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Lattice design for an ultra-low emittance synchrotron light source
Author’s Accepted Manuscript Lattice Design for an ultra-low Emittance Synchrotron Light Source E. Ahmadi, S.M. Jazayeri, J. Rahighi, M. Mollabashi www.elsevier.com/locate/nima
PII: DOI: Reference:
S0168-9002(17)30472-2 http://dx.doi.org/10.1016/j.nima.2017.04.021 NIMA59809
To appear in: Nuclear Inst. and Methods in Physics Research, A Received date: 17 December 2016 Revised date: 13 April 2017 Accepted date: 14 April 2017 Cite this article as: E. Ahmadi, S.M. Jazayeri, J. Rahighi and M. Mollabashi, Lattice Design for an ultra-low Emittance Synchrotron Light Source, Nuclear Inst. and Methods in Physics Research, A, http://dx.doi.org/10.1016/j.nima.2017.04.021 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Lattice Design for an ultra-low Emittance Synchrotron Light Source
1 2
E. Ahmadia*, S.M. Jazayeria, J. Rahighib, M. Mollabashia
3 4
a
b
Department of Physics, Iran University of Science and Technology, Tehran, Iran
5
Iranian Light Source Facility (ILSF), Institute for Research in Fundamental Sciences (IPM), Tehran, Iran
6 7 8
*
9
Corresponding author. [email protected]
10
Abstract
11
In this paper, we investigate a lattice design for an ultra-low emittance storage ring with intermediate energy of 3 GeV and circumference of 576 m. We present the design results for a seven-band achromat lattice with the natural emittance of 90 pm-rad. The base line is based on 20 straight sections with the length of 6 m. The minimum distance between different magnets is 10 cm and it has been tried to keep the lattice structure as simple as possible. The brilliance of radiated synchrotron radiation from a 2.24 m In-vacuum undulator installed in the lattice is [ ] . We will address the challenging points of ⁄ ⁄ ⁄ ⁄ ⁄ designed ultra-low emittance storage ring lattice such as strong nonlinearity and collective effects. The dynamic aperture of the lattice is comparatively sufficient for off-axis injection and the lifetime is relatively large without using any harmonic cavity. Since the proposed lattice structure in some aspects is similar to MAX IV storage ring, we will make some comparison between two lattices in different sections.
12 13 14 15 16 17 18 19 20 21 22 23 24 25
Keywords: Storage ring, Ultra-low emittance, MBA lattice, Dynamic aperture.
26
1. Introduction
27
Storage rings are the principle sources of high brightness photon beams driving the majority of x-ray science experiment in the world today. There has been remarkable progress in the developing these sources over the last two decades. Existing third generation light sources continue to upgrade their capabilities, while new light sources storage rings coming on line with ever-improved performances. Since one of the most important factors for the users of a synchrotron radiation is brightness which in the storage ring is determined by the emittance, our aim in this paper is to investigate and present the design results for a lattice in a 3GeV range that can provide high brilliance photons near soft X-ray diffraction limited. The diffraction limited light sources performance for a given wave length is met
28 29 30 31 32 33 34 35
when
⁄(
). Accordingly, for hard X-ray radiation at 0.1 nm,
while for soft X-
36
ray at 1 nm the diffraction limit is 80 pm rad. For the best recently existing storage rings, the horizontal electron beam emittance are nearly three orders of magnitude larger from diffraction limited for soft X-ray.
37 38 39
One simple and robust method to achieve ultralow emittance is the use of a multibend achromat (MBA) lattice [1, 2]. The MBA exploits the inverse cubic dependence of emittance on the number of bending magnets. By choosing a very small bending angle per dipole, the emittance can be dramatically reduced. Introducing a vertically focusing gradient in the dipoles causes more reduction in the emittance (the emittance scales inversely with the horizontal damping partition Jx) while the dispersion is limited to small values without requiring any extra space for vertically focusing quadrupoles. This concept has been employed in the designing the storage ring of several facilities such as of MAX IV [3] and Sirius [4]. Hybrid multiband achromat [HMBA] is another effective way to decrease emittance. It was employed for the first time in the upgrade plan of ESRF [5]. HMBA concept have some advantages relative to MBA structure, which make it a suitable candidate for the future 4th generation light sources [6]. Our designed lattice is based on MBA concept. In order to further push down the horizontal emittance, we resort to other well-known methods. For example, we used shorter dipoles at the achromat ends where the dispersion function is matched to zero in the straight section. The mentioned criteria for designing a low emittance MBA lattice can be find in reference [2]. Considering the mentioned criteria to reach the ultra-low emittance, a seven-bend achromt lattice was selected. By this selection, the circumference of storage ring is 576 m and its emittance is 90 pm-rad. The symmetry of lattice is 20 and the length of straight sections is 6 m. In this paper, we will address the challenging points of the designed ultra-low emittance lattice. Strong nonlinearity is a most demanding aspect which drastically limited the momentum acceptance of lattice and causes short life time and small dynamic aperture. Another challenging point in the ultra-low emittance rings is short electron bunches which is in consequence of small momentum compaction factor [7]. The short bunches result in strong collective effects and their corresponding instabilities.
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In Section 2, we present the lattice design, linear optics and radiation property of designed lattice. In this section, we will make a comparison among the parameters of designed lattice and MAX IV existing storage ring lattice. In Section 3, we will address the nonlinear optimization of designed lattice. Section 4 explains the tolerance of lattice to misalign and field errors. In this section, we present the results of closed orbit correction and the effects of machine errors on the dynamic aperture. Section 5 includes the lifetime and energy acceptance calculations. Section 6 is devoted to Intra Beam Scattering (IBS) calculation. In this section we explain how to mitigate undesired effect of IBS. And finally, in section 7, we evaluate the threshold current of different single bunch instabilities produced in the storage ring.
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2. Lattice design, linear optics and radiation properties
72
The storage ring lattice consists of 20 seven-bend achromats separated by 6 m straight sections for IDs. Each of the achromats consists of seven unit cells and two matching cells. The unit cells have a 2.8° bending magnet, while the matching cells at the ends of the achromat have a 2° bending magnet. A schematic view of one achromat is shown in Fig.1.
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Fig 1. Structure of one achromat of storage ring lattice with the emittance of 90 pm-rad
77
The matching cells contain dedicated quadrupole doublets in order to match the achromat optics to the ID in the straight sections. Since the vertical focusing is performed by the gradient dipoles, dedicated
78 79
quadrupoles are, apart from ID matching; only required for horizontal focusing. As mentioned in [2], the emittance of an MBA achromatic cell is mainly determined by the bending angles of the internal unit cells. The lattice functions of one-unit cell are shown in Fig. 2. The specification of different elements in the unit cell are listed in Table 1. Considering the selected phase advance between focusing quadrupoles and gradient dipole in the unit cell, the corresponding emittance is 97 pm rad. On the other hand, the emittance of matching cell is 63 pm rad. The lattice functions of matching cell is depicted in Fig.3. The specification of different elements in the matching cell is presented in Table 2. Matching the lattice functions between the unit and matching cells define the emittance of whole lattice. The emittance of lattice after fine matching is 90 pm rad. Fig. 4 shows the lattice functions for one super period of the lattice. Since there are similarities between the lattice structure of designed lattice and MAX IV existing storage ring, in Table 3, we have made comparison between the main parameters of two rings. The parameters listed for MAX IV are from reference [3].
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Fig 2. The arrangement of magnets and lattice functions in one-unit cell of an achromat. The blue, red and green blocks indicate dipole, quadrupole and sextupole magnets respectively. The unit cell starts and ends at the half of sextupole.
