 BPMs. In each measurement, 8192 turns of beam positions of a single electron bunch were recorded at these BPMs. The closed orbit was then obtained by averaging the 8192 positions at each BPM. For high bunch currents (> 1 mA), different gain settings were applied to the BPM electronics. Therefore, there was a current-dependent effect on the orbits due to instrumentation error. In Section 4, we will discuss the methods to obtain the orbit distortion after eliminating this systematic effect. Once the orbit distortion 𝛥𝑦 was obtained, we did a fit of 𝛥𝑦 as a function of 𝑠 using Eq. (2) and find the average kick of the monopole wake or the kick factor for the dipole wake. This method has been successfully applied to measure local impedance of narrow-gap undulator in APS [16], ELETTRA [17], NSLS-II [19], and DLS [24].
2. Methods When a bunch travels through a vacuum chamber, each particle experiences a vertical kick 𝛥𝑦′ due to the transverse wake field induced by the bunch: 𝑒𝑞 𝛥𝑦′ = 𝑊⟂ , (1) 𝐸 where 𝑊⟂ is the transverse wake potential, 𝑞 is the total bunch charge, 𝑒 is the electron charge, and 𝐸 is the beam energy. The average kick acting on the centroid of the bunch is similar to a dipole error and distorts the closed orbit. Suppose the bunch gets an average vertical kick 𝛥𝑦′ at location 𝑠0 ; the change in the closed orbit 𝛥𝑦(𝑠) at location 𝑠 will be √ 𝛥𝑦′ 𝛽𝑦 (𝑠)𝛽𝑦0 𝛥𝑦(𝑠) = cos[|𝛥𝜙𝑦 (𝑠)| − 𝜋𝑄𝑦 ], (2) 2 sin 𝜋𝑄𝑦
3. Numerical simulations The ACE3P electromagnetic simulation suite was used to calculate the wake fields [9]. The three-dimensional models of the scrapers and the undulator chamber were constructed using the finite element mesh toolkit CUBIT [25]. The CUBIT models are shown in Fig. 2. The CUBIT models were imported to the time domain wake field solver T3P [9] to calculate the longitudinal wake. We ran T3P at the National Energy Research Scientific Computing Center [26]. The charged bunch was modeled as a Gaussian distribution with 𝜎𝑧 = 10 mm, as calculated from the lattice model at the experimentally measured synchrotron tune. Longitudinal wakes were computed with the drive bunch at each of three vertical positions (−𝛥𝑦, 0, 𝛥𝑦) where 𝛥𝑦 = 0.5 mm is the mesh size of the CUBIT model. T3P calculate the longitudinal wake of the witness particle at a specific vertical position. A Taylor expansion of the longitudinal wake in terms of the vertical displacement of the drive bunch and witness particle, and then application of the Panofsky– Wenzel theorem [27], gives the transverse wake. The wake terms are
where 𝑠 is the location of the bunch in the accelerator, 𝛥𝜙𝑦 (𝑠) is the vertical phase advance from 𝑠0 to 𝑠, 𝑄𝑦 is the vertical tune, and 𝛽𝑦 (𝑠) and 𝛽𝑦0 are the vertical betatron functions at 𝑠 and 𝑠0 , respectively. In general, 𝑊⟂ (𝑧) is a function of 𝑧, the particle’s longitudinal position relative to the bunch centroid. Hence, the wake field can produce a yz coupling in the beam. The contribution to the yz tilt 𝜃𝑦𝑧 manifests in an increase in the vertical beam size [20]. Since each particle in the bunch experiences a different kick from the wake, the bunch as a whole will receive an average kick from the wake fields. 251
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Fig. 2. CUBIT half model of the scrapers including artificial tapers (a) with only top scraper inserted and (c) with both scrapers inserted. (e) CUBIT inverse model of the undulator chamber. Transverse wakes of (b) the asymmetric scrapers, (d) the symmetric scrapers, and (f) the undulator chamber: monopole (red), dipole (green), and quadrupolar (blue) term. In (b), the dipole and quadrupolar terms have been multiplied by a factor of 𝛥𝑦 = 0.5 mm to compare with the transverse monopole term. Dashed line: bunch distribution (𝜎𝑧 = 10 mm, arbitrary scale). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) Table 1 CESR machine parameters.
shown in Fig. 2. It is worth noting here that we used T3P to calculate the short-range wake field in order to predict the effect of the wake on the particles in the bunch; thus our predictions do not include long range effects from wakes that endure for multiple turns. As a note, we have successfully used the same transverse monopole wake from the asymmetric scraper configuration in the tracking simulation to explain the increased vertical beam size behavior due to 𝑦𝑧 coupling in our previous paper [20]. Fig. 2a shows the half model of the scrapers with top scraper inserted and bottom scraper retracted. The model includes artificial tapers which match the 7 mm gap of the closed scrapers. The artificial tapers are used in order to satisfy the constraints of the Weiland integration method used by T3P. Wake fields for a separate model with just the artificial tapers were also obtained, and subtracted from the wake fields from the scraper+tapers model to obtain the wake fields for the scraper alone. As Fig. 2b shows clearly, the transverse monopole wake dominates the dipole and quadrupolar wakes and it depends nonlinearly on particle longitudinal position (𝑧). Using Eq. (3), we find that the average monopole wake amplitude from the asymmetric scrapers (top scraper inserted) 𝑊⟂ is 0.80 V/pC. Fig. 2c shows the CUBIT half model of the scrapers, including artificial tapers, with both scrapers fully inserted. The calculated dipole and quadrupolar wake fields from this model are shown in Fig. 2d. Because of the up/down symmetry, the transverse monopole wake is zero. We created a similar CUBIT model with both scrapers retracted and obtained the corresponding dipole and quadrupolar wake terms, which are much smaller than those from the model with both scrapers inserted. In order to compare with the kick factor extracted from the data, we used the dipole wake field with both scrapers inserted after subtracting the wake fields from the model with both scrapers retracted to calculate the kick factor due to inserting the scrapers: 220.1 V/(pC⋅m). Fig. 2e displays the CUBIT full model of the undulator chamber with tapers. An inverse model was used to satisfy constraints of the Weiland integration method used in T3P. In this model, the chamber
Beam energy (GeV)
𝐸
2.085
Circumference (m) Transverse damping time (ms) Momentum compaction Nominal RF voltage (MV) Synchrotron tune Horizontal tune Vertical tune Horizontal emittance (nm·rad) Vertical emittance (pm·rad)
𝐿 𝜏 𝛼𝑝 𝑉𝑅𝐹 𝑄𝑠 𝑄𝑥 𝑄𝑦 𝜖𝑎 𝜖𝑏
768 56.6 0.0068 5.8 0.067 14.573 9.629 ∼3 10–20
gap (∼5 mm) is at the end, with tapers to a regular-height beam chamber in the middle. Because the chamber has up/down symmetry, the transverse monopole wake is again zero. The dipole wake is shown in Fig. 2f; the calculated kick factor from the dipole wake is 255.3 V/(pC⋅m). 4. Experiments All the measurements were done with a single bunch of electrons stored at 2.1 GeV in CESR. The nominal parameters of the lattice are listed in Table 1. A tuning procedure which corrects the lattice betatron functions, 𝑥𝑦 coupling, and vertical dispersion was initially applied to achieve low vertical emittance (𝜖𝑦 ≈ 15 pm⋅rad) in the accelerator [22]. Two vacuum components, the scrapers and the undulator chamber, were investigated and information about the impedance was extracted. Two scraper configurations (asymmetric and symmetric) were studied. 4.1. Asymmetric scrapers The vertical scrapers were located in the north region of the ring as shown in Fig. 1. The gap between top and bottom scrapers is 7 mm when both scrapers are fully inserted and 50 mm with both retracted. 252
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Fig. 4. The measured orbit bump at 150 cu at 4 mA with scrapers retracted. Dashed green line: scraper location.
