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Lattice design and beam dynamics in a compact X-ray source based on Compton scattering
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Nuclear Instruments and Methods in Physics Research A 592 (2008) 1–8 www.elsevier.com/locate/nima
Lattice design and beam dynamics in a compact X-ray source based on Compton scattering Peicheng Yu, Wenhui Huang Department of Engineering Physics, Tsinghua University, Beijing 100084, PR China Received 15 October 2007; received in revised form 20 March 2008; accepted 21 March 2008 Available online 4 April 2008
Abstract We present a feasibility study of a particular X-ray source based on Compton scattering. In particular, we focus on the pulse mode of its operation, in which electron beams are injected with the frequency of 50–60 Hz. We propose to construct a compact storage ring with a circumference of 12 m, as well as a lattice to provide stable operation in the pulse mode for electron–laser interaction. We develop a computer code to simulate beam dynamics in the pulse mode. Intra-beam scattering and Compton scattering are included in the simulation, and their effects on beam emittance and stability are discussed. This source provides X-ray beam in the pulse mode with an intensity of 1.7 1012 photons/s and spectral brightness of 1010 photons/s/0.1%BW/mm2/mrad2 in the energy range from 20 to 80 keV. These parameters meet the requirements for angiography as well as other technological and scientific applications that require high brightness and pulse nature of X-ray. r 2008 Elsevier B.V. All rights reserved. Keywords: LESR; Compton scattering; X-ray; Beam dynamics; Lattice design
1. Introduction Hard X-rays of 20–80 keV are now very useful for biological and medical applications. Currently, there is a growing interest in developing compact hard X-ray sources based on Compton scattering. The idea of utilizing the laser-electron storage ring (LESR) for the purpose of increasing frequency and luminosity of hard X-rays by means of Compton scattering was proposed by Huang and Ruth in 1998 [1]. The basic principle of this scheme is Compton scattering off a low-energy electron beam stored in a storage ring with an intense laser pulse stored in an optical storage system to produce the desired photon spectrum. In a head-on collision, the maximal scattered photon energy can be represented by Eg, max ¼ 4g2Elas, where g is the relative energy and Elas is the energy of the laser photon. The micropulse length of the scattered photon is almost the same as that of the electron beam. For its compactness and low construction cost, this scheme is an alternative proposal to the traditional method of Corresponding author.
E-mail address: [email protected] (P. Yu). 0168-9002/$ - see front matter r 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2008.03.114
generating X-ray photons using a superconducting wiggler on synchrotron radiation source. In this proposal, the main dynamic features of the electron beam involve Compton scattering (CS), synchrotron radiation (SR), and intra-beam scattering (IBS). Due to the quantum nature of Compton scattering, quantum excitation which increases beam emittance should be taken into consideration in ring design. Likewise, IBS becomes a significant factor for emittance growth when the beam energy is a few MeV to a few hundred MeV. There are two basic schemes for the LESR [2]. The first scheme keeps beams in the ring until steady-state parameters are reached [3,4]. In such a storage ring, the long-term stability of photon intensity can be achieved. The LESR in this scheme is designed with a controlled momentum compaction factor, by means of which one can achieve large energy acceptance and keep the long-term stable motion of electron beam with large energy spread. The intensity of the X-ray is stable due to the use of an electron beam with steady-state parameters. The second scheme uses non-steady-state parameters of electron beams. In this scheme, beams are injected more frequently and dumped before steady parameters are
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reached in the ring [5]. Due to periodic injections of electron beam, the ring produces X-ray in the pulse mode. Compared with the steady-state mode, the pulse mode has certain advantages. Damping time of such compact storage ring is of the order of 1 s, thus, the electron beam parameters are close to the initial values rather than steady-state values when they are dumped. Since electron beam parameters deteriorate in the ring, we obtain a higher intensity of scattered photon if beams are dumped long before they reach steady-state parameters. Another reason is that from the simulation results of this paper, longitudinal beam size is the bottleneck of stable operation. Hence, long-term stability requires low momentum compaction factor, which makes lattice design of LESR more difficult. The pulse mode is able to avoid this problem, because electron beams are dumped before their longitudinal size become too long for stable operation. We discuss this pulse mode in this paper and present a lattice design for the scheme. We likewise present theoretical analysis and numerical simulation for beam dynamics involving Compton scattering, synchrotron radiation, and intra-beam scattering. 2. LESR lattice design 2.1. Main requirements In the laboratory frame, the number of photons, Nx, generated in collision between a laser pulse and an electron pulse is determined by the following expression, assuming that electron beams and laser density have Gaussian distribution [13]:
where ac is the momentum compaction factor, h is the harmonic number, and d is the total energy spread; lRF and VRF are the wave length and voltage of the RF cavity, respectively; and Ee is the energy of electron. From Eq. (2), we can see that a small momentum compaction factor is important when designing the LESR lattice [2]. For on-axis single-turn injection, we plan to use a traveling wave design kicker and a Lambertson-style septum for the injection scheme [5]. As a result, we need one long dispersion-free straight section where we will place the injection system. The RF cavity as well as IP should also be placed into the dispersion-free section to achieve stability. Hence, the lattice should have long dispersion-free straight sections. As such, the lattice designed for the LESR should have the following features:
sN e N g qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Nx ¼ 2p s2ey þ s2ly 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðs2ex þ s2lx Þ cos2 ða=2Þ þ ðs2ez þ s2lz Þ sin2 ða=2Þ