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Table 1. The specification of different elements in the unit cell presented in Fig.2.
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name S4 D1 Q3 D2 S5 D3 BD D3 S5 D2 Q3 D1 S4
type Sextupole Drift Quadrupole Drift Sextupole Drift Dipole Drift Sextupole Drift Quadrupole Drift Sextupole
Length (m) 0.125 0.1275 0.20 0.2335 0.15 0.26 1.10 0.26 0.15 0.2335 0.20 0.1275 0.125
B0 [T] 0.44 -
B' [T/m] 38.20 -8.48 38.20 -
B'' [T/m2] 453.18 -507.83 -507.83 453.18 99
Fig 3. The arrangement of magnets and lattice functions in the matching cell of an achromat. The blue, red and green blocks indicate dipole, quadrupole and sextupole magnets respectively. The matching cell starts at the beginning of long drift and ends at the half of sextupole.
100 101 102 103
Table 2. The specification of different elements in the matching cell presented in Fig.3. name D_ID S1 D1
type Drift Sextupole Drift
Length (m) 3.057 0.10 0.10
B0 [T] -
B' [T/m] -
B'' [T/m2] 131.90 -
104
Q1 D2 S2 D3 Q2 D4 BD D5 S3 D6 Q3 D7 S4
Quadrupole Drift Sextupole Drift Quadrupole Drift Dipole Drift Sextupole Drift Quadrupole Drift Sextupole
0.25 0.13 0.10 0.13 0.20 0.36 0.8 0.2483 0.2 0.12 0.25 0.10 0.125
0.44 -
34.94 25.68 -8.48 38.61 -
-108.68 -228.56 453.18 105 106 107
Fig 4. The lattice functions in one super period of storage ring lattice.
108 109 Table 3. Comparison between the main parameters of designed lattice and MAX IV existing storage ring Parameter
Unit
Designed lattice
MAX IV
Beam energy
GeV
3
3
pm-rad
90
320
Circumference
m
576
528
Lattice structure
-
7BA
7 BA
-/m
20/6.0
20/5.0
Natural energy spread
-
6.08×10-4
7.7×10-4
Momentum compaction factor
-
1.19×10-4
3.07×10-4
MeV
0.317
0.360
)
-
58.76/13.24
42.20/14.28
Natural chromaticity (H/V)
-
-121.23/-80.98
-49.8/-43.9
Corrected Chromaticity
-
+1/+1
+1/+1
Horizontal damping partition Jx
-
1.46
1.86
MHz
100
100
µm
38.56/1.71
68.94/3.23
Natural horizontal beam emittance
Number/length of straight sections
Radiation loss per turn (bare lattice) Tune (
,
RF frequency Beam size at straight section (H/V)
110 111
Beam divergence at straight section (H/V)
µrad
2.31/0.52
4.60/0.98
Maximum beam current
mA
400
500 112 113
In order to evaluate the radiation property of designed lattice, we have calculated the brilliance of radiated photons from typical Insertion Devices (ID). The specifications of IDs inserted in the lattice are given in Table 4 [8]. Fig.5 shows the brilliance of synchrotron radiation calculated by SPECTRA [9] at 400 mA current.
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Table 4. Specifications of IDs used in the brilliance calculation [8].
120 121
Parameter
SCW
EPU
IVU
20
130
140
Period length (cm)
6
8
1.6
Magnetic field (T)
3.50/0.00
0.90/0.50
0.85/0.00
Length of ID (m)
1.2
2.4
2.24
K parameter
19.61/0.000
6.725/3.736
1.270/0.000
Number periods
of
ID
122 Fig 5. The brilliance of designed lattice for some typical IDs and low field dipole. The beam current is considered 400 mA.
123 124 125
3. Nonlinear optics
126
Analysis of the charged particle dynamics in the presence of nonlinearities is generally tackled via Hamiltonian perturbation approach [10, 11]. The Hamiltonian of the particle in the transverse plane of the storage ring lattice is decomposed into a series of different orders, corresponding with different resonance driving terms. The resonance driving terms are suppressed by properly optimizing the harmonic sextupole strength. This approach has been extensively applied in various electron storage rings. [12–15].
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Hamiltonian, which describes particle motion in the transverse plane up to third order in co-ordinates, consists of five geometric resonance driving terms. this terms vary linearly with integrated strength of the sextupoles. The sensitivity of dynamic aperture changes with these terms and therefore, finding the strength of these terms for obtaining good dynamic aperture is difficult [14,15]. This is a major
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obstacle in the initial choice of weight factors for different resonance terms to optimize the strength of the harmonic sextupoles. Selection of weight factor to different resonance driving terms to obtain the suitable sextupole strength for enlarging the dynamic aperture is based on the experience and also demands extensive iterative studies. We used OPA [16] code to assign weight factors to different resonance driving terms to enlarge the dynamic aperture. Initial weight factors to different driving term are assigned based on lattice sensitivity against different driving terms with sextupole strength and frequency map analysis. Totally 11 families of sextupoles have been employed for nonlinear optimization. Two families of sextupoles are devoted for chromaticity correction. In order to make the strength of chromatic sextupoles smaller, we employee the family of sextupoles for this purpose in which the dispersion function has the maximum values. The chromaticity is set to (+1,+1) in both horizontal and vertical directions. The variation of tune with energy after chromaticity correction up to ±5% energy deviation is depicted in Fig.6. The corresponding chromatic tune footprint in the resonance diagram is shown in Fig.7. In order to calculate the chromatic tune footprint, the particles are tracked in different energy deviation step (0.5%) for 2048 turns. The chromatic tune footprint is relatively small and does not cross the dangerous resonance line except for -4.5% which is close the intersection point of half-integer and third order resonance lines.
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Fig 6. The variation of horizontal and vertical tune with energy. The chromaticity is corrected to (+1,+1) in both horizontal and vertical direction. The dashed and solid curves shows the vertical and horizontal tune shift vertically. The OPA code has been utilized for the calculation.
153 154 155
Fig.7. The chromatic tune footprint up to ±4.5% energy deviation. The working point of the lattice (58.76,13.24) is indicated by square. The plus sign shows the result of tracking (2048 turns) for each energy deviation step (0.5%). The OPA code has been employed for the calculation.
156 157 158
For suppressing resonance deriving terms, we utilize nine family of sextupoles. Harmonic sextupole strength optimization was performed using the computer codes OPA [16]. Harmonic sextupoles optimization results in relatively small tune shift with amplitude. Fig.8 and Fig.9 shows the variation of tunes with horizontal and vertical amplitude excursion.
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Fig.8. The variation of tune with amplitude for horizontal excursion. The dashed and solid curves shows the vertical and horizontal tune shift with horizontal amplitude respectively.
163 164
Fig.9. The variation of tune with amplitude for vertical excursion. The dashed and solid curves shows the vertical and horizontal tune shift with vertical amplitude respectively.
165 166
After optimizing chromatic and resonance driving terms, we evaluate the dynamic aperture of designed lattice. Fig.10 shows the results of dynamic aperture tracking for different energy deviations in the middle of one straight section. The dynamic aperture has been calculated by Elegant code [17].
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Fig.10. The results of dynamic aperture tracking for different energy deviation in the middle of one straight section. The dynamic aperture has been calculated by Elegant code, and the particles have been tracked for 2000 turns.
170 171 172
.