Fig. 3. Measured orbit difference 𝛥𝑦 at (a) 1.7 mA, (b) 2.7 mA, and (c) 3.7 mA due to insertion of the top scraper. Blue circles: measured data; solid red lines: fit; dashed green line: scraper location.
Before measurements, the beam was carefully centered between the two fully-inserted scrapers. With only the top scraper inserted, the wake generated by the asymmetric scraper configuration is mainly the vertical monopole wake. The bunch receives a centroid kick from the average wake field 𝑊⟂ when passing through the scrapers. TBT beam positions at 100 BPMs were recorded for 8192 turns at three different bunch currents: ∼1.7, ∼2.7, and ∼3.7 mA. Since there is a current-dependent tune shift, in order to avoid a resonance effect through the current range, the ring fractional tunes were carefully set at 𝑄𝑥 = 0.573, 𝑄𝑦 = 0.629, 𝑄𝑠 = 0.067 at low current (∼1 mA). Analysis of these TBT BPM data confirms the current-dependent tune shift and provides the closed orbit at the different currents. The BPM data were collected with both scrapers retracted or with only the top scraper fully inserted. The closed orbit distortion 𝛥𝑦 due to the wake from the asymmetric scraper configuration (top scraper inserted) was obtained by subtracting the closed orbit without scrapers inserted (𝑦𝑜𝑜 ) from the closed orbit with top scraper fully inserted (𝑦𝑖𝑜 ) at the same currents: 𝛥𝑦 = 𝑦𝑖𝑜 − 𝑦𝑜𝑜 .
Fig. 5. Measured orbit difference 𝛥𝑦 when the beam was displaced from the center by (a) −1.04 mm, (b) −1.89 mm, and (c) 2.42 mm at the scrapers, with the scrapers fully inserted. Blue circles: measured data; solid red lines: fit; dashed green line: scraper location.
(6) unchanged. As the measured bump at 150 computer units (cu) in Fig. 4 shows, the bump was well closed outside the scraper region with an RMS of 0.022 mm, which is about 1.8% of the bump peak value. In addition, five sextupoles at the bump location were turned off to eliminate nonlinear coupling from the sextupoles. The origin (𝑦 = 0, the gap center) is inferred from the bump settings at which beam loss was observed (via a decrease in the stored beam lifetime). The TBT BPM data were taken for 4096 turns with a single electron bunch at 4 mA at various bump settings with both scrapers fully inserted. Similar measurements were also taken with both scrapers retracted. The closed orbit distortion 𝛥𝑦 due to the dipole wake was then obtained by subtracting the closed orbit without scrapers inserted (𝑦𝑜𝑜 ) from the closed orbit with both scrapers fully inserted (𝑦𝑖𝑖 ) at different 𝑦 displacements:
The 𝛥𝑦 values obtained at ∼1.7, ∼2.7, and ∼3.7 mA are shown in Fig. 3. We can see that the orbit distortion increases as the current increases due to the linear dependence of the wake on beam current. With the design Twiss parameters of the lattice, we can fit the measured data using Eqs. (1) and (2). The average wake field 𝑊⟂ is the only fitting parameter. The best fits are shown as red lines in Fig. 3. The 𝑊⟂ from the best fits are found to be 0.86, 0.76, and 0.75 V/pC when beam current is ∼1.7, ∼2.7, and ∼3.7 mA, respectively. As we can see, the average 𝑊⟂ (0.79 ± 0.06 V/pC) agrees very well with the value of 0.80 V/pC extracted from the numerical simulation in Section 3. 4.2. Symmetric scrapers With both scrapers fully inserted, the transverse monopole wake is zero because of the up/down symmetry. If the beam is displaced from the gap center, the beam will get a kick from the dipole wake (Eq. (4)) which distorts the closed orbit. Three vertical correctors around the scrapers were selected to create a vertical displacement bump at the scraper location. This bump only displaces the vertical position of the beam at the scraper location and keeps 𝑦 positions everywhere else
𝛥𝑦 = 𝑦𝑖𝑖 − 𝑦𝑜𝑜 .
(7)
The measured closed orbit distortion due to the dipole wake at three bump settings with three displacements of 𝑦 = −1.04, −1.89, and 2.42 mm are displayed in Fig. 5. As Fig. 5a and b show, the larger the displacement the larger the distortion of the closed orbit. When the 253
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Nuclear Inst. and Methods in Physics Research, A 927 (2019) 250–256
Table 2 Bump measurements at the undulator chamber. Figure
𝑑 (mm)
𝛥𝐼 (mA)
𝑘⟂ (V/(pC⋅m))
a b c d
1.36 1.36 −0.72 −1.00
1 3 3 3
1347.7 935.4 981.4 841.7
(2-1) (4-1) (4-1) (4-1)
displacement reverses sign, so does the sign of the distortion, as seen in Fig. 5b and c. Using the design Twiss parameters of the lattice and Eqs. (2) and (4), we did a fit of the measured orbit distortion and found the kick factor 𝑘⟂ of the symmetric scrapers. Here 𝑘⟂ is the only fitting parameter. From the best fits, the kick factors found from the three data sets in Fig. 5a, b, and c are 211.2, 228.7, and 246.2 V/(pC⋅m), respectively. As we can see, the average value [228.7 ± 17.5 V/(pC⋅m)] agrees reasonably well with the value of 220.1 V/(pC⋅m) from numerical simulation in Section 3. The scrapers are two stainless-steel cylinders with a diameter of 50 mm. Estimate of the kick factor from the resistive wall impedance of the scrapers is about 9.1 V/(pC⋅m) [6,10], which improves the agreement between the measurement and the calculation.
Fig. 6. Measured orbit difference 𝛥𝑦 due to displacement of the beam at the undulator with different currents. (a) 𝛥𝐼 = 1 mA, 𝛥𝑦𝑢𝑛𝑑 = 1.36 mm, (b) 𝛥𝐼 = 3 mA, 𝛥𝑦𝑢𝑛𝑑 = 1.36 mm, (c) 𝛥𝐼 = 3 mA, 𝛥𝑦𝑢𝑛𝑑 = −0.72 mm, (d) 𝛥𝐼 = 3 mA, 𝛥𝑦𝑢𝑛𝑑 = −1.00 mm. Blue circles: measured data; solid red lines: fit; dashed green line: undulator location.