system to allow the electron beam to go through it, which makes it more difficult to build the optical system. When head-on collision is difficult to realize, a small collision angle is necessary, which makes the term (sez2+slz2) sin2(a/2) a significant part that determines the total yield of scattered photons. Therefore, longitudinal beam size should necessarily be short for the optimization of the photon yield. Longitudinal beam size sez is determined by the following expression [2]: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ac hE e sez ¼ lRF d (2) 2peV RF j cos fj
(1)
where sex, slx, sey, sly, and slz are the RMS transverse and longitudinal sizes of beam and laser pulse, respectively; Ne and Ng are the number of electrons and the number of laser photons in a single pulse, respectively; a is the collision angle, with a ¼ 0 corresponding to head-on collision; and s is the Compton Scattering cross-section. We can see from Eq. (1) that in order to achieve a high quantity of scattered photons the transverse beam size at the interaction point (IP) should be adequately small, which means the beta function should be small at IP. Hence, with low beta insertion involved, the natural chromaticity of the storage ring becomes significantly large and strong sextupoles are placed in the dispersive area to correct chromaticity. The dynamic aperture of the ring is diminished as a result. In order to implement a head-on collision, we have to place a hole in the center of each mirror of the optical
The IP should have low beta function. The lattice should have a long dispersion-free section in which to place the injection system, RF cavity and IP. Strong sextupoles are introduced to correct the natural chromaticity of the storage ring; therefore, the dynamic aperture should be compensated using harmonic sextupoles. The photon yield depends strongly on longitudinal beam size in non-head-on collision; hence, a small momentum compaction factor is necessary in order to achieve high yield of scattered photon.
2.2. The LESR lattice The most common scheme for the LESR is a racetrack design with two long straight sections [2,5]. The placement for our LESR is illustrated in Fig. 1. This scheme is based on the lattice design for the NESTOR ring in Ukraine [2]. The injection system is placed in one straight section, while the IP and RF cavity are placed in the opposite one [5]. We use the typical DBA structure with two quadrupoles in the arc area, and in this way it becomes convenient to place sextupoles in the dispersive section. The circumference of the ring is 11.92 m. There are 4 bending magnets, 16 quadrupoles, and 10 sextupoles. Bending radius is 0.38 m and bending angle is 901. The
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Fig. 1. LESR racetrack scheme with two long straight sections. The long straight sections are dispersion free. Groups of Sextupoles (S1 and S2) are placed in the arc area to correct natural chromaticity. Harmonic sextupoles (SX1, SX2 and SX3) are placed in one of the long straight sections to enhance dynamic aperture.
Table 1 Parameters of laser-electron storage ring in THU Parameters Circumference (m) Beam current (per bunch) (mA)
Fig. 2. The horizontal beta function and dispersion function of the storage ring. This lattice is based on the ring lattice for the NESTOR ring [2].
maximal absolute value k of quadrupoles is 58.9. Both long straight sections have been made dispersion-free; thus, two families of sextupoles are placed in the arc area to correct chromaticity, while harmonic sextupoles are placed in one of the long straight sections to enlarge dynamic aperture. The lattice parameters are illustrated in Fig. 2. The beta function at IP are bx ¼ 3.2 cm and by ¼ 5.3 cm, while the maximal value of beta function across the ring is bmax ¼ bx, max ¼ 8.63 m. The tune of the ring is Qx ¼ 2.78 and Qy ¼ 1.61. Other ring parameters are presented in Table 1. 2.3. Dynamic aperture (DA) We use harmonic sextupoles in the dispersive area to enlarge the dynamic aperture with the help of software
Value 11.92 25
Tunes Horizontal Vertical
2.78 1.61
Beta function at IP Horizontal (cm) Vertical (cm)
3.2 5.3
Mean energy loss per turn Synchrotron radiation (eV) Compton scattering (laser energy 1 mJ) (eV) RF voltage RF frequency Harmonic number Momentum compaction factor Energy acceptance Injection frequency (Hz)
1.42 0.188 300 kV 1.2 GHz 48 0.079 3.2% 50–60
Horizontal beam size at IP Without IBS (mm) With IBS (400,000 turns) (mm)
32 63
Natural chromaticity Horizontal Vertical
4.3 3.28
Quadrupole parameters Number of quadrupoles Maximum strength
16 58.9
Dipoles parameters Number of bending magnets Bending radius (m) Bending angle (deg) Magnet field index
4 0.38 90 4.1
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OPA [11]. DA at IP without harmonic sextupoles is 0.4 mm, while DA at the azimuths with maximal beta functions (around Q2, see Fig. 2) is 5 mm. We found that DA of the LESR lattice is sufficiently enhanced upon adding harmonic sextupoles in the long straight section. In Fig. 3, we present tracing of 2500 particles for 50,000 turns at IP with momentum deviation of 2%; in Fig. 4, we present the same tracing at the azimuths with maximal beta function. The transverse beam size at IP is 0.032 mm at startup, and from the simulation, we know that beam size after 400,000 turns is about 0.063 mm, while dynamic aperture is about 0.80 mm. Beam size at the azimuths with
maximal beta function is 0.41 mm at startup, and 0.81 mm after 400,000 turns, while dynamic aperture is 10 mm. Hence, dynamic apertures are 10 times the transverse size of the electron beam, which are large enough for practical lattice design and stable operation in the pulse mode. 2.4. The optical storage system The basic concept of the optical storage system for the Compton scattering X-ray source has been described in Refs. [3,5]. The optical storage system consists of an enhancement cavity resonantly driven by a mode-locked laser. A basic optical cavity configuration is a two-mirror Fabry–Perot interferometer in which power is coupled through the backside of one partially transmissive mirror. The circulation time equals the revolution time of electron in the storage ring; therefore the length of optical cavity is determined by the circumference of the storage ring. We plan to use Ti:sapphire laser to generate a laser pulse of 30 ps duration, with a pulse-repetition rate f75 MHz. 3. Beam dynamics in the pulse mode The main factors influencing the storage ring include intra-beam scattering (IBS), Compton scattering (CS), and synchrotron radiation (SR). In this paper, these factors are discussed for the pulse mode of LESR in THU. The electron beam and laser parameters for the storage ring are listed in Table 2.