173
In order to confirm the results of dynamic aperture tracking and explore in detail the nonlinear behavior of electrons, we conducted frequency map analysis [18]. Fig.11 shows the diffusion map for bare lattice in the middle of one straight section. The color bar shows the tune diffusion
174 175 176
(
√
). In order to calculated frequency map analysis, we have employed Elegant code,
178
and tracked the on energy electrons for 2000 turns. Fig.11. The diffusion map of on energy electrons in the middle of on straight section. The color bar √
shows the tune diffusion
177
In the area with red color the electrons are more stable
179 180
while in the blue region the electrons behave chaotically.
181
According to the diffusion map of on energy electrons, by increasing the amplitude, particles cross many resonance lines. Crossing the different resonance lines can be seen obviously from the tune footprint with amplitude [See Fig.12]. To calculate tune footprint, we tracked the particles for 2048 turns in different amplitude with the step of 0.5 mm. The calculation has been done by OPA code. Particles after crossing the half integer resonance line in the amplitude behave more chaotically and finally lost at . The same story is correct for positive horizontal direction. The particles cross the half integer resonance line at and eventually lost at .
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Fig. 12. The tune footprint with amplitude. The electrons have been tracked for 2048 turns for each steps (0.5 mm) of amplitude. The working point is indicated by a filled black circle. The OPA code utilized for calculation.
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4. Errors studying
194
In this section, we describe the tolerance of lattice to misalignment and field errors. Misalignment of different magnets and beam position monitors (BPM) will affect the closed orbit and consequently the quality of emitted photons will be degraded by closed orbit distortion. In order to study error sensitivity of the electron orbits, different types of the expected misalignments and field errors were imposed randomly in the lattice. The value of errors are given in Table 5. The numbers for misalignment are similar the errors given in reference [19]. The error distribution is Gaussian and it is truncated at .
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Table 5. RMS misalignment and field errors of different magnets and BPMs. , and relative field errors of dipoles, quadrupoles and sextupoles respectively.
are the
error source
BPM
Dipole ( )
quadrupole ( )
Sextupole (
Relative Field Error Transverse displacement ( Tilting error(
)
)
)
±60
±60
±60
±40
±100
±100
±100
±100 205
We employed 280 horizontal and vertical corrector magnets to correct the distorted orbit. The correctors are considered as additional coils in some family of sextupoles. In order to monitor the electron orbits, we have used 280 BPMs. Based on singular value decomposition (SVD) and employing the Elegant code, the orbit of electron is corrected to acceptable values. Fig.13 and Fig.14 shows the closed orbit distortion before and after orbit correction in x and y directions respectively.
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The values of correctors kick in the x and y directions are depicted in the Fig.15. The calculation is based on 100 machine samples.
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Fig 13. Closed orbit distortion for different seeds before correction in horizontal (left) and vertical (right) directions
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Fig 14. Closed orbit distortion for different seeds after correction in horizontal (left) and vertical (right) directions. For the vertical direction, the orbit is corrected to 100 µm and for horizontal direction, the orbit is corrected to 25 µm.
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Fig 15. The kick of correctors for different seeds in horizontal (left) and vertical (right) directions. The required kick for the correctors of both horizontal and vertical directions is 100 µrad.
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It is expected that misalignment and multipole errors significantly shrink the dynamic aperture and consequently decrease the lifetime. In order to study the effect of multipole errors, we utilized the multipole errors of different magnets giving in reference [20]. Fig.16 shows the result of dynamic aperture tracking of on energy electrons for 2000 turns in the presence of misalignment and multipole errors. We have used AT [21] code to calculate the effect of errors on the dynamic aperture. Dynamic aperture calculation has been conducted after applying closed orbit correction. Although shrinking dynamic aperture due to residual closed orbit distortion and multipole errors is significant, it is still sufficient for off-axis injection.
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Fig 16. The effect of misalignment and multipole errors on the dynamic aperture. By employing AT code, the tracking has been done for 2000 turn and 10 different seeds. The stars sign indicate the boundary of dynamic aperture for different seeds. The regions without any star marker show the net effect of multipole errors on dynamic aperture shrinkage.
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5. Life time issues
235
The demanding design feature of diffraction limited storage rings, ultra-low emittance and small gap undulator vacuum chambers, causes Touscheck scattering and gas scattering to play a major limitation role for beam lifetime [22]. We calculate the Touscheck lifetime based on Piwinski formalism [23]. By employing Elegant code, a 6-D tracking procedure is conducted to determine the energy acceptance of lattice. The nonlinear synchrotron oscillation due to large second- order momentum compaction factor is included in the energy acceptance calculations. A small vertical ID with 5 mm full gap is imposed in the tracking procedure. Fig.17 shows the result of momentum acceptance tracking. Using the result of momentum acceptance tracking, the Touscheck and inelastic gas scattering lifetime are calculated. In order to calculate Touscheck lifetime, we assume that 80% of 192 buckets in the ring are filled with 5 nC charge per bunch. The elastic gas scattering depends on the ring physical acceptance which in our lattice it is limited by the low gap in-vacuum undulators with gap of 5 mm. The variation of vacuum and total lifetime in different pressures are shown in Fig.18. At the standard pressure of 1.5 nTorr, the total life time is about 7 hours. It is obvious from Fig.18 that
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at the pressures larger than 1.6 nTorr, unlike to third generation light sources, the vacuum lifetime dominant over Touscheck lifetime. For the MAX IV storage ring [24], the total
lifetime is above 10 hours. In the MAX IV case, the gas scattering (elastic) lifetime dominant over Touscheck lifetime as well.
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Fig 17. The result of momentum acceptance tracking with 1.1 MV RF peak voltages. The blue curves show lattice momentum acceptance and red curves indicates the RF acceptance. With the cavity peak voltage of 1.1 MV, the RF acceptance is 5.6%. Elegant code has been utilized for the tracking calculation.
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Fig 18. The total and vacuum (elastic and inelastic gas scattering) lifetime versus pressure. The red, blue and green curves indicate Touscheck, vacuum and total lifetime respectively.
260 261 262
6. Intra beam scattering
263
Intra beam scattering (IBS) limits the achievable horizontal emittance in the storage rings. It acts more power full in the storage rings with ultra-low emittances [19]. By employing 6D tracking code, Elegant [17], we have calculated the effect of IBS on the emittance and energy spread growth. Fig. 19 shows the variation of emittance and energy spread versus beam current. In the consequence of IBS effect, the zero current emittance (90 pm-rad) increased to 262 pm-rad.
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Fig 19. The effect of IBS on equilibrium emittance and energy spread growth in different currents. In the consequence of IBS effect, the zero current emittance (90 pm-rad) increased to 262 pm-rad.
269 270
Employing Landau cavity is an effective way to mitigate the unwanted effects of IBS on the emittance and energy spread growth. By using Elegant code, we have simulated the effect of Landau cavity on the horizontal emittance and energy spread in different bunch lengthening factor. The bunchlengthening factor is the ratio of bunch length after employing Landau cavity to zero current bunch length. The zero current bunch length is 5.86 mm. The variation of horizontal emittance and energy spread versus different bunch lengthening factor are shown in Fig.20. The calculations are conducted at 400mA. We observed that after employing Landau cavity and making the bunch length 5.5 times of the zero current bunch length, it is possible to bring back the horizontal emittance to 169 pm-rad. The zero current natural emittance is 90 pm-rad. In order to make a comparison between the results of IBS effect on our proposed lattice parameter and MAX IV storage ring, we bring here the numbers giving in the references [3, 24] for the effect of IBS on emittance growth. The zero current emittance of MAX IV storage ring ( 320 pm-rad) in the consequence of IBS blowup increases to 466 pm-rad. After applying Landau cavity, the emittance decreased to 364 pm-rad. Emittance blowup from IBS with and without bunch lengthening from Landau cavity is calculated assuming 500 mA stored current.