4.3. Narrow-gap undulator chamber As Eqs. (6) and (7) indicated, by subtracting the closed orbit with scrapers retracted, the systematic errors from instruments or bumps can be eliminated and the closed orbit distortion purely due to the wake fields is obtained. This is possible since the scrapers are movable. On the other hand, for the narrow-gap undulator chamber, the gap cannot be changed. To eliminate the instrumental systematic error, especially the current-dependent effect, the closed orbit distortion was obtained using the following equation [18]: 𝛥𝑦 = (𝑦1𝑑 − 𝑦10 ) − (𝑦2𝑑 − 𝑦20 ),
tapers (3.38 m), half aperture (2.25 mm), and bunch length (𝜎𝑧 = 10 mm), we estimated the kick factor from the resistive wall impedance of the undulator chamber to be about 464.7 V/(pC⋅m) [10]. Hence the estimated total kick factor from both geometry and resistive wall is 763.1 V/(pC⋅m), which is closer to the measured value, but does not eliminate the discrepancy completely. A third effect is that the orbit distortion data were somewhat noisier when the measurements were done at low currents or small displacements, which produces a larger uncertainty in the fitted kick factors. As seen in Table 2, the four measurements at the undulator location indeed show a relatively large scatter in the measured kick factors. A fourth consideration is that the beam trajectory through the chamber may not be exactly parallel to the chamber center line. If so, the actual beam displacement at the entrance or exit of the chamber will be larger, and the kick will be correspondingly bigger. From the orbit bump data, we find the angle through the undulator chamber varies from 8 to 16 𝜇rad depending on the bump setting and the current. Using the average angle, the difference of the vertical beam position at the entrance and the exit of the undulator is ∼50 μm. Using Eq. (4) and Table 2, we estimate the uncertainty in the kick due to this angle effect is 4 to 7%, corresponding to a change in the kick factor of up to 100 V/(pC⋅m). A fifth contribution is that other elements of the vacuum chamber within the orbit bump may contribute to the measured kick factor. There are three BPMs, two sliding joints with bellows, two RF cavities with a pair of tapers, and one electrostatic separator within the orbit bump created by four vertical steerings. The CESR BPM buttons are embedded in the CESR vacuum pipe. Their contributions to the measured kick factor should be negligible. From T3P simulation, the kick factors of a single sliding joint, a single RF cavity, a pair of tapers and one separator are calculated to be 0.36, 0.35, 8.39, and 6.82 V/(pC⋅m), respectively. However, the separator is near one of the two outer steerings and the vertical displacement is about 15% of that at the bump center. The two RF cavities are also within the range where the vertical displacement is changing. One RF cavity has the 80% of the full displacement of the bump where the other one has 50%. Therefore, considering the relative vertical displacement effect, the total contribution to the measured kick factor from these elements is about 7.70 V/(pC⋅m).
(8)
where 𝑦1𝑑 and 𝑦10 are the closed orbits measured at one current 𝐼1 with a vertical displacement of 𝑑 and 0 at the undulator, respectively, and 𝑦2𝑑 and 𝑦20 are the closed orbits measured at another current 𝐼2 with the same vertical displacement of 𝑑 and 0 at the undulator, respectively. The fitting equation for the 𝛥𝑦 is slightly modified: √ 𝛽𝑦 (𝑠)𝛽𝑦0 𝑒𝛥𝑞 𝑘⟂ 𝑑 cos[|𝛥𝜙𝑦 (𝑠)| − 𝜋𝑄𝑦 ]. (9) 𝛥𝑦(𝑠) = 𝐸 2 sin 𝜋𝑄𝑦 Here 𝛥𝑞 is the bunch charge difference between two measurements and 𝑑 is the vertical displacement at the undulator chamber for the two measurements. Thus, the bump experiments at the undulator chamber were done by displacing the beam vertically to several 𝑦 positions for three currents: 1 mA, 2 mA, and 4 mA. A four-corrector closed bump was created at the undulator chamber location to displace the beam. Fig. 6 shows four measured and modeled orbit distortions. The errors in the measured orbit data are much noiser than those from the scrapers. It may be due to the fact that less TBT data (2048 turns) were taken during the undulator experiment. The measurement conditions and the fitted kick factors are summarized in Table 2. The average kick factor from the four measurements is 1026.5 ± 221.9 V/(pC⋅m), which is much larger than the calculated value [255.3 V/(pC⋅m)] from numerical simulations. There are a number of effects that may contribute to the discrepancy. First, although the gap of the undulator chamber was machined to be 5 mm, the realistic gap is about 4.5 mm under vacuum because of thin chamber wall (∼0.5 mm). The kick factor of the undulator chamber with a gap of 4.5 mm is estimated using T3P to be 298.4 V/(pC⋅m), 43 V/(pC⋅m) higher than that of a 5 mm-gap undulator chamber. Secondly, the numerical simulation does not include the contribution from the resistive wall wake. Using the formulas discussed in Ref. [6] with Aluminum conductivity (3.5×107 S/m), actual chamber length without 254
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Nuclear Inst. and Methods in Physics Research, A 927 (2019) 250–256
Acknowledgments
A final consideration is that, as Fig. 2e shows, we use an inverse model for the undulator chamber in the wake field calculation. As a result, the tapers in the model may interact in ways that are different from the real undulator chamber (given that the aperture between the tapers and the distance between the tapers are both different between model and reality). This may produce some discrepancy between the kick factor of the model and that of the actual undulator chamber. Simulation with the actual undulator model using a different simulation tool such as GdfidL [24] will be a good check. We did cross-check the T3P results against the analytic approach to calculating the kick factor of a flat collimator assuming linear tapers [28]. The analytic kick factors are 346.1 and 405.4 V/(pC⋅m) for a 5 mm-gap and 4.5 mmgap undulator chamber, respectively, which are about 36% higher than those T3P values, indicating the T3P results from the inverse undulator model may underestimate the real kick factors. However, one should be aware that the analytic formula is approximate. Additionally, the analytic formula does not take into account the non-linear tapering for wake reduction and does not accurately represent the large-aperture pipe cross-section. The vertical betatron-tune shifts were measured before and after installation of the undulator chamber at 5.3 GeV. The tune shift solely due to the undulator chamber was extracted to be −93 Hz/mA [10]. Using the design Twiss parameter 𝛽𝑦 = 7.5 m, we inferred the kick factor from this tune shift measurement: 823.5 V/(pC⋅m), which is also larger than the value computed by the numerical simulation. Thus it appears that the numerical and theoretical calculation underestimates the kick factors obtained from both the closed orbit distortion measurements and the tune shift measurements. This is consistent with a recent paper which reports a similar trend in most other storage rings that the measured impedance is larger than the predicted impedance from numerical simulation [29]. That we find good agreement between measurements and predictions for the vertical scrapers and poorer agreement for the narrow gap chamber may provide a clue to the deficiencies of the impedance calculations. Recently, one sixth of the CESR ring has been upgraded to accommodate more undulator beamlines. Five similar narrow-gap undulator chambers were installed in CESR for X-ray users at 6 GeV. At this high beam energy, the distortion of the closed orbit due to wake fields will be three times less than that at 2 GeV. However, care must still be taken to ensure the beam trajectory through the center of the narrow-gap chambers to eliminate the transverse dipole wake field. We also studied the coupled-bunch instability due to resistive wall impedance [30]. Moderate instability growth rates were found for various bunch patterns that can be damped by the already installed bunch-by-bunch feedback. On the other hand, obtaining the impedance of these narrow-gap chambers and understanding further effects could be important to the operation of CESR at low energies (< 2 GeV). We plan to conduct more studies using the technique discussed above and the method in Ref [14] to obtain the local impedance of these narrowgap undulator chambers. These studies may shed more light on the discrepancy between the measurements and simulation results.