Fig. 3. Dynamic aperture at interaction point is about 0.8 mm, while beam size is about 0.074 mm after 400,000 turns in the ring. Therefore, the dynamic aperture at IP is large enough for stable operation. We trace 2500 particles for 50,000 turns, and the lost particles are marked by x. Momentum deviation is 2%.
3.1. Synchrotron radiation Due to the low electron energy, the effect of synchrotron radiation is relatively small compared with large storage ring. In this storage ring, energy loss per turn per electron is USR ¼ 1.42 eV, and the number of photons emitted per turn per electron is N¯ ¼ 6.48. Since the number of synchrotron photons is relatively small, a complete Monte Carlo method has to be applied to simulate SR. Table 2 Parameters of the electron beam, laser pulse, and their interaction Parameters Electron Energy (MeV) Beam charge (nC) Transverse emittance (mrad) Beam pulse length (ps) Transverse beam size at startup (mm)
Fig. 4. Dynamic aperture at the azimuths with maximal beta function (around Q2, see Fig. 2) is about 10 mm, while the beam size here is about 1.2 mm after 400,000 turns in the ring. Therefore, dynamic aperture at the azimuths with maximal beta function is large enough for stable operation. We trace 2500 particles for 50,000 turns, and the lost particles are marked by x. Momentum deviation is 2%.
Laser Pulse energy (mJ) Laser photon wavelength (nm) Rayleigh length (mm) Pulse length (fs) Transverse spot size at IP (mm) Collision angle (deg) Scattered photon energy (keV) Scattered photon pulse length (ps)
Value
30–60 1 2 108 10 32 1–1000 800 1 100 20 3 20–80 10
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The features of SR have been well described in Ref. [12]. For this particular ring, SR will be discussed in detail in part 4. 3.2. Compton scattering
where E0 is the rest energy of electron, Ee refers to the energy of the electron beam, and Elas the energy of laser photon. Meanwhile, reE2.82 1015 m is the classical radius of electron, and g is the relative energy of electron. The angular distribution of the scattered photon can be obtained by (4)
where y is the scattering angle of the photon, and Ex is the energy of the scattered photon. From Eqs. (3) and (4), we can calculate the transverse emittance change due to laser beam. The transverse emittance change can be regarded as transverse recoil of the laser photon; hence, the increase in transverse emittance is given by Z bx;y dx;y 1 E x sin y 2 dN x ¼ dE x (5) Ee dt 2T rev 2 dE x where bx, y is the beta function at IP, dNx/dEx is the energy distribution of the scattered photon derived from Eqs. (1) and (3), and Trev is the revolution time of electron around the ring. As we can see in Eq. (1), transverse beam size determines photon yield in head-on collision; while in non-head-on collision (31 collision for this ring) longitudinal beam length plays an important part, therefore making energy spread of electron beam an important parameter. We can obtain energy spread due to interaction with the laser photon as represented by the equation below: Z 1 dhs2E i 1 h20 i 1 1 2 dN x ¼ ¼ E dE x (6) dt 2 dt 2T rev 0 2 x dE x where e0 is the amplitude of longitudinal oscillation. Therefore, given the parameters of laser and electron beams, we can calculate emittance change of the electron beam due to Compton scattering. The average damping rates to the transverse emittance and energy spread have been described in Ref. [1]. The magnitude of the damping rates can be evaluated by the following expression: 1 dx;y x;y dt 2 1 dsE s2E dt
SR ¼ T1rev U CSEþU e SR ¼ T2rev U CSEþU e
where UCS and USR are the average energy loss of electron per turn due to Compton scattering and synchrotron radiation respectively. 3.3. Intra-beam scattering
The Compton scattering cross-section is described by [7] " 2 ds pr2e 1 E 20 E ¼ dE 2 g2 E las 4g2 E 2las E e E # E0 E Ee E Ee þ þ (3) Ee gE las E e E Ee E
Ex 1 þ b cos a ¼ E las 1 b cos y
5
(7)
As mentioned in part II, intra-beam scattering is an important factor in such a low-energy range. IBS has been described well in Ref. [8]. Factors that influence IBS of an electron beam include electron energy and lattice design. M. Venturini [10] had applied the equations in Ref. [8] to calculate IBS emittance growing rates. However, in this paper, another way of calculating IBS growth rates developed by S. Nagaitsev [9] is applied, which is identical to Ref. [8], yet more efficient for programming and calculation. Calculation on emittance growth due to IBS has been included in the simulation code. 3.4. Beam dynamics in the pulse mode We calculate the values of IBS growth rate, CS quantum excitation rate and damping rate at startup of every injection period, at the laser pulse energy of 1 mJ, initial relative energy spread 0.2% and initial transverse emittance 2 108 mrad. In transverse direction, IBS growth rate is three orders of magnitude larger than the damping effect; in longitudinal direction, IBS growth rate and CS quantum excitation rate have magnitudes of the same order, while damping effect is two orders of magnitude smaller than IBS. Hence, from the calculation, we conclude that IBS is the dominant factor in transverse direction, while IBS and CS play important roles in longitudinal direction. This is different from the situation in the steady mode, in which damping effect counterbalances CS quantum excitation and IBS effect. In the pulse mode, IBS is the most important effect that influences beam dynamics. 4. Simulation of beam dynamics The electron beam is represented by macro particles following a Gaussian distribution. We include IBS, SR, and CS in our simulation. The electron beam energy is set to be 50 MeV in the simulation. As mentioned in part III, less than ten synchrotron photons are emitted per electron per turn in this storage ring; hence the widely used description for synchrotron radiation of ‘‘one big fat photon,’’ drawn from a Gaussian distribution, is inadequate in this case [6]. Therefore, a complete Monte Carlo method is used in which electron macro particles are traced particle-by-particle to simulate their emissions of synchrotron photons. The procedure of Compton scattering is likewise traced particle-by-particle, and a complete Monte Carlo method is used to simulate the interaction. Since we are only interested in the dynamics of electron beam rather than the temporal and spatial distribution of the scattered