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Fig 20. Horizontal emittance and energy spread as a function of bunch lengthening factor via Landau cavity tuning at 400 mA beam current. The red and blue lines show energy spread and horizontal emittance respectively.
286 287 288 289
7. Instabilities and impedance
290
It is expected that instabilities emerging from collective effects become more significant in the ultralow emittance storage rings. The self-induced wakefields due to the resistive wall impedance, chamber geometric effects, and coherent synchrotron radiation (CSR) will result in instabilities at some threshold currents. The small momentum compaction factor, short bunch length and small synchrotron tune are the common characteristic of any low emittance storage rings [25]. On the other hand, because of strong sextupoles and quadrupole strength, there is no choice to use small aperture magnets in low emittance rings. All mentioned properties make the low emittance storage rings more vulnerable to instabilities arisen from wake fields. In this section, we will briefly evaluate the tolerance of designed lattice against the single-bunch microwave instability excited by coherent synchrotron radiation (CSR) and the single-bunch transverse mode coupling instability due to wall resistance.
291 292 293 294 295 296 297 298 299 300 301
In order to estimate the threshold current for single-bunch microwave instability, We consider the CSR wakefield generated by an electron moving on a circular orbit of radius in the middle of two (perfectly conducting) parallel plates that are separated by a distance 2h. For the CSR wake, the beam dynamics depend only on two dimensionless parameters, the shielding parameter and scattering parameter [26]
302 303 304 305 306
(1)
307
Where are the number of electrons, synchrotron tune, energy spread and bunch length respectively. The threshold associated with CSR is given by
308 309
(
(2)
310
supposing an elliptical vacuum chamber in the arcs with cross section of (20mm,13mm) , and , we find the shielding parameter 36.1399 and threshold number of electrons in each bunch . The corresponding threshold current for 154 number of bunches is which is close to maximum design current of ring. Small aperture magnets and low gap insertion devices in low emittance storage rings make instabilities excited with resistance wall more important. Transverse mode coupling instability (TMCI) is one of the instabilities which dominantly excited by resistance wall. To calculate the threshold of this instability in our lattice, we use the similar calculations conducted in references [26, 27]. The kick factor for a Gaussian bunch pathing through a circular vacuum chamber is given by
311 312 313 314 315 316 317 318 319 320
,
)
(
)
√
(3)
321
the bunch length. The
322 323
(4)
324
Where is the circumference of storage ring. In the dominator, the sum is done on the different segments of the lattice. We suppose that the whole ring is composed of two parts, the arcs and low gap undulators. The straight sections are filled with 17 low gap undulators. The related parameters for calculating are given in table 6. In the multibunche mode, threshold current is . For
325 326 327 328
With the radius of vacuum chamber, TMCI threshold current is given by ( ) ∑
,
the conductivity and
154 bunchs, we find the threshold current about 1.87 A where it is much larger than the designed current of ring.
329 330 331
Table 6. The different parameters (length, shape, radius, material and Betateron oscillation) in the arcs and straight sections utilized for current threshold calculation Shape
(
)[mm,mm]
332 333
[ ]
Type
Length[m]
Metal
Arcs
499.5
Elliptical
(20,13)
Al
8.8
Low gap undulators
76.5
Elliptical
(20,2.5)
Cu
3.3 334
8. Conclusion
335
In this paper, we presented the lattice design and beam dynamic calculations for an ultra-low emittance storage ring with intermediate energy and circumference. According to design results, unlikely the other proposed diffraction limited storage rings, circumference of designed ring is significantly small and it is possible to push natural emittance more down by using more complicate concepts for magnet design and construction. The total lifetime without Landau cavity is about 7 hours and the dynamic aperture in presence of misalignment and field errors is relatively enough for conventional off axis injection. The effect of IBS on emittance and energy spread growth was calculated. The results of employing Landau cavity to alleviate unwanted effect of IBS was also presented. We derived analytically the threshold current for single bunch instabilities and according to the results, it is expected that these instabilities do not have big impact on the beam stability and quality.
336 337 338 339 340 341 342 343 344 345 346
Acknowledgements
347
The authors would like to especially thank H.Wiedemann for his several helpful comments on this research. This paper is supported by Iranian Light Source Facility.
348 349
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350
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[14] Laurent S. Nadolski, Methods and tools to simulate, Optimize and analyze non- linear Dynamics in low Emittance Storage Rings, in: ICFA Beam Dynamics Newsletter, 57, (2012).
376 377
[15] BNL (Conceptual Design Report), National Synchrotron Light Source-II, BNL, USA, (2006).
378
[16] A.Streun, OPA Version 3.39,PSI,Switzerland, (2012).
379
[17] M. Borland, Elegant: a flexible sdds-compliant code for accelerator simulation, Advanced photon Source, Argonne National Laboratory, USA. Report No. LS-287, (2000).
380 381
[18] C. Steier, et al, Measuring and optimizing the momentum aperture in a particle accelerator, PHYSICAL REVIEW E, VOLUME 65, 056506, (2001).
382 383
[19]. S. C. Leemann, et al, Beam dynamics and expected performance of Sweden’s new storage-ring light source: MAX IV, Physical Review Special Topics - Accelerators and Beams 12, 120701 (2009).
384 385
[20] Sirius project, https://wiki-sirius.lnls.br/mediawiki/index.php/Main_Page.
386
[21] A. Terebilo, “Accelerator Modeling with MATLAB Accelerator Toolbox”, in Proc. PAC’01, p3202-3.
387 388
[22] M.Borland, et al, Lattice design challenges for fourth-generation storage-ring light sources, Journal of Synchrotron Radiation, (2014).
389 390
[23] A.Piwinski, The Touscheck Effect in strong focusing storage rings, DESY 98-179, November (1998).
391 392
[24] S. C. Leemann, Interplay of Touschek scattering, intrabeam scattering, and rf cavities in ultralow-emittance storage rings, Physical review special topics - accelerators and beams 17, 050705 (2014)
393 394 395
[25] R. Nagaoka and K. L. F. Bane, Collective effects in a diffraction-limited storage ring, Journal of Synchrotron Radiation, 2014.
396 397
[26] Y. Cai, et al, Ultimate storage ring based on fourth-order geometric achromats, Physical review special topics - accelerators and beams 15, 054002 (2012)
398 399
[27] S. Krinsky, Simulation of Transverse Instabilities in the NSLS II Storage Ring, Brookhaven Report No. 75019-2005-IR, (2005).
400 401
402 Figure 1.
403 404
405 Figure 2.
406 407 408 409
410
Figure 3.
411
412
Figure 4.
413
EPU 1st harmonic
1E+22 Brilliance (Photons/[sec*mm^2 mrad^2 0.1% BW])
EPU 3rd harmonic
1E+21
EPU 5th harmonic IVU 1st harmonic
1E+20
IVU 3rd harmonic IVU 5th harmonic
1E+19
SCW Bending
1E+18 1E+17 1E+16 1E+15 1E+14 0.1
1
Energy (keV)
Figure 5.
10
100
414
415
416
Figure 6.
417
418
419
420
Figure.7.
58.8
13.36
𝜈_x
𝜈_x
13.34
𝜈_𝑦
58.7
13.32
58.65
13.3
58.6
13.28
58.55
13.26
58.5
13.24
58.45
𝜈_𝑦
58.75
13.22 0
2
4
6
Δx(mm)
8
10
421
Figure.8.