The authors thank Mike Billing, Mike Ehrlichman, Nate Rider, and Jim Shanks for valuable discussions, and Cho Ng (SLAC) for his assistance with T3P. This research was supported by NSF, United States PHYS-1068662 and PHYS-1416318. References [1] Y.C. Chae, How many IVUs Can we install without sacrificing 16-mA operation, Tech. Rep. ANL/APS/LS-318, Advanced Phton Source, Argonne National Laboratory, Argonne, IL, 2009. [2] P. Brunelle, R. Nagaoka, R. Sreedharan, Measurement and analysis of the impact of transverse incoherent wakefields in a light source storage ring, Phys. Rev. ST Accel. Beams 19 (2016) 044401. [3] S.T. Wang, D.L. Rubin, J.P. Shanks, CESR lattice for two beam operations with narrow gap undulators at CHESS, in: Proceedings of International Particle Accelerator Conference 2016, Busan, Korea, 2016, pp. 2968–2971. [4] A.W. Chao, Physics of Collective Beam Instabilities in High Energy Accelerators, John Wiley and Sons, 1993. [5] G.V. Stupakov, Wake and Impedance, Tech. Rep. SLAC-PUB-8683, SLAC, Stanford, CA, 2000. [6] P. Tenenbaum, Collimator Wakefield Calculations for ILC-TRC Report, Tech. Rep. LCC-0101, Stanford Linear Accelerator Center, Stanford, CA 94309, USA, 2002. [7] K. Bane, S. Krinsky, Impedance of the NSLS Prototype Small-Gap Undulator Vacuum Chamber, Tech. Rep. BNL-48792, Brookhaven National Laboratory, Brookhaven, NY, USA, 1993. [8] W. Bruns, The GdfidL Electromagnetic Field simulator, www.gdfidl.de. [9] K. Ko, et al., Advances in parallel electronmagnetic codes for accelerator science and development, in: Proceedings of LINAC2010, Tsukuba, Japan, 2010, p. 1028. [10] M.G. Billing, W. Hartung, Transverse impedance of undulator, vertical scraper and mask chambers, 2016, unpublished. [11] D. Brandt, P. Castro, K. Cornelis, A. Hofmann, G. Morpurgo, G. Sabbi, J. Wenninger, B. Zotter, Measurements of impedance distributions and instability thresholds in LEP, in: Proceedings of the Particle Accelerator Conference 1995, Dallas, USA, 1995, p. 570. [12] V. Sajaev, Transverse impedance distribution measurements using the response matrix fit method at APS, in: Proceedings of the Particle Accelerator Conference 2003, Portland, USA, 1995, p. 417. [13] M. Carla, G. Benedetti, T. Gunzel, U. Iriso, Z. Marti, Local transverse coupling impedance measurements in a synchrotron light source from turn-by-turn acquisitions, Phys. Rev. ST Accel. Beams 19 (2016) 121002. [14] M.P. Ehrlichman, S.T. Wang, J.P. Shanks, Measurement of transverse impedance of specific components in CESR Using BPM measurements of pinged bunches, in: Proceedings of International Particle Accelerator Conference 2016, Vancouver, Canada, 2018, p. THPAF020. [15] V. Kiselev, V. Smaluk, A method for measurement of transverse impedance distribution along storage ring, in: Proceedings of DIPAC 1999, Chester, UK, 1999, pp. 202–204. [16] L. Emery, G. Decker, J. Galayda, Local bump method for measurement of transverse impedance of narrow-gap ID chambers in storage rings, in: Proceedings of the Particle Accelerator Conference 2001, Chicago, USA, 2001, pp. 1823–1825. [17] L. Tosi, E. Karantzoulis, V. Smaluk, Measurements of the impedance introduced by the vertical scraper at ELETTRA and its effects, in: Proceedings of EPAC 2002, Paris, France, 2002, pp. 1568–1570. [18] V. Kiselev, V. Smaluk, Measurement of local impedance by an orbit bump method, Nucl. Instrum. Methods Phys. Res. A 525 (2004) 433–438. [19] V. Smaluk, X. Yang, A. Blednykh, Y. Tian, K. Ha, AC Orbit bump method of local impedance measurement, Nucl. Instrum. Methods Phys. Res. A 871 (2017) 59–62, http://dx.doi.org/10.1016/j.nima.2017.07.067. [20] S.T. Wang, J.D. Perrin, S. Poprocki, W. Hartung, D.L. Rubin, Vertical-longitudinal coupling induced by wakefields at the Cornell Electron-Positron Storage Ring Test Accelerator, Phys. Rev. ST Accel. Beams 21 (2018) 024401. [21] M.G. Billing, et al., Beam position monitoring system at CESR, J. Instrum. 12 (2017) T09005. [22] J. Shanks, D.L. Rubin, D. Sagan, Low emittance tuning at the Cornell electron storage ring test accelerator, Phys. Rev. ST Accel. Beams 17 (2014) 044003. [23] A. Chatterjee, K. Blaser, W. Hartung, D.L. Rubin, S.T. Wang, Fast ion instability at the Cornell Electron Storage Ring Test Accelerator, Phys. Rev. ST Accel. Beams 18 (2015) 064402. [24] V. Smaluk, R. Fielder, A. Blednykh, G. Rehm, R. Bartolini, Coupling impedance of an in-vacuum undulator: Measurement, simulation, and analytical estimation, Phys. Rev. ST Accel. Beams 17 (2014) 074402. [25] CUBIT 14.1, Sandia Nationa Laboratories, Albuquerque, NM, cubit.sandia.gov. [26] NERSC, Lawrence Berkeley National Laboratory, Berkeley, CA, www.nersc.gov.
5. Conclusion We measured the current-dependent and displacement-dependent closed orbit distortion due to the transverse wake fields induced from asymmetric and symmetric scraper configurations and a narrow-gap undulator chamber. By fitting the orbit distortion with a simple function, we found the average transverse wake from the asymmetric scrapers and the kick factor of the symmetric scrapers; they agreed reasonably well with the calculated values from numerical simulations. However, there is a large discrepancy between the kick factor of the narrow-gap chamber measured from the orbit distortion and the numerical simulation. Possible explanations have been discussed. Our data demonstrate that the wake fields not only affect the beam emittance (incoherently) but also distort the closed orbit of the beam (coherently). 255
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Nuclear Inst. and Methods in Physics Research, A 927 (2019) 250–256 [29] V. Smaluk, Impedance computations and beam-based measurements: A problem of discrepancy, Nucl. Instrum. Methods Phys. Res. A 888 (2018) 22–30, http: //dx.doi.org/10.1016/j.nima.2018.01.047. [30] S.T. Wang, M.G. Billing, S. Poprocki, D.L. Rubin, D.C. Sagan, Resistive wall instability and impedance studies of narrow undulator chamber in CHESS-U, in: Proceedings of International Particle Accelerator Conference 2017, Copenhagen, Denmark, 2017, pp. 3204–3207.
[27] W.K.H. Panofsky, W.A. Wenzel, Some considerations concerning the transverse defection of charged particles in radio-frequency fields, Rev. Sci. Instr. 27 (1956) 967. [28] P. Tenenbaum, K.L.F. Bane, L. Eriksson, J. Irwin, R.K. Jobe, D. McCormick, C.K. Ng, T.O. Raubenheimer, M.C. Ross, G. Stupakov, D. Walz, D. Onoprienko, I. Zagorodnov, Direct measurement of the transverse wakefields of tapered collimators, Phys. Rev. ST Accel. Beams 10 (2007) 034401.