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photons, Compton scattering cross-section is integrated all over the interaction area to determine the interaction chances of electron macro particles with laser pulse in a single turn. Thus, the interaction between the laser and electron beams is therefore considered as one single kick at IP per turn. Furthermore, angular distribution of synchrotron radiation and Compton scattering are taken into consideration to simulate the transverse emittance change of beam in the ring. IBS evaluation is carried out using the method described in Ref. [9]. After the particle goes through SR, CS, IBS, and RF cavity, it is multiplied by a transport matrix to simulate its movement around the ring lattice. 4.1. Simulation of transverse emittance The simulation results of transverse emittance at the stored laser energy of 1 mJ are shown in Fig. 5. As can be seen, emittance grows about four times in horizontal direction. We found that IBS is a dominant factor at such a low energy, and covers up the effects from SR and CS. The laser energy stored in the optical cavity has the potential to increase in the future. Therefore, we simulate horizontal emittance at different laser energy in order to identify the effect of CS to the beam dynamics, and to examine the stability and dynamic features of the electron beam. The result is shown in Fig. 6. We can see that, although the cooling effect of laser becomes stronger as laser energy increases, the emittance change in the horizontal direction is dominated by the effect of IBS. Therefore, in transverse direction, IBS is the dominant factor. Hence, increasing photon yield by means of optimizing our lattice to reduce the effect of IBS can be considered an effective method.
Fig. 5. Transverse emittance change in the pulse mode is presented. The simulation is carried out with electron energy of 50 MeV and laser energy of 1 mJ. IBS, SR and CS are all included in the simulation. As is shown in the figure, transverse emittance grows about four times in the horizontal direction.
Fig. 6. Horizontal emittance growth at different laser energy is presented. Simulation is carried out with electron energy of 50 MeV. We can see that laser cooling can be witnessed as laser energy increases. However, IBS effect dominates the emittance change in the horizontal direction.
4.2. Simulation of relative energy spread and scattered photon intensity We simulate relative energy spread and scattered photon intensity at the stored laser energy of 1 mJ, as shown in Fig. 7. The absolute value of photon intensity is the number of scattered photons integrated over photon energy. In order to identify the contribution of IBS, CS and SR separately, we accomplish simulation in three different ways: with all factors; CS and SR; and SR only. We can see from Fig. 7 that at this energy level IBS and CS have excitation rates of the same order, while SR contributes insignificantly to energy spread. The line corresponding to all factors shows that energy spread grows about 1.6 times, and scattered photon intensity drops to 60% of its value at the end of the pulse due to the increase in transverse and longitudinal beam size. At startup the longitudinal beam size is 3 mm and X-ray intensity is 1.7 1012 photons/s, while at the end of the pulse longitudinal beam size is 9 mm and X-ray intensity is 1.0 1012 photons/s. We also enhanced laser energy to see the effect of CS, as shown in Fig. 8. It is predictable that when laser energy goes up CS becomes the dominant factor in longitudinal direction, because the excitation rate of the energy spread is in proportion with scattering chance. Noticeably, we find that when laser energy stored in the optical cavity is 1000 mJ, energy spread does not follow a linear increase with number of turns. This is because when the energy spread goes too high, the beam exceeds RF bucket, thereby resulting in longitudinal beam instability. This can be witnessed in Fig. 9, in which we witness an abrupt drop of scattered photon intensity in line corresponding to 1000 mJ. We know that energy acceptance is determined by [2]
sRF
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2eV RF pac hE e
(8)
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Fig. 7. Simulation results of the energy spread and scattered photon intensity at laser energy of 1 mJ and electron energy of 50 MeV. The absolute value of the scattered photons intensity is the number of scattered photon integrated over the photon energy. We accomplish the simulation in three different ways: IBS, CS & SR; CS & SR; SR only. As shown in the figure, SR is trivial in the longitudinal direction, while the excitation rates of IBS and CS are of the same order at laser energy 1 mJ. The energy spread grows about 1.6 times. Scattered photon intensity drops to 60% of its value at the end of the pulse.