422
423
424
58.92
13.55
58.9
𝜈_𝑥
13.5
58.88
𝜈_𝑦
13.45 13.4 13.35
58.84
13.3
58.82
𝜈_𝑦
𝜈_x
58.86
13.25
58.8
13.2
58.78
13.15
58.76
13.1
58.74
13.05 0
1
2
3
Δy (mm)
Figure.9.
4
5
425
426
427
428
Figure.10.
429
430
Figure.11.
431
432
Figure. 12.
433
434
435
Figure 13.
436 437
438
Figure 14.
439 440
441
Figure 15.
442 443
444 Figure 16.
445 446
447 Figure 17.
448 449 450
451 Figure 18.
452
453
454 455
Figure 19.
456 280
0.0012 Horizontal Emittance
260
Energy spread
240
ε_x(pm-rad)
220 200 0.0008 180 160
Energy spread
0.001
0.0006
140 120 100
0.0004 1
2
3
4
5
Bunch lengthening factor 457 Figure 20.
458 459 460 461 462 463 464
PII: DOI: Reference:
S0168-9002(17)30472-2 http://dx.doi.org/10.1016/j.nima.2017.04.021 NIMA59809
To appear in: Nuclear Inst. and Methods in Physics Research, A Received date: 17 December 2016 Revised date: 13 April 2017 Accepted date: 14 April 2017 Cite this article as: E. Ahmadi, S.M. Jazayeri, J. Rahighi and M. Mollabashi, Lattice Design for an ultra-low Emittance Synchrotron Light Source, Nuclear Inst. and Methods in Physics Research, A, http://dx.doi.org/10.1016/j.nima.2017.04.021 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Lattice Design for an ultra-low Emittance Synchrotron Light Source
1 2
E. Ahmadia*, S.M. Jazayeria, J. Rahighib, M. Mollabashia
3 4
a
b
Department of Physics, Iran University of Science and Technology, Tehran, Iran
5
Iranian Light Source Facility (ILSF), Institute for Research in Fundamental Sciences (IPM), Tehran, Iran
6 7 8
*
9
Corresponding author. [email protected]
10
Abstract
11
In this paper, we investigate a lattice design for an ultra-low emittance storage ring with intermediate energy of 3 GeV and circumference of 576 m. We present the design results for a seven-band achromat lattice with the natural emittance of 90 pm-rad. The base line is based on 20 straight sections with the length of 6 m. The minimum distance between different magnets is 10 cm and it has been tried to keep the lattice structure as simple as possible. The brilliance of radiated synchrotron radiation from a 2.24 m In-vacuum undulator installed in the lattice is [ ] . We will address the challenging points of ⁄ ⁄ ⁄ ⁄ ⁄ designed ultra-low emittance storage ring lattice such as strong nonlinearity and collective effects. The dynamic aperture of the lattice is comparatively sufficient for off-axis injection and the lifetime is relatively large without using any harmonic cavity. Since the proposed lattice structure in some aspects is similar to MAX IV storage ring, we will make some comparison between two lattices in different sections.
12 13 14 15 16 17 18 19 20 21 22 23 24 25
Keywords: Storage ring, Ultra-low emittance, MBA lattice, Dynamic aperture.
26
1. Introduction
27
Storage rings are the principle sources of high brightness photon beams driving the majority of x-ray science experiment in the world today. There has been remarkable progress in the developing these sources over the last two decades. Existing third generation light sources continue to upgrade their capabilities, while new light sources storage rings coming on line with ever-improved performances. Since one of the most important factors for the users of a synchrotron radiation is brightness which in the storage ring is determined by the emittance, our aim in this paper is to investigate and present the design results for a lattice in a 3GeV range that can provide high brilliance photons near soft X-ray diffraction limited. The diffraction limited light sources performance for a given wave length is met
28 29 30 31 32 33 34 35
when
⁄(
). Accordingly, for hard X-ray radiation at 0.1 nm,
while for soft X-
36
ray at 1 nm the diffraction limit is 80 pm rad. For the best recently existing storage rings, the horizontal electron beam emittance are nearly three orders of magnitude larger from diffraction limited for soft X-ray.
37 38 39
One simple and robust method to achieve ultralow emittance is the use of a multibend achromat (MBA) lattice [1, 2]. The MBA exploits the inverse cubic dependence of emittance on the number of bending magnets. By choosing a very small bending angle per dipole, the emittance can be dramatically reduced. Introducing a vertically focusing gradient in the dipoles causes more reduction in the emittance (the emittance scales inversely with the horizontal damping partition Jx) while the dispersion is limited to small values without requiring any extra space for vertically focusing quadrupoles. This concept has been employed in the designing the storage ring of several facilities such as of MAX IV [3] and Sirius [4]. Hybrid multiband achromat [HMBA] is another effective way to decrease emittance. It was employed for the first time in the upgrade plan of ESRF [5]. HMBA concept have some advantages relative to MBA structure, which make it a suitable candidate for the future 4th generation light sources [6]. Our designed lattice is based on MBA concept. In order to further push down the horizontal emittance, we resort to other well-known methods. For example, we used shorter dipoles at the achromat ends where the dispersion function is matched to zero in the straight section. The mentioned criteria for designing a low emittance MBA lattice can be find in reference [2]. Considering the mentioned criteria to reach the ultra-low emittance, a seven-bend achromt lattice was selected. By this selection, the circumference of storage ring is 576 m and its emittance is 90 pm-rad. The symmetry of lattice is 20 and the length of straight sections is 6 m. In this paper, we will address the challenging points of the designed ultra-low emittance lattice. Strong nonlinearity is a most demanding aspect which drastically limited the momentum acceptance of lattice and causes short life time and small dynamic aperture. Another challenging point in the ultra-low emittance rings is short electron bunches which is in consequence of small momentum compaction factor [7]. The short bunches result in strong collective effects and their corresponding instabilities.
40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61
In Section 2, we present the lattice design, linear optics and radiation property of designed lattice. In this section, we will make a comparison among the parameters of designed lattice and MAX IV existing storage ring lattice. In Section 3, we will address the nonlinear optimization of designed lattice. Section 4 explains the tolerance of lattice to misalign and field errors. In this section, we present the results of closed orbit correction and the effects of machine errors on the dynamic aperture. Section 5 includes the lifetime and energy acceptance calculations. Section 6 is devoted to Intra Beam Scattering (IBS) calculation. In this section we explain how to mitigate undesired effect of IBS. And finally, in section 7, we evaluate the threshold current of different single bunch instabilities produced in the storage ring.
62 63 64 65 66 67 68 69 70 71
2. Lattice design, linear optics and radiation properties
72
The storage ring lattice consists of 20 seven-bend achromats separated by 6 m straight sections for IDs. Each of the achromats consists of seven unit cells and two matching cells. The unit cells have a 2.8° bending magnet, while the matching cells at the ends of the achromat have a 2° bending magnet. A schematic view of one achromat is shown in Fig.1.
73 74 75 76
Fig 1. Structure of one achromat of storage ring lattice with the emittance of 90 pm-rad
77
The matching cells contain dedicated quadrupole doublets in order to match the achromat optics to the ID in the straight sections. Since the vertical focusing is performed by the gradient dipoles, dedicated
78 79
quadrupoles are, apart from ID matching; only required for horizontal focusing. As mentioned in [2], the emittance of an MBA achromatic cell is mainly determined by the bending angles of the internal unit cells. The lattice functions of one-unit cell are shown in Fig. 2. The specification of different elements in the unit cell are listed in Table 1. Considering the selected phase advance between focusing quadrupoles and gradient dipole in the unit cell, the corresponding emittance is 97 pm rad. On the other hand, the emittance of matching cell is 63 pm rad. The lattice functions of matching cell is depicted in Fig.3. The specification of different elements in the matching cell is presented in Table 2. Matching the lattice functions between the unit and matching cells define the emittance of whole lattice. The emittance of lattice after fine matching is 90 pm rad. Fig. 4 shows the lattice functions for one super period of the lattice. Since there are similarities between the lattice structure of designed lattice and MAX IV existing storage ring, in Table 3, we have made comparison between the main parameters of two rings. The parameters listed for MAX IV are from reference [3].