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Local impedance measurements from closed orbit distortion due to wake fields at CESR S.T. Wang ∗, J.D. Perrin, S. Poprocki, W. Hartung 1 , D.L. Rubin Cornell Laboratory for Accelerator-based Sciences and Education, Cornell University, Ithaca, NY 14853, United States
ARTICLE Keywords: Storage rings Wake fields Beam dynamics
INFO
ABSTRACT Short range transverse wake fields are excited by charged particles passing through a vacuum chamber. The wake fields can act back on a bunch of particles and alter its trajectory. We have measured the distortion of the closed orbit due to transverse wake fields in the Cornell Electron–positron Storage Ring. A transverse monopole wake field is induced by an asymmetric impedance associated with the insertion of a single vertical scraper. A transverse dipole wake field is generated by displacing the bunch from the symmetry axis of the chamber. The orbit is offset vertically through the gap of a pair of symmetrically inserted scrapers and also in a narrow-gap undulator chamber. A fit to the distorted orbits yields the average kick of the transverse monopole wake of the asymmetric scrapers and the kick factor of the symmetric scrapers; the scraper results are in reasonable agreement with the values from numerical simulation of the wake fields. The kick factor extracted from the closed orbit distortion of the dipole wake induced by the narrow-gap undulator chamber is significantly greater than that predicted by the numerical simulation. Possible sources of the discrepancy are discussed.
1. Introduction Wake fields can distort the bunch shape, induce instability, and limit beam intensity, thus compromising the performance of particle accelerators. The strength of the wake fields increases linearly with beam current, making them an important consideration for the design of high-current colliders and light sources. In a synchrotron light source, transverse wake fields become an important source of beam emittance dilution at high currents. Insertion devices with narrow vertical gaps, and high impedance, are often used to generate high-brightness X-ray beams [1–3]. The wake fields in these devices can dilute the emittance and limit the total charge in a bunch or train of bunches. The interaction between moving charged particles and the induced wake fields in a vacuum chamber are characterized by the wake functions, which are determined by the geometry and electromagnetic properties of the chamber. Analytic formulas to calculate the impedance (Fourier transform of the wake function) are limited to vacuum components with simple geometries, such as a round pipe [4], small pillbox cavity [5], tapered collimator [6], and a small-gap undulator chamber [7]. For complex geometries, numerical simulations such as GdfidL [8] or T3P [9] are typically used. Beam-based impedance measurement techniques are used in various store rings, including measurements of current-dependent bunch lengthening for the longitudinal impedance and current-dependent ∗ 1
betatron-tune shift for the transverse impedance. These techniques characterize the global impedance of a storage ring. To measure the local impedance of a vacuum component, one approach is to measure the global impedance before and after installation of a vacuum chamber, as the difference will be due to the impedance of the new chamber [10]. However, this method requires two separate measurements at two different times, so that different machine conditions and instrumentation errors may complicate the analysis. Making using of the fact that the transverse wake field can change both the betatron phase advance and the closed orbit, two additional methods have been developed to characterize the local distribution of impedances: measuring the current-dependent phase advance [11–14] and the closed orbit distortion [15–19]. The Cornell Electron–positron Storage Ring (CESR), built on the Cornell University campus, stores counter-rotating beams of electrons and positrons. Positrons circulate in the clockwise direction and electrons in the counter-clockwise direction as shown in Fig. 1. An undulator chamber with a vertical gap of ∼5 mm and length of ∼3.5 m was installed in CESR to accommodate two undulators for X-ray users. Because of its narrow gap, the undulator chamber produces a significant transverse wake which can affect the beam. Current-dependent betatron tune shifts measured before and after the installation of the vacuum chamber give an estimate of the transverse impedance of the chamber [10].
Corresponding author. E-mail address: [email protected] (S.T. Wang). Present address: Facility for Rare Isotope Beams, Michigan State University, East Lansing, MI 48824, United States.
https://doi.org/10.1016/j.nima.2019.02.060 Received 26 November 2018; Received in revised form 4 February 2019; Accepted 20 February 2019 Available online 23 February 2019 0168-9002/© 2019 Elsevier B.V. All rights reserved.
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Nuclear Inst. and Methods in Physics Research, A 927 (2019) 250–256
The average kick 𝛥𝑦′ on the bunch causes the distortion of the closed orbit of the bunch as Eq. (2) described. The average kick is defined in Eq. (1) for an average wake field 𝑊⟂ . For the asymmetric scraper configuration, 𝑊⟂ is dominated by the transverse monopole wake and 𝑊⟂ is ∞
𝑊⟂ =
∫−∞ 𝑓 (𝑧)𝑊⟂ (𝑧)𝑑𝑧 ∞
∫−∞ 𝑓 (𝑧)𝑑𝑧
,
(3)
where 𝑓 (𝑧) is the distribution function of the bunch particles in the longitudinal direction. Typically the longitudinal distribution is approximately a Gaussian. For the cases with the beam displaced between the symmetrically inserted scrapers or at the undulator chamber, the transverse monopole wake is zero because of the top-down symmetry and the transverse dipole wake term dominates. The average wake field is then (4)
𝑊⟂ = 𝑘⟂ 𝑦,
where 𝑦 is the displacement of the beam from the chamber center and 𝑘⟂ is the kick factor, which is defined by the following equation in the limit where the vertical beam size is much less than the displacement 𝑦:
Fig. 1. CESR layout showing the locations of undulators and scrapers.
A pair of adjustable vertical scrapers with a minimum gap of 7 mm were installed in CESR in order to intercept the injected particles with large vertical oscillations. Studies have been done to explore the effects of both the undulator chamber and the scrapers [20]. It was found that the vertical beam size increases significantly due to wake fields generated by an asymmetric scraper configuration or by displacing the beam off the chamber center at the undulator or the scrapers. In this paper, we extend our investigation to the closed orbit distortion caused by the transverse monopole wake from an asymmetric scraper configuration and the dipole wake induced by displacing the beam at the undulator chamber or the scrapers. Using the same analysis method described in [18,19], we obtain the average vertical kick of the asymmetric scraper configuration and the kick factors from the scrapers and the undulator chamber. The values from the scrapers agree reasonably well with model values extracted from numerical simulations using T3P. On the other hand, there is a large discrepancy between the kick factor of the narrow-gap undulator chamber measured from the closed orbit distortion and the numerical simulation. Possible explanations are discussed.
∞
𝑘⟂ =
∫−∞ 𝑓 (𝑧)𝑊𝑑 (𝑧)𝑑𝑧 ∞
∫−∞ 𝑓 (𝑧)𝑑𝑧
,
(5)
where 𝑊𝑑 (𝑧) is the dipole wake function. As discussed in Section 3, we used numerical simulations to compute the wake functions for the asymmetric and symmetric scraper configurations and for the undulator chamber. The average kick of the transverse monopole wake from an asymmetric scraper configuration and the kick factors of the symmetric scrapers and the undulator chambers are calculated from those wakes. The CESR beam position monitors (BPM) have the capability of measuring the turn-by-turn (TBT) and bunch-by-bunch beam positions with a resolution of 10 μm [21]. That capability has been an important and useful tool for low emittance tuning [22] and studying collective effects such as fast ion instability [23]. In CESR, there are 100 BPMs. In each measurement, 8192 turns of beam positions of a single electron bunch were recorded at these BPMs. The closed orbit was then obtained by averaging the 8192 positions at each BPM. For high bunch currents (> 1 mA), different gain settings were applied to the BPM electronics. Therefore, there was a current-dependent effect on the orbits due to instrumentation error. In Section 4, we will discuss the methods to obtain the orbit distortion after eliminating this systematic effect. Once the orbit distortion 𝛥𝑦 was obtained, we did a fit of 𝛥𝑦 as a function of 𝑠 using Eq. (2) and find the average kick of the monopole wake or the kick factor for the dipole wake. This method has been successfully applied to measure local impedance of narrow-gap undulator in APS [16], ELETTRA [17], NSLS-II [19], and DLS [24].