Fig. 8. We show energy spread in this figure. Simulation is carried out at the electron energy of 50 MeV with all effects turned on. We can see that energy spread grows in direct proportion with scattering chance. We observe that with the line corresponding to laser pulse of 1000 mJ, energy spread does not follow the linear increase at around 100,000 turns. This indicates that the beam has exceeded the RF bucket.
We have to either raise RF voltage or reduce momentum compaction factor in order to enlarge energy acceptance. We tried both ways, and the results show that the abrupt drop is eliminated by both means. Yet it is quite difficult to raise RF voltage in such a compact ring design; thus, we have to reduce momentum compaction factor in the lattice design. 4.3. Summary of simulation results In this study, we found that SR does not contribute much to the emittance change either in the transverse or in the longitudinal direction. IBS is fixed when the ring lattice and beam energy are both fixed, while CS may become
Fig. 9. Scattered photon intensity in different laser energy with electron energy of 50 MeV. The absolute value of the scattered photon intensity is the number of scattered photons integrated over photon energy. The intensity decreases quite rapidly to about 60% of its peak at the end of the photon pulse. As we can see from the line corresponding to 1000 mJ, there is an abrupt drop, which indicates that the longitudinal instability has occurred.
strong with the increase of laser energy. However, IBS covers up the effects from CS and SR, and is dominant in the transverse direction. Due to the high energy of scattered photon and small collision angle, emittance growth due to CS is quite large. In the longitudinal direction, CS has an excitation rate with the same order as that of IBS. These results are in accordance with the theoretical analysis in part 3. As mentioned above, when the laser energy goes up to 1000 mJ, it is high enough to cause the energy spread to increase rapidly that longitudinal instability is achieved even in the pulse mode.
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5. Summary We present a lattice that provides an electron beam with a transverse size of electron beam of 32 mm, and X-ray beam intensity of 1.7 1012 photons/s. Since dynamic aperture is intolerable due to large natural chromaticities caused by low beta insertion, we placed groups of harmonic sextupoles to enlarge DA. In calculation and simulation, we know that in the transverse direction emittance grows mainly due to IBS, while in the longitudinal direction CS and IBS have excitation rates of the same order. Moreover, CS has the potential of becoming stronger when laser energy stored in optical storage system goes higher. Both longitudinal size and longitudinal stability require for small momentum compaction factor, which adds to the difficulty of lattice design. Meanwhile, IBS can also be diminished with better lattice design. Therefore, finding a better lattice with smaller momentum compaction factor and less IBS can be the next step of this scheme.
colleagues, Yingchao Du and Dao Xiang, for helpful discussions. This work was supported by the Chinese National Foundation of Natural Sciences under Contract no. 10735050. Reference [1] [2] [3] [4] [5] [6]
[7] [8] [9] [10]
Acknowledgments [11]
The authors are greatly indebted to Professor Peter Gladkikh for many helpful discussions and advice on lattice design. The authors would also like to thank their
[12] [13]
Z. Huang, R.D. Ruth, Phys. Rev. Lett. 80 (1998) 976. Peter Gladkikh, Phys. Rev. Spec. Top. Accel. Beams 8 (2005) 050702. E. Bulyak, et al., Nucl. Instr. and Meth. A 487 (2002) 241. J. Urakawa, et al., Nucl. Instr. and Meth. A 532 (2004) 388. R.J. Loewen, Ph.D. Thesis, Stanford, 2003. Shane Koscielniak, Automated self-convolution applied to synchrotron radiation spectra for use in longitudinal particle tracking, /http://www.triumf.ca/people/koscielniak/synkro-aps.pdfS. IanC. Hsu, Energy measurement of relativistic electron beams by laser Compton scattering, Phys. Rev. E. 54 (5) (1996). J.D. Bjorken, S.K. Mtingwa, Intrabeam Scattering Part. Accel. 13 (1983) 115. Sergei Nagaitsev, Phys. Rev. Spec. Top. Accel. Beams 8 (2005) 064403. Marco Venturini, Study of intrabeam scattering in low-energy electron rings, In: Proceedings of the 2001 Particle Accelerator, p. 2961. Andreas Streun, Practical guidelines for lattice design, SLS-TMETA-1999-0014. H. Wiedemann, Synchrotron Radiation, Springer, Berlin, 2003. J. Yang, M. Washio, A. Endo, T. Hori, Nucl. Instr. Meth. Phys. Res. A 428 (1999) 556.