80 81 82 83 84 85 86 87 88 89 90 91 92
Fig 2. The arrangement of magnets and lattice functions in one-unit cell of an achromat. The blue, red and green blocks indicate dipole, quadrupole and sextupole magnets respectively. The unit cell starts and ends at the half of sextupole.
93 94 95 96
Table 1. The specification of different elements in the unit cell presented in Fig.2.
97 98
name S4 D1 Q3 D2 S5 D3 BD D3 S5 D2 Q3 D1 S4
type Sextupole Drift Quadrupole Drift Sextupole Drift Dipole Drift Sextupole Drift Quadrupole Drift Sextupole
Length (m) 0.125 0.1275 0.20 0.2335 0.15 0.26 1.10 0.26 0.15 0.2335 0.20 0.1275 0.125
B0 [T] 0.44 -
B' [T/m] 38.20 -8.48 38.20 -
B'' [T/m2] 453.18 -507.83 -507.83 453.18 99
Fig 3. The arrangement of magnets and lattice functions in the matching cell of an achromat. The blue, red and green blocks indicate dipole, quadrupole and sextupole magnets respectively. The matching cell starts at the beginning of long drift and ends at the half of sextupole.
100 101 102 103
Table 2. The specification of different elements in the matching cell presented in Fig.3. name D_ID S1 D1
type Drift Sextupole Drift
Length (m) 3.057 0.10 0.10
B0 [T] -
B' [T/m] -
B'' [T/m2] 131.90 -
104
Q1 D2 S2 D3 Q2 D4 BD D5 S3 D6 Q3 D7 S4
Quadrupole Drift Sextupole Drift Quadrupole Drift Dipole Drift Sextupole Drift Quadrupole Drift Sextupole
0.25 0.13 0.10 0.13 0.20 0.36 0.8 0.2483 0.2 0.12 0.25 0.10 0.125
0.44 -
34.94 25.68 -8.48 38.61 -
-108.68 -228.56 453.18 105 106 107
Fig 4. The lattice functions in one super period of storage ring lattice.
108 109 Table 3. Comparison between the main parameters of designed lattice and MAX IV existing storage ring Parameter
Unit
Designed lattice
MAX IV
Beam energy
GeV
3
3
pm-rad
90
320
Circumference
m
576
528
Lattice structure
-
7BA
7 BA
-/m
20/6.0
20/5.0
Natural energy spread
-
6.08×10-4
7.7×10-4
Momentum compaction factor
-
1.19×10-4
3.07×10-4
MeV
0.317
0.360
)
-
58.76/13.24
42.20/14.28
Natural chromaticity (H/V)
-
-121.23/-80.98
-49.8/-43.9
Corrected Chromaticity
-
+1/+1
+1/+1
Horizontal damping partition Jx
-
1.46
1.86
MHz
100
100
µm
38.56/1.71
68.94/3.23
Natural horizontal beam emittance
Number/length of straight sections
Radiation loss per turn (bare lattice) Tune (
,
RF frequency Beam size at straight section (H/V)
110 111
Beam divergence at straight section (H/V)
µrad
2.31/0.52
4.60/0.98
Maximum beam current
mA
400
500 112 113
In order to evaluate the radiation property of designed lattice, we have calculated the brilliance of radiated photons from typical Insertion Devices (ID). The specifications of IDs inserted in the lattice are given in Table 4 [8]. Fig.5 shows the brilliance of synchrotron radiation calculated by SPECTRA [9] at 400 mA current.
114 115 116 117 118 119
Table 4. Specifications of IDs used in the brilliance calculation [8].
120 121
Parameter
SCW
EPU
IVU
20
130
140
Period length (cm)
6
8
1.6
Magnetic field (T)
3.50/0.00
0.90/0.50
0.85/0.00
Length of ID (m)
1.2
2.4
2.24
K parameter
19.61/0.000
6.725/3.736
1.270/0.000
Number periods
of
ID
122 Fig 5. The brilliance of designed lattice for some typical IDs and low field dipole. The beam current is considered 400 mA.
123 124 125
3. Nonlinear optics
126
Analysis of the charged particle dynamics in the presence of nonlinearities is generally tackled via Hamiltonian perturbation approach [10, 11]. The Hamiltonian of the particle in the transverse plane of the storage ring lattice is decomposed into a series of different orders, corresponding with different resonance driving terms. The resonance driving terms are suppressed by properly optimizing the harmonic sextupole strength. This approach has been extensively applied in various electron storage rings. [12–15].
127 128 129 130 131 132
Hamiltonian, which describes particle motion in the transverse plane up to third order in co-ordinates, consists of five geometric resonance driving terms. this terms vary linearly with integrated strength of the sextupoles. The sensitivity of dynamic aperture changes with these terms and therefore, finding the strength of these terms for obtaining good dynamic aperture is difficult [14,15]. This is a major
133 134 135 136
obstacle in the initial choice of weight factors for different resonance terms to optimize the strength of the harmonic sextupoles. Selection of weight factor to different resonance driving terms to obtain the suitable sextupole strength for enlarging the dynamic aperture is based on the experience and also demands extensive iterative studies. We used OPA [16] code to assign weight factors to different resonance driving terms to enlarge the dynamic aperture. Initial weight factors to different driving term are assigned based on lattice sensitivity against different driving terms with sextupole strength and frequency map analysis. Totally 11 families of sextupoles have been employed for nonlinear optimization. Two families of sextupoles are devoted for chromaticity correction. In order to make the strength of chromatic sextupoles smaller, we employee the family of sextupoles for this purpose in which the dispersion function has the maximum values. The chromaticity is set to (+1,+1) in both horizontal and vertical directions. The variation of tune with energy after chromaticity correction up to ±5% energy deviation is depicted in Fig.6. The corresponding chromatic tune footprint in the resonance diagram is shown in Fig.7. In order to calculate the chromatic tune footprint, the particles are tracked in different energy deviation step (0.5%) for 2048 turns. The chromatic tune footprint is relatively small and does not cross the dangerous resonance line except for -4.5% which is close the intersection point of half-integer and third order resonance lines.
137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152
Fig 6. The variation of horizontal and vertical tune with energy. The chromaticity is corrected to (+1,+1) in both horizontal and vertical direction. The dashed and solid curves shows the vertical and horizontal tune shift vertically. The OPA code has been utilized for the calculation.
153 154 155
Fig.7. The chromatic tune footprint up to ±4.5% energy deviation. The working point of the lattice (58.76,13.24) is indicated by square. The plus sign shows the result of tracking (2048 turns) for each energy deviation step (0.5%). The OPA code has been employed for the calculation.
156 157 158
For suppressing resonance deriving terms, we utilize nine family of sextupoles. Harmonic sextupole strength optimization was performed using the computer codes OPA [16]. Harmonic sextupoles optimization results in relatively small tune shift with amplitude. Fig.8 and Fig.9 shows the variation of tunes with horizontal and vertical amplitude excursion.
159 160 161 162
Fig.8. The variation of tune with amplitude for horizontal excursion. The dashed and solid curves shows the vertical and horizontal tune shift with horizontal amplitude respectively.