2. Methods When a bunch travels through a vacuum chamber, each particle experiences a vertical kick 𝛥𝑦′ due to the transverse wake field induced by the bunch: 𝑒𝑞 𝛥𝑦′ = 𝑊⟂ , (1) 𝐸 where 𝑊⟂ is the transverse wake potential, 𝑞 is the total bunch charge, 𝑒 is the electron charge, and 𝐸 is the beam energy. The average kick acting on the centroid of the bunch is similar to a dipole error and distorts the closed orbit. Suppose the bunch gets an average vertical kick 𝛥𝑦′ at location 𝑠0 ; the change in the closed orbit 𝛥𝑦(𝑠) at location 𝑠 will be √ 𝛥𝑦′ 𝛽𝑦 (𝑠)𝛽𝑦0 𝛥𝑦(𝑠) = cos[|𝛥𝜙𝑦 (𝑠)| − 𝜋𝑄𝑦 ], (2) 2 sin 𝜋𝑄𝑦
3. Numerical simulations The ACE3P electromagnetic simulation suite was used to calculate the wake fields [9]. The three-dimensional models of the scrapers and the undulator chamber were constructed using the finite element mesh toolkit CUBIT [25]. The CUBIT models are shown in Fig. 2. The CUBIT models were imported to the time domain wake field solver T3P [9] to calculate the longitudinal wake. We ran T3P at the National Energy Research Scientific Computing Center [26]. The charged bunch was modeled as a Gaussian distribution with 𝜎𝑧 = 10 mm, as calculated from the lattice model at the experimentally measured synchrotron tune. Longitudinal wakes were computed with the drive bunch at each of three vertical positions (−𝛥𝑦, 0, 𝛥𝑦) where 𝛥𝑦 = 0.5 mm is the mesh size of the CUBIT model. T3P calculate the longitudinal wake of the witness particle at a specific vertical position. A Taylor expansion of the longitudinal wake in terms of the vertical displacement of the drive bunch and witness particle, and then application of the Panofsky– Wenzel theorem [27], gives the transverse wake. The wake terms are
where 𝑠 is the location of the bunch in the accelerator, 𝛥𝜙𝑦 (𝑠) is the vertical phase advance from 𝑠0 to 𝑠, 𝑄𝑦 is the vertical tune, and 𝛽𝑦 (𝑠) and 𝛽𝑦0 are the vertical betatron functions at 𝑠 and 𝑠0 , respectively. In general, 𝑊⟂ (𝑧) is a function of 𝑧, the particle’s longitudinal position relative to the bunch centroid. Hence, the wake field can produce a yz coupling in the beam. The contribution to the yz tilt 𝜃𝑦𝑧 manifests in an increase in the vertical beam size [20]. Since each particle in the bunch experiences a different kick from the wake, the bunch as a whole will receive an average kick from the wake fields. 251
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Fig. 2. CUBIT half model of the scrapers including artificial tapers (a) with only top scraper inserted and (c) with both scrapers inserted. (e) CUBIT inverse model of the undulator chamber. Transverse wakes of (b) the asymmetric scrapers, (d) the symmetric scrapers, and (f) the undulator chamber: monopole (red), dipole (green), and quadrupolar (blue) term. In (b), the dipole and quadrupolar terms have been multiplied by a factor of 𝛥𝑦 = 0.5 mm to compare with the transverse monopole term. Dashed line: bunch distribution (𝜎𝑧 = 10 mm, arbitrary scale). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) Table 1 CESR machine parameters.
shown in Fig. 2. It is worth noting here that we used T3P to calculate the short-range wake field in order to predict the effect of the wake on the particles in the bunch; thus our predictions do not include long range effects from wakes that endure for multiple turns. As a note, we have successfully used the same transverse monopole wake from the asymmetric scraper configuration in the tracking simulation to explain the increased vertical beam size behavior due to 𝑦𝑧 coupling in our previous paper [20]. Fig. 2a shows the half model of the scrapers with top scraper inserted and bottom scraper retracted. The model includes artificial tapers which match the 7 mm gap of the closed scrapers. The artificial tapers are used in order to satisfy the constraints of the Weiland integration method used by T3P. Wake fields for a separate model with just the artificial tapers were also obtained, and subtracted from the wake fields from the scraper+tapers model to obtain the wake fields for the scraper alone. As Fig. 2b shows clearly, the transverse monopole wake dominates the dipole and quadrupolar wakes and it depends nonlinearly on particle longitudinal position (𝑧). Using Eq. (3), we find that the average monopole wake amplitude from the asymmetric scrapers (top scraper inserted) 𝑊⟂ is 0.80 V/pC. Fig. 2c shows the CUBIT half model of the scrapers, including artificial tapers, with both scrapers fully inserted. The calculated dipole and quadrupolar wake fields from this model are shown in Fig. 2d. Because of the up/down symmetry, the transverse monopole wake is zero. We created a similar CUBIT model with both scrapers retracted and obtained the corresponding dipole and quadrupolar wake terms, which are much smaller than those from the model with both scrapers inserted. In order to compare with the kick factor extracted from the data, we used the dipole wake field with both scrapers inserted after subtracting the wake fields from the model with both scrapers retracted to calculate the kick factor due to inserting the scrapers: 220.1 V/(pC⋅m). Fig. 2e displays the CUBIT full model of the undulator chamber with tapers. An inverse model was used to satisfy constraints of the Weiland integration method used in T3P. In this model, the chamber
Beam energy (GeV)
𝐸
2.085
Circumference (m) Transverse damping time (ms) Momentum compaction Nominal RF voltage (MV) Synchrotron tune Horizontal tune Vertical tune Horizontal emittance (nm·rad) Vertical emittance (pm·rad)
𝐿 𝜏 𝛼𝑝 𝑉𝑅𝐹 𝑄𝑠 𝑄𝑥 𝑄𝑦 𝜖𝑎 𝜖𝑏
768 56.6 0.0068 5.8 0.067 14.573 9.629 ∼3 10–20
gap (∼5 mm) is at the end, with tapers to a regular-height beam chamber in the middle. Because the chamber has up/down symmetry, the transverse monopole wake is again zero. The dipole wake is shown in Fig. 2f; the calculated kick factor from the dipole wake is 255.3 V/(pC⋅m). 4. Experiments All the measurements were done with a single bunch of electrons stored at 2.1 GeV in CESR. The nominal parameters of the lattice are listed in Table 1. A tuning procedure which corrects the lattice betatron functions, 𝑥𝑦 coupling, and vertical dispersion was initially applied to achieve low vertical emittance (𝜖𝑦 ≈ 15 pm⋅rad) in the accelerator [22]. Two vacuum components, the scrapers and the undulator chamber, were investigated and information about the impedance was extracted. Two scraper configurations (asymmetric and symmetric) were studied. 4.1. Asymmetric scrapers The vertical scrapers were located in the north region of the ring as shown in Fig. 1. The gap between top and bottom scrapers is 7 mm when both scrapers are fully inserted and 50 mm with both retracted. 252
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Fig. 4. The measured orbit bump at 150 cu at 4 mA with scrapers retracted. Dashed green line: scraper location.