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Lattice design and beam dynamics in a compact X-ray source based on Compton scattering Peicheng Yu, Wenhui Huang Department of Engineering Physics, Tsinghua University, Beijing 100084, PR China Received 15 October 2007; received in revised form 20 March 2008; accepted 21 March 2008 Available online 4 April 2008
Abstract We present a feasibility study of a particular X-ray source based on Compton scattering. In particular, we focus on the pulse mode of its operation, in which electron beams are injected with the frequency of 50–60 Hz. We propose to construct a compact storage ring with a circumference of 12 m, as well as a lattice to provide stable operation in the pulse mode for electron–laser interaction. We develop a computer code to simulate beam dynamics in the pulse mode. Intra-beam scattering and Compton scattering are included in the simulation, and their effects on beam emittance and stability are discussed. This source provides X-ray beam in the pulse mode with an intensity of 1.7 1012 photons/s and spectral brightness of 1010 photons/s/0.1%BW/mm2/mrad2 in the energy range from 20 to 80 keV. These parameters meet the requirements for angiography as well as other technological and scientific applications that require high brightness and pulse nature of X-ray. r 2008 Elsevier B.V. All rights reserved. Keywords: LESR; Compton scattering; X-ray; Beam dynamics; Lattice design
1. Introduction Hard X-rays of 20–80 keV are now very useful for biological and medical applications. Currently, there is a growing interest in developing compact hard X-ray sources based on Compton scattering. The idea of utilizing the laser-electron storage ring (LESR) for the purpose of increasing frequency and luminosity of hard X-rays by means of Compton scattering was proposed by Huang and Ruth in 1998 [1]. The basic principle of this scheme is Compton scattering off a low-energy electron beam stored in a storage ring with an intense laser pulse stored in an optical storage system to produce the desired photon spectrum. In a head-on collision, the maximal scattered photon energy can be represented by Eg, max ¼ 4g2Elas, where g is the relative energy and Elas is the energy of the laser photon. The micropulse length of the scattered photon is almost the same as that of the electron beam. For its compactness and low construction cost, this scheme is an alternative proposal to the traditional method of Corresponding author.
E-mail address: [email protected] (P. Yu). 0168-9002/$ - see front matter r 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2008.03.114
generating X-ray photons using a superconducting wiggler on synchrotron radiation source. In this proposal, the main dynamic features of the electron beam involve Compton scattering (CS), synchrotron radiation (SR), and intra-beam scattering (IBS). Due to the quantum nature of Compton scattering, quantum excitation which increases beam emittance should be taken into consideration in ring design. Likewise, IBS becomes a significant factor for emittance growth when the beam energy is a few MeV to a few hundred MeV. There are two basic schemes for the LESR [2]. The first scheme keeps beams in the ring until steady-state parameters are reached [3,4]. In such a storage ring, the long-term stability of photon intensity can be achieved. The LESR in this scheme is designed with a controlled momentum compaction factor, by means of which one can achieve large energy acceptance and keep the long-term stable motion of electron beam with large energy spread. The intensity of the X-ray is stable due to the use of an electron beam with steady-state parameters. The second scheme uses non-steady-state parameters of electron beams. In this scheme, beams are injected more frequently and dumped before steady parameters are
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reached in the ring [5]. Due to periodic injections of electron beam, the ring produces X-ray in the pulse mode. Compared with the steady-state mode, the pulse mode has certain advantages. Damping time of such compact storage ring is of the order of 1 s, thus, the electron beam parameters are close to the initial values rather than steady-state values when they are dumped. Since electron beam parameters deteriorate in the ring, we obtain a higher intensity of scattered photon if beams are dumped long before they reach steady-state parameters. Another reason is that from the simulation results of this paper, longitudinal beam size is the bottleneck of stable operation. Hence, long-term stability requires low momentum compaction factor, which makes lattice design of LESR more difficult. The pulse mode is able to avoid this problem, because electron beams are dumped before their longitudinal size become too long for stable operation. We discuss this pulse mode in this paper and present a lattice design for the scheme. We likewise present theoretical analysis and numerical simulation for beam dynamics involving Compton scattering, synchrotron radiation, and intra-beam scattering. 2. LESR lattice design 2.1. Main requirements In the laboratory frame, the number of photons, Nx, generated in collision between a laser pulse and an electron pulse is determined by the following expression, assuming that electron beams and laser density have Gaussian distribution [13]:
where ac is the momentum compaction factor, h is the harmonic number, and d is the total energy spread; lRF and VRF are the wave length and voltage of the RF cavity, respectively; and Ee is the energy of electron. From Eq. (2), we can see that a small momentum compaction factor is important when designing the LESR lattice [2]. For on-axis single-turn injection, we plan to use a traveling wave design kicker and a Lambertson-style septum for the injection scheme [5]. As a result, we need one long dispersion-free straight section where we will place the injection system. The RF cavity as well as IP should also be placed into the dispersion-free section to achieve stability. Hence, the lattice should have long dispersion-free straight sections. As such, the lattice designed for the LESR should have the following features:
sN e N g qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Nx ¼ 2p s2ey þ s2ly 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðs2ex þ s2lx Þ cos2 ða=2Þ þ ðs2ez þ s2lz Þ sin2 ða=2Þ
system to allow the electron beam to go through it, which makes it more difficult to build the optical system. When head-on collision is difficult to realize, a small collision angle is necessary, which makes the term (sez2+slz2) sin2(a/2) a significant part that determines the total yield of scattered photons. Therefore, longitudinal beam size should necessarily be short for the optimization of the photon yield. Longitudinal beam size sez is determined by the following expression [2]: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ac hE e sez ¼ lRF d (2) 2peV RF j cos fj
(1)
where sex, slx, sey, sly, and slz are the RMS transverse and longitudinal sizes of beam and laser pulse, respectively; Ne and Ng are the number of electrons and the number of laser photons in a single pulse, respectively; a is the collision angle, with a ¼ 0 corresponding to head-on collision; and s is the Compton Scattering cross-section. We can see from Eq. (1) that in order to achieve a high quantity of scattered photons the transverse beam size at the interaction point (IP) should be adequately small, which means the beta function should be small at IP. Hence, with low beta insertion involved, the natural chromaticity of the storage ring becomes significantly large and strong sextupoles are placed in the dispersive area to correct chromaticity. The dynamic aperture of the ring is diminished as a result. In order to implement a head-on collision, we have to place a hole in the center of each mirror of the optical
The IP should have low beta function. The lattice should have a long dispersion-free section in which to place the injection system, RF cavity and IP. Strong sextupoles are introduced to correct the natural chromaticity of the storage ring; therefore, the dynamic aperture should be compensated using harmonic sextupoles. The photon yield depends strongly on longitudinal beam size in non-head-on collision; hence, a small momentum compaction factor is necessary in order to achieve high yield of scattered photon.