163 164
Fig.9. The variation of tune with amplitude for vertical excursion. The dashed and solid curves shows the vertical and horizontal tune shift with vertical amplitude respectively.
165 166
After optimizing chromatic and resonance driving terms, we evaluate the dynamic aperture of designed lattice. Fig.10 shows the results of dynamic aperture tracking for different energy deviations in the middle of one straight section. The dynamic aperture has been calculated by Elegant code [17].
167 168 169
Fig.10. The results of dynamic aperture tracking for different energy deviation in the middle of one straight section. The dynamic aperture has been calculated by Elegant code, and the particles have been tracked for 2000 turns.
170 171 172
.
173
In order to confirm the results of dynamic aperture tracking and explore in detail the nonlinear behavior of electrons, we conducted frequency map analysis [18]. Fig.11 shows the diffusion map for bare lattice in the middle of one straight section. The color bar shows the tune diffusion
174 175 176
(
√
). In order to calculated frequency map analysis, we have employed Elegant code,
178
and tracked the on energy electrons for 2000 turns. Fig.11. The diffusion map of on energy electrons in the middle of on straight section. The color bar √
shows the tune diffusion
177
In the area with red color the electrons are more stable
179 180
while in the blue region the electrons behave chaotically.
181
According to the diffusion map of on energy electrons, by increasing the amplitude, particles cross many resonance lines. Crossing the different resonance lines can be seen obviously from the tune footprint with amplitude [See Fig.12]. To calculate tune footprint, we tracked the particles for 2048 turns in different amplitude with the step of 0.5 mm. The calculation has been done by OPA code. Particles after crossing the half integer resonance line in the amplitude behave more chaotically and finally lost at . The same story is correct for positive horizontal direction. The particles cross the half integer resonance line at and eventually lost at .
182 183 184 185 186 187 188 189
Fig. 12. The tune footprint with amplitude. The electrons have been tracked for 2048 turns for each steps (0.5 mm) of amplitude. The working point is indicated by a filled black circle. The OPA code utilized for calculation.
190 191 192 193
4. Errors studying
194
In this section, we describe the tolerance of lattice to misalignment and field errors. Misalignment of different magnets and beam position monitors (BPM) will affect the closed orbit and consequently the quality of emitted photons will be degraded by closed orbit distortion. In order to study error sensitivity of the electron orbits, different types of the expected misalignments and field errors were imposed randomly in the lattice. The value of errors are given in Table 5. The numbers for misalignment are similar the errors given in reference [19]. The error distribution is Gaussian and it is truncated at .
195 196 197 198 199 200 201 202 203 204
Table 5. RMS misalignment and field errors of different magnets and BPMs. , and relative field errors of dipoles, quadrupoles and sextupoles respectively.
are the
error source
BPM
Dipole ( )
quadrupole ( )
Sextupole (
Relative Field Error Transverse displacement ( Tilting error(
)
)
)
±60
±60
±60
±40
±100
±100
±100
±100 205
We employed 280 horizontal and vertical corrector magnets to correct the distorted orbit. The correctors are considered as additional coils in some family of sextupoles. In order to monitor the electron orbits, we have used 280 BPMs. Based on singular value decomposition (SVD) and employing the Elegant code, the orbit of electron is corrected to acceptable values. Fig.13 and Fig.14 shows the closed orbit distortion before and after orbit correction in x and y directions respectively.
206 207 208 209 210
The values of correctors kick in the x and y directions are depicted in the Fig.15. The calculation is based on 100 machine samples.
211 212 213
Fig 13. Closed orbit distortion for different seeds before correction in horizontal (left) and vertical (right) directions
214 215 216
Fig 14. Closed orbit distortion for different seeds after correction in horizontal (left) and vertical (right) directions. For the vertical direction, the orbit is corrected to 100 µm and for horizontal direction, the orbit is corrected to 25 µm.
217 218 219
Fig 15. The kick of correctors for different seeds in horizontal (left) and vertical (right) directions. The required kick for the correctors of both horizontal and vertical directions is 100 µrad.
220 221 222
It is expected that misalignment and multipole errors significantly shrink the dynamic aperture and consequently decrease the lifetime. In order to study the effect of multipole errors, we utilized the multipole errors of different magnets giving in reference [20]. Fig.16 shows the result of dynamic aperture tracking of on energy electrons for 2000 turns in the presence of misalignment and multipole errors. We have used AT [21] code to calculate the effect of errors on the dynamic aperture. Dynamic aperture calculation has been conducted after applying closed orbit correction. Although shrinking dynamic aperture due to residual closed orbit distortion and multipole errors is significant, it is still sufficient for off-axis injection.
223 224 225 226 227 228 229 230
Fig 16. The effect of misalignment and multipole errors on the dynamic aperture. By employing AT code, the tracking has been done for 2000 turn and 10 different seeds. The stars sign indicate the boundary of dynamic aperture for different seeds. The regions without any star marker show the net effect of multipole errors on dynamic aperture shrinkage.
231 232 233 234
5. Life time issues
235
The demanding design feature of diffraction limited storage rings, ultra-low emittance and small gap undulator vacuum chambers, causes Touscheck scattering and gas scattering to play a major limitation role for beam lifetime [22]. We calculate the Touscheck lifetime based on Piwinski formalism [23]. By employing Elegant code, a 6-D tracking procedure is conducted to determine the energy acceptance of lattice. The nonlinear synchrotron oscillation due to large second- order momentum compaction factor is included in the energy acceptance calculations. A small vertical ID with 5 mm full gap is imposed in the tracking procedure. Fig.17 shows the result of momentum acceptance tracking. Using the result of momentum acceptance tracking, the Touscheck and inelastic gas scattering lifetime are calculated. In order to calculate Touscheck lifetime, we assume that 80% of 192 buckets in the ring are filled with 5 nC charge per bunch. The elastic gas scattering depends on the ring physical acceptance which in our lattice it is limited by the low gap in-vacuum undulators with gap of 5 mm. The variation of vacuum and total lifetime in different pressures are shown in Fig.18. At the standard pressure of 1.5 nTorr, the total life time is about 7 hours. It is obvious from Fig.18 that
236 237 238 239 240 241 242 243 244 245 246 247 248 249 250
at the pressures larger than 1.6 nTorr, unlike to third generation light sources, the vacuum lifetime dominant over Touscheck lifetime. For the MAX IV storage ring [24], the total
lifetime is above 10 hours. In the MAX IV case, the gas scattering (elastic) lifetime dominant over Touscheck lifetime as well.
251 252 253
Fig 17. The result of momentum acceptance tracking with 1.1 MV RF peak voltages. The blue curves show lattice momentum acceptance and red curves indicates the RF acceptance. With the cavity peak voltage of 1.1 MV, the RF acceptance is 5.6%. Elegant code has been utilized for the tracking calculation.
254 255 256 257 258 259
Fig 18. The total and vacuum (elastic and inelastic gas scattering) lifetime versus pressure. The red, blue and green curves indicate Touscheck, vacuum and total lifetime respectively.
260 261 262
6. Intra beam scattering
263
Intra beam scattering (IBS) limits the achievable horizontal emittance in the storage rings. It acts more power full in the storage rings with ultra-low emittances [19]. By employing 6D tracking code, Elegant [17], we have calculated the effect of IBS on the emittance and energy spread growth. Fig. 19 shows the variation of emittance and energy spread versus beam current. In the consequence of IBS effect, the zero current emittance (90 pm-rad) increased to 262 pm-rad.