Fig. 3. Measured orbit difference 𝛥𝑦 at (a) 1.7 mA, (b) 2.7 mA, and (c) 3.7 mA due to insertion of the top scraper. Blue circles: measured data; solid red lines: fit; dashed green line: scraper location.
Before measurements, the beam was carefully centered between the two fully-inserted scrapers. With only the top scraper inserted, the wake generated by the asymmetric scraper configuration is mainly the vertical monopole wake. The bunch receives a centroid kick from the average wake field 𝑊⟂ when passing through the scrapers. TBT beam positions at 100 BPMs were recorded for 8192 turns at three different bunch currents: ∼1.7, ∼2.7, and ∼3.7 mA. Since there is a current-dependent tune shift, in order to avoid a resonance effect through the current range, the ring fractional tunes were carefully set at 𝑄𝑥 = 0.573, 𝑄𝑦 = 0.629, 𝑄𝑠 = 0.067 at low current (∼1 mA). Analysis of these TBT BPM data confirms the current-dependent tune shift and provides the closed orbit at the different currents. The BPM data were collected with both scrapers retracted or with only the top scraper fully inserted. The closed orbit distortion 𝛥𝑦 due to the wake from the asymmetric scraper configuration (top scraper inserted) was obtained by subtracting the closed orbit without scrapers inserted (𝑦𝑜𝑜 ) from the closed orbit with top scraper fully inserted (𝑦𝑖𝑜 ) at the same currents: 𝛥𝑦 = 𝑦𝑖𝑜 − 𝑦𝑜𝑜 .
Fig. 5. Measured orbit difference 𝛥𝑦 when the beam was displaced from the center by (a) −1.04 mm, (b) −1.89 mm, and (c) 2.42 mm at the scrapers, with the scrapers fully inserted. Blue circles: measured data; solid red lines: fit; dashed green line: scraper location.
(6) unchanged. As the measured bump at 150 computer units (cu) in Fig. 4 shows, the bump was well closed outside the scraper region with an RMS of 0.022 mm, which is about 1.8% of the bump peak value. In addition, five sextupoles at the bump location were turned off to eliminate nonlinear coupling from the sextupoles. The origin (𝑦 = 0, the gap center) is inferred from the bump settings at which beam loss was observed (via a decrease in the stored beam lifetime). The TBT BPM data were taken for 4096 turns with a single electron bunch at 4 mA at various bump settings with both scrapers fully inserted. Similar measurements were also taken with both scrapers retracted. The closed orbit distortion 𝛥𝑦 due to the dipole wake was then obtained by subtracting the closed orbit without scrapers inserted (𝑦𝑜𝑜 ) from the closed orbit with both scrapers fully inserted (𝑦𝑖𝑖 ) at different 𝑦 displacements:
The 𝛥𝑦 values obtained at ∼1.7, ∼2.7, and ∼3.7 mA are shown in Fig. 3. We can see that the orbit distortion increases as the current increases due to the linear dependence of the wake on beam current. With the design Twiss parameters of the lattice, we can fit the measured data using Eqs. (1) and (2). The average wake field 𝑊⟂ is the only fitting parameter. The best fits are shown as red lines in Fig. 3. The 𝑊⟂ from the best fits are found to be 0.86, 0.76, and 0.75 V/pC when beam current is ∼1.7, ∼2.7, and ∼3.7 mA, respectively. As we can see, the average 𝑊⟂ (0.79 ± 0.06 V/pC) agrees very well with the value of 0.80 V/pC extracted from the numerical simulation in Section 3. 4.2. Symmetric scrapers With both scrapers fully inserted, the transverse monopole wake is zero because of the up/down symmetry. If the beam is displaced from the gap center, the beam will get a kick from the dipole wake (Eq. (4)) which distorts the closed orbit. Three vertical correctors around the scrapers were selected to create a vertical displacement bump at the scraper location. This bump only displaces the vertical position of the beam at the scraper location and keeps 𝑦 positions everywhere else
𝛥𝑦 = 𝑦𝑖𝑖 − 𝑦𝑜𝑜 .
(7)
The measured closed orbit distortion due to the dipole wake at three bump settings with three displacements of 𝑦 = −1.04, −1.89, and 2.42 mm are displayed in Fig. 5. As Fig. 5a and b show, the larger the displacement the larger the distortion of the closed orbit. When the 253
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Nuclear Inst. and Methods in Physics Research, A 927 (2019) 250–256
Table 2 Bump measurements at the undulator chamber. Figure
𝑑 (mm)
𝛥𝐼 (mA)
𝑘⟂ (V/(pC⋅m))
a b c d
1.36 1.36 −0.72 −1.00
1 3 3 3
1347.7 935.4 981.4 841.7
(2-1) (4-1) (4-1) (4-1)
displacement reverses sign, so does the sign of the distortion, as seen in Fig. 5b and c. Using the design Twiss parameters of the lattice and Eqs. (2) and (4), we did a fit of the measured orbit distortion and found the kick factor 𝑘⟂ of the symmetric scrapers. Here 𝑘⟂ is the only fitting parameter. From the best fits, the kick factors found from the three data sets in Fig. 5a, b, and c are 211.2, 228.7, and 246.2 V/(pC⋅m), respectively. As we can see, the average value [228.7 ± 17.5 V/(pC⋅m)] agrees reasonably well with the value of 220.1 V/(pC⋅m) from numerical simulation in Section 3. The scrapers are two stainless-steel cylinders with a diameter of 50 mm. Estimate of the kick factor from the resistive wall impedance of the scrapers is about 9.1 V/(pC⋅m) [6,10], which improves the agreement between the measurement and the calculation.
Fig. 6. Measured orbit difference 𝛥𝑦 due to displacement of the beam at the undulator with different currents. (a) 𝛥𝐼 = 1 mA, 𝛥𝑦𝑢𝑛𝑑 = 1.36 mm, (b) 𝛥𝐼 = 3 mA, 𝛥𝑦𝑢𝑛𝑑 = 1.36 mm, (c) 𝛥𝐼 = 3 mA, 𝛥𝑦𝑢𝑛𝑑 = −0.72 mm, (d) 𝛥𝐼 = 3 mA, 𝛥𝑦𝑢𝑛𝑑 = −1.00 mm. Blue circles: measured data; solid red lines: fit; dashed green line: undulator location.