2.2. The LESR lattice The most common scheme for the LESR is a racetrack design with two long straight sections [2,5]. The placement for our LESR is illustrated in Fig. 1. This scheme is based on the lattice design for the NESTOR ring in Ukraine [2]. The injection system is placed in one straight section, while the IP and RF cavity are placed in the opposite one [5]. We use the typical DBA structure with two quadrupoles in the arc area, and in this way it becomes convenient to place sextupoles in the dispersive section. The circumference of the ring is 11.92 m. There are 4 bending magnets, 16 quadrupoles, and 10 sextupoles. Bending radius is 0.38 m and bending angle is 901. The
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Fig. 1. LESR racetrack scheme with two long straight sections. The long straight sections are dispersion free. Groups of Sextupoles (S1 and S2) are placed in the arc area to correct natural chromaticity. Harmonic sextupoles (SX1, SX2 and SX3) are placed in one of the long straight sections to enhance dynamic aperture.
Table 1 Parameters of laser-electron storage ring in THU Parameters Circumference (m) Beam current (per bunch) (mA)
Fig. 2. The horizontal beta function and dispersion function of the storage ring. This lattice is based on the ring lattice for the NESTOR ring [2].
maximal absolute value k of quadrupoles is 58.9. Both long straight sections have been made dispersion-free; thus, two families of sextupoles are placed in the arc area to correct chromaticity, while harmonic sextupoles are placed in one of the long straight sections to enlarge dynamic aperture. The lattice parameters are illustrated in Fig. 2. The beta function at IP are bx ¼ 3.2 cm and by ¼ 5.3 cm, while the maximal value of beta function across the ring is bmax ¼ bx, max ¼ 8.63 m. The tune of the ring is Qx ¼ 2.78 and Qy ¼ 1.61. Other ring parameters are presented in Table 1. 2.3. Dynamic aperture (DA) We use harmonic sextupoles in the dispersive area to enlarge the dynamic aperture with the help of software
Value 11.92 25
Tunes Horizontal Vertical
2.78 1.61
Beta function at IP Horizontal (cm) Vertical (cm)
3.2 5.3
Mean energy loss per turn Synchrotron radiation (eV) Compton scattering (laser energy 1 mJ) (eV) RF voltage RF frequency Harmonic number Momentum compaction factor Energy acceptance Injection frequency (Hz)
1.42 0.188 300 kV 1.2 GHz 48 0.079 3.2% 50–60
Horizontal beam size at IP Without IBS (mm) With IBS (400,000 turns) (mm)
32 63
Natural chromaticity Horizontal Vertical
4.3 3.28
Quadrupole parameters Number of quadrupoles Maximum strength
16 58.9
Dipoles parameters Number of bending magnets Bending radius (m) Bending angle (deg) Magnet field index
4 0.38 90 4.1
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OPA [11]. DA at IP without harmonic sextupoles is 0.4 mm, while DA at the azimuths with maximal beta functions (around Q2, see Fig. 2) is 5 mm. We found that DA of the LESR lattice is sufficiently enhanced upon adding harmonic sextupoles in the long straight section. In Fig. 3, we present tracing of 2500 particles for 50,000 turns at IP with momentum deviation of 2%; in Fig. 4, we present the same tracing at the azimuths with maximal beta function. The transverse beam size at IP is 0.032 mm at startup, and from the simulation, we know that beam size after 400,000 turns is about 0.063 mm, while dynamic aperture is about 0.80 mm. Beam size at the azimuths with
maximal beta function is 0.41 mm at startup, and 0.81 mm after 400,000 turns, while dynamic aperture is 10 mm. Hence, dynamic apertures are 10 times the transverse size of the electron beam, which are large enough for practical lattice design and stable operation in the pulse mode. 2.4. The optical storage system The basic concept of the optical storage system for the Compton scattering X-ray source has been described in Refs. [3,5]. The optical storage system consists of an enhancement cavity resonantly driven by a mode-locked laser. A basic optical cavity configuration is a two-mirror Fabry–Perot interferometer in which power is coupled through the backside of one partially transmissive mirror. The circulation time equals the revolution time of electron in the storage ring; therefore the length of optical cavity is determined by the circumference of the storage ring. We plan to use Ti:sapphire laser to generate a laser pulse of 30 ps duration, with a pulse-repetition rate f75 MHz. 3. Beam dynamics in the pulse mode The main factors influencing the storage ring include intra-beam scattering (IBS), Compton scattering (CS), and synchrotron radiation (SR). In this paper, these factors are discussed for the pulse mode of LESR in THU. The electron beam and laser parameters for the storage ring are listed in Table 2.
Fig. 3. Dynamic aperture at interaction point is about 0.8 mm, while beam size is about 0.074 mm after 400,000 turns in the ring. Therefore, the dynamic aperture at IP is large enough for stable operation. We trace 2500 particles for 50,000 turns, and the lost particles are marked by x. Momentum deviation is 2%.