264 265 266 267 268
Fig 19. The effect of IBS on equilibrium emittance and energy spread growth in different currents. In the consequence of IBS effect, the zero current emittance (90 pm-rad) increased to 262 pm-rad.
269 270
Employing Landau cavity is an effective way to mitigate the unwanted effects of IBS on the emittance and energy spread growth. By using Elegant code, we have simulated the effect of Landau cavity on the horizontal emittance and energy spread in different bunch lengthening factor. The bunchlengthening factor is the ratio of bunch length after employing Landau cavity to zero current bunch length. The zero current bunch length is 5.86 mm. The variation of horizontal emittance and energy spread versus different bunch lengthening factor are shown in Fig.20. The calculations are conducted at 400mA. We observed that after employing Landau cavity and making the bunch length 5.5 times of the zero current bunch length, it is possible to bring back the horizontal emittance to 169 pm-rad. The zero current natural emittance is 90 pm-rad. In order to make a comparison between the results of IBS effect on our proposed lattice parameter and MAX IV storage ring, we bring here the numbers giving in the references [3, 24] for the effect of IBS on emittance growth. The zero current emittance of MAX IV storage ring ( 320 pm-rad) in the consequence of IBS blowup increases to 466 pm-rad. After applying Landau cavity, the emittance decreased to 364 pm-rad. Emittance blowup from IBS with and without bunch lengthening from Landau cavity is calculated assuming 500 mA stored current.
271 272 273 274 275 276 277 278 279 280 281 282 283 284 285
Fig 20. Horizontal emittance and energy spread as a function of bunch lengthening factor via Landau cavity tuning at 400 mA beam current. The red and blue lines show energy spread and horizontal emittance respectively.
286 287 288 289
7. Instabilities and impedance
290
It is expected that instabilities emerging from collective effects become more significant in the ultralow emittance storage rings. The self-induced wakefields due to the resistive wall impedance, chamber geometric effects, and coherent synchrotron radiation (CSR) will result in instabilities at some threshold currents. The small momentum compaction factor, short bunch length and small synchrotron tune are the common characteristic of any low emittance storage rings [25]. On the other hand, because of strong sextupoles and quadrupole strength, there is no choice to use small aperture magnets in low emittance rings. All mentioned properties make the low emittance storage rings more vulnerable to instabilities arisen from wake fields. In this section, we will briefly evaluate the tolerance of designed lattice against the single-bunch microwave instability excited by coherent synchrotron radiation (CSR) and the single-bunch transverse mode coupling instability due to wall resistance.
291 292 293 294 295 296 297 298 299 300 301
In order to estimate the threshold current for single-bunch microwave instability, We consider the CSR wakefield generated by an electron moving on a circular orbit of radius in the middle of two (perfectly conducting) parallel plates that are separated by a distance 2h. For the CSR wake, the beam dynamics depend only on two dimensionless parameters, the shielding parameter and scattering parameter [26]
302 303 304 305 306
(1)
307
Where are the number of electrons, synchrotron tune, energy spread and bunch length respectively. The threshold associated with CSR is given by
308 309
(
(2)
310
supposing an elliptical vacuum chamber in the arcs with cross section of (20mm,13mm) , and , we find the shielding parameter 36.1399 and threshold number of electrons in each bunch . The corresponding threshold current for 154 number of bunches is which is close to maximum design current of ring. Small aperture magnets and low gap insertion devices in low emittance storage rings make instabilities excited with resistance wall more important. Transverse mode coupling instability (TMCI) is one of the instabilities which dominantly excited by resistance wall. To calculate the threshold of this instability in our lattice, we use the similar calculations conducted in references [26, 27]. The kick factor for a Gaussian bunch pathing through a circular vacuum chamber is given by
311 312 313 314 315 316 317 318 319 320
,
)
(
)
√
(3)
321
the bunch length. The
322 323
(4)
324
Where is the circumference of storage ring. In the dominator, the sum is done on the different segments of the lattice. We suppose that the whole ring is composed of two parts, the arcs and low gap undulators. The straight sections are filled with 17 low gap undulators. The related parameters for calculating are given in table 6. In the multibunche mode, threshold current is . For
325 326 327 328
With the radius of vacuum chamber, TMCI threshold current is given by ( ) ∑
,
the conductivity and
154 bunchs, we find the threshold current about 1.87 A where it is much larger than the designed current of ring.
329 330 331
Table 6. The different parameters (length, shape, radius, material and Betateron oscillation) in the arcs and straight sections utilized for current threshold calculation Shape
(
)[mm,mm]
332 333
[ ]
Type
Length[m]
Metal
Arcs
499.5
Elliptical
(20,13)
Al
8.8
Low gap undulators
76.5
Elliptical
(20,2.5)
Cu
3.3 334
8. Conclusion
335
In this paper, we presented the lattice design and beam dynamic calculations for an ultra-low emittance storage ring with intermediate energy and circumference. According to design results, unlikely the other proposed diffraction limited storage rings, circumference of designed ring is significantly small and it is possible to push natural emittance more down by using more complicate concepts for magnet design and construction. The total lifetime without Landau cavity is about 7 hours and the dynamic aperture in presence of misalignment and field errors is relatively enough for conventional off axis injection. The effect of IBS on emittance and energy spread growth was calculated. The results of employing Landau cavity to alleviate unwanted effect of IBS was also presented. We derived analytically the threshold current for single bunch instabilities and according to the results, it is expected that these instabilities do not have big impact on the beam stability and quality.
336 337 338 339 340 341 342 343 344 345 346
Acknowledgements
347
The authors would like to especially thank H.Wiedemann for his several helpful comments on this research. This paper is supported by Iranian Light Source Facility.
348 349
References
350
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374 375
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398 399
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400 401
402 Figure 1.
403 404
405 Figure 2.
406 407 408 409
410
Figure 3.
411
412
Figure 4.
413
EPU 1st harmonic
1E+22 Brilliance (Photons/[sec*mm^2 mrad^2 0.1% BW])
EPU 3rd harmonic
1E+21
EPU 5th harmonic IVU 1st harmonic
1E+20
IVU 3rd harmonic IVU 5th harmonic
1E+19
SCW Bending
1E+18 1E+17 1E+16 1E+15 1E+14 0.1
1
Energy (keV)
Figure 5.
10
100
414
415
416
Figure 6.
417
418
419
420
Figure.7.
58.8
13.36
𝜈_x
𝜈_x
13.34
𝜈_𝑦
58.7
13.32
58.65
13.3
58.6
13.28
58.55
13.26
58.5
13.24
58.45
𝜈_𝑦
58.75
13.22 0
2
4
6
Δx(mm)
8
10
421
Figure.8.
422
423
424
58.92
13.55
58.9
𝜈_𝑥
13.5
58.88
𝜈_𝑦
13.45 13.4 13.35
58.84
13.3
58.82
𝜈_𝑦
𝜈_x
58.86
13.25
58.8
13.2
58.78
13.15
58.76
13.1
58.74
13.05 0
1
2
3
Δy (mm)
Figure.9.
4
5
425
426
427
428
Figure.10.
429
430
Figure.11.
431
432
Figure. 12.
433
434
435
Figure 13.
436 437
438
Figure 14.
439 440
441
Figure 15.
442 443
444 Figure 16.
445 446
447 Figure 17.
448 449 450
451 Figure 18.
452
453
454 455
Figure 19.
456 280
0.0012 Horizontal Emittance
260
Energy spread
240
ε_x(pm-rad)
220 200 0.0008 180 160
Energy spread
0.001
0.0006
140 120 100
0.0004 1
2
3
4
5
Bunch lengthening factor 457 Figure 20.
458 459 460 461 462 463 464