4.3. Narrow-gap undulator chamber As Eqs. (6) and (7) indicated, by subtracting the closed orbit with scrapers retracted, the systematic errors from instruments or bumps can be eliminated and the closed orbit distortion purely due to the wake fields is obtained. This is possible since the scrapers are movable. On the other hand, for the narrow-gap undulator chamber, the gap cannot be changed. To eliminate the instrumental systematic error, especially the current-dependent effect, the closed orbit distortion was obtained using the following equation [18]: 𝛥𝑦 = (𝑦1𝑑 − 𝑦10 ) − (𝑦2𝑑 − 𝑦20 ),
tapers (3.38 m), half aperture (2.25 mm), and bunch length (𝜎𝑧 = 10 mm), we estimated the kick factor from the resistive wall impedance of the undulator chamber to be about 464.7 V/(pC⋅m) [10]. Hence the estimated total kick factor from both geometry and resistive wall is 763.1 V/(pC⋅m), which is closer to the measured value, but does not eliminate the discrepancy completely. A third effect is that the orbit distortion data were somewhat noisier when the measurements were done at low currents or small displacements, which produces a larger uncertainty in the fitted kick factors. As seen in Table 2, the four measurements at the undulator location indeed show a relatively large scatter in the measured kick factors. A fourth consideration is that the beam trajectory through the chamber may not be exactly parallel to the chamber center line. If so, the actual beam displacement at the entrance or exit of the chamber will be larger, and the kick will be correspondingly bigger. From the orbit bump data, we find the angle through the undulator chamber varies from 8 to 16 𝜇rad depending on the bump setting and the current. Using the average angle, the difference of the vertical beam position at the entrance and the exit of the undulator is ∼50 μm. Using Eq. (4) and Table 2, we estimate the uncertainty in the kick due to this angle effect is 4 to 7%, corresponding to a change in the kick factor of up to 100 V/(pC⋅m). A fifth contribution is that other elements of the vacuum chamber within the orbit bump may contribute to the measured kick factor. There are three BPMs, two sliding joints with bellows, two RF cavities with a pair of tapers, and one electrostatic separator within the orbit bump created by four vertical steerings. The CESR BPM buttons are embedded in the CESR vacuum pipe. Their contributions to the measured kick factor should be negligible. From T3P simulation, the kick factors of a single sliding joint, a single RF cavity, a pair of tapers and one separator are calculated to be 0.36, 0.35, 8.39, and 6.82 V/(pC⋅m), respectively. However, the separator is near one of the two outer steerings and the vertical displacement is about 15% of that at the bump center. The two RF cavities are also within the range where the vertical displacement is changing. One RF cavity has the 80% of the full displacement of the bump where the other one has 50%. Therefore, considering the relative vertical displacement effect, the total contribution to the measured kick factor from these elements is about 7.70 V/(pC⋅m).
(8)
where 𝑦1𝑑 and 𝑦10 are the closed orbits measured at one current 𝐼1 with a vertical displacement of 𝑑 and 0 at the undulator, respectively, and 𝑦2𝑑 and 𝑦20 are the closed orbits measured at another current 𝐼2 with the same vertical displacement of 𝑑 and 0 at the undulator, respectively. The fitting equation for the 𝛥𝑦 is slightly modified: √ 𝛽𝑦 (𝑠)𝛽𝑦0 𝑒𝛥𝑞 𝑘⟂ 𝑑 cos[|𝛥𝜙𝑦 (𝑠)| − 𝜋𝑄𝑦 ]. (9) 𝛥𝑦(𝑠) = 𝐸 2 sin 𝜋𝑄𝑦 Here 𝛥𝑞 is the bunch charge difference between two measurements and 𝑑 is the vertical displacement at the undulator chamber for the two measurements. Thus, the bump experiments at the undulator chamber were done by displacing the beam vertically to several 𝑦 positions for three currents: 1 mA, 2 mA, and 4 mA. A four-corrector closed bump was created at the undulator chamber location to displace the beam. Fig. 6 shows four measured and modeled orbit distortions. The errors in the measured orbit data are much noiser than those from the scrapers. It may be due to the fact that less TBT data (2048 turns) were taken during the undulator experiment. The measurement conditions and the fitted kick factors are summarized in Table 2. The average kick factor from the four measurements is 1026.5 ± 221.9 V/(pC⋅m), which is much larger than the calculated value [255.3 V/(pC⋅m)] from numerical simulations. There are a number of effects that may contribute to the discrepancy. First, although the gap of the undulator chamber was machined to be 5 mm, the realistic gap is about 4.5 mm under vacuum because of thin chamber wall (∼0.5 mm). The kick factor of the undulator chamber with a gap of 4.5 mm is estimated using T3P to be 298.4 V/(pC⋅m), 43 V/(pC⋅m) higher than that of a 5 mm-gap undulator chamber. Secondly, the numerical simulation does not include the contribution from the resistive wall wake. Using the formulas discussed in Ref. [6] with Aluminum conductivity (3.5×107 S/m), actual chamber length without 254
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Acknowledgments
A final consideration is that, as Fig. 2e shows, we use an inverse model for the undulator chamber in the wake field calculation. As a result, the tapers in the model may interact in ways that are different from the real undulator chamber (given that the aperture between the tapers and the distance between the tapers are both different between model and reality). This may produce some discrepancy between the kick factor of the model and that of the actual undulator chamber. Simulation with the actual undulator model using a different simulation tool such as GdfidL [24] will be a good check. We did cross-check the T3P results against the analytic approach to calculating the kick factor of a flat collimator assuming linear tapers [28]. The analytic kick factors are 346.1 and 405.4 V/(pC⋅m) for a 5 mm-gap and 4.5 mmgap undulator chamber, respectively, which are about 36% higher than those T3P values, indicating the T3P results from the inverse undulator model may underestimate the real kick factors. However, one should be aware that the analytic formula is approximate. Additionally, the analytic formula does not take into account the non-linear tapering for wake reduction and does not accurately represent the large-aperture pipe cross-section. The vertical betatron-tune shifts were measured before and after installation of the undulator chamber at 5.3 GeV. The tune shift solely due to the undulator chamber was extracted to be −93 Hz/mA [10]. Using the design Twiss parameter 𝛽𝑦 = 7.5 m, we inferred the kick factor from this tune shift measurement: 823.5 V/(pC⋅m), which is also larger than the value computed by the numerical simulation. Thus it appears that the numerical and theoretical calculation underestimates the kick factors obtained from both the closed orbit distortion measurements and the tune shift measurements. This is consistent with a recent paper which reports a similar trend in most other storage rings that the measured impedance is larger than the predicted impedance from numerical simulation [29]. That we find good agreement between measurements and predictions for the vertical scrapers and poorer agreement for the narrow gap chamber may provide a clue to the deficiencies of the impedance calculations. Recently, one sixth of the CESR ring has been upgraded to accommodate more undulator beamlines. Five similar narrow-gap undulator chambers were installed in CESR for X-ray users at 6 GeV. At this high beam energy, the distortion of the closed orbit due to wake fields will be three times less than that at 2 GeV. However, care must still be taken to ensure the beam trajectory through the center of the narrow-gap chambers to eliminate the transverse dipole wake field. We also studied the coupled-bunch instability due to resistive wall impedance [30]. Moderate instability growth rates were found for various bunch patterns that can be damped by the already installed bunch-by-bunch feedback. On the other hand, obtaining the impedance of these narrow-gap chambers and understanding further effects could be important to the operation of CESR at low energies (< 2 GeV). We plan to conduct more studies using the technique discussed above and the method in Ref [14] to obtain the local impedance of these narrowgap undulator chambers. These studies may shed more light on the discrepancy between the measurements and simulation results.
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5. Conclusion We measured the current-dependent and displacement-dependent closed orbit distortion due to the transverse wake fields induced from asymmetric and symmetric scraper configurations and a narrow-gap undulator chamber. By fitting the orbit distortion with a simple function, we found the average transverse wake from the asymmetric scrapers and the kick factor of the symmetric scrapers; they agreed reasonably well with the calculated values from numerical simulations. However, there is a large discrepancy between the kick factor of the narrow-gap chamber measured from the orbit distortion and the numerical simulation. Possible explanations have been discussed. Our data demonstrate that the wake fields not only affect the beam emittance (incoherently) but also distort the closed orbit of the beam (coherently). 255
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