3.1. Synchrotron radiation Due to the low electron energy, the effect of synchrotron radiation is relatively small compared with large storage ring. In this storage ring, energy loss per turn per electron is USR ¼ 1.42 eV, and the number of photons emitted per turn per electron is N¯ ¼ 6.48. Since the number of synchrotron photons is relatively small, a complete Monte Carlo method has to be applied to simulate SR. Table 2 Parameters of the electron beam, laser pulse, and their interaction Parameters Electron Energy (MeV) Beam charge (nC) Transverse emittance (mrad) Beam pulse length (ps) Transverse beam size at startup (mm)
Fig. 4. Dynamic aperture at the azimuths with maximal beta function (around Q2, see Fig. 2) is about 10 mm, while the beam size here is about 1.2 mm after 400,000 turns in the ring. Therefore, dynamic aperture at the azimuths with maximal beta function is large enough for stable operation. We trace 2500 particles for 50,000 turns, and the lost particles are marked by x. Momentum deviation is 2%.
Laser Pulse energy (mJ) Laser photon wavelength (nm) Rayleigh length (mm) Pulse length (fs) Transverse spot size at IP (mm) Collision angle (deg) Scattered photon energy (keV) Scattered photon pulse length (ps)
Value
30–60 1 2 108 10 32 1–1000 800 1 100 20 3 20–80 10
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The features of SR have been well described in Ref. [12]. For this particular ring, SR will be discussed in detail in part 4. 3.2. Compton scattering
where E0 is the rest energy of electron, Ee refers to the energy of the electron beam, and Elas the energy of laser photon. Meanwhile, reE2.82 1015 m is the classical radius of electron, and g is the relative energy of electron. The angular distribution of the scattered photon can be obtained by (4)
where y is the scattering angle of the photon, and Ex is the energy of the scattered photon. From Eqs. (3) and (4), we can calculate the transverse emittance change due to laser beam. The transverse emittance change can be regarded as transverse recoil of the laser photon; hence, the increase in transverse emittance is given by Z bx;y dx;y 1 E x sin y 2 dN x ¼ dE x (5) Ee dt 2T rev 2 dE x where bx, y is the beta function at IP, dNx/dEx is the energy distribution of the scattered photon derived from Eqs. (1) and (3), and Trev is the revolution time of electron around the ring. As we can see in Eq. (1), transverse beam size determines photon yield in head-on collision; while in non-head-on collision (31 collision for this ring) longitudinal beam length plays an important part, therefore making energy spread of electron beam an important parameter. We can obtain energy spread due to interaction with the laser photon as represented by the equation below: Z 1 dhs2E i 1 h20 i 1 1 2 dN x ¼ ¼ E dE x (6) dt 2 dt 2T rev 0 2 x dE x where e0 is the amplitude of longitudinal oscillation. Therefore, given the parameters of laser and electron beams, we can calculate emittance change of the electron beam due to Compton scattering. The average damping rates to the transverse emittance and energy spread have been described in Ref. [1]. The magnitude of the damping rates can be evaluated by the following expression: 1 dx;y x;y dt 2 1 dsE s2E dt
SR ¼ T1rev U CSEþU e SR ¼ T2rev U CSEþU e
where UCS and USR are the average energy loss of electron per turn due to Compton scattering and synchrotron radiation respectively. 3.3. Intra-beam scattering
The Compton scattering cross-section is described by [7] " 2 ds pr2e 1 E 20 E ¼ dE 2 g2 E las 4g2 E 2las E e E # E0 E Ee E Ee þ þ (3) Ee gE las E e E Ee E
Ex 1 þ b cos a ¼ E las 1 b cos y
5
(7)
As mentioned in part II, intra-beam scattering is an important factor in such a low-energy range. IBS has been described well in Ref. [8]. Factors that influence IBS of an electron beam include electron energy and lattice design. M. Venturini [10] had applied the equations in Ref. [8] to calculate IBS emittance growing rates. However, in this paper, another way of calculating IBS growth rates developed by S. Nagaitsev [9] is applied, which is identical to Ref. [8], yet more efficient for programming and calculation. Calculation on emittance growth due to IBS has been included in the simulation code. 3.4. Beam dynamics in the pulse mode We calculate the values of IBS growth rate, CS quantum excitation rate and damping rate at startup of every injection period, at the laser pulse energy of 1 mJ, initial relative energy spread 0.2% and initial transverse emittance 2 108 mrad. In transverse direction, IBS growth rate is three orders of magnitude larger than the damping effect; in longitudinal direction, IBS growth rate and CS quantum excitation rate have magnitudes of the same order, while damping effect is two orders of magnitude smaller than IBS. Hence, from the calculation, we conclude that IBS is the dominant factor in transverse direction, while IBS and CS play important roles in longitudinal direction. This is different from the situation in the steady mode, in which damping effect counterbalances CS quantum excitation and IBS effect. In the pulse mode, IBS is the most important effect that influences beam dynamics. 4. Simulation of beam dynamics The electron beam is represented by macro particles following a Gaussian distribution. We include IBS, SR, and CS in our simulation. The electron beam energy is set to be 50 MeV in the simulation. As mentioned in part III, less than ten synchrotron photons are emitted per electron per turn in this storage ring; hence the widely used description for synchrotron radiation of ‘‘one big fat photon,’’ drawn from a Gaussian distribution, is inadequate in this case [6]. Therefore, a complete Monte Carlo method is used in which electron macro particles are traced particle-by-particle to simulate their emissions of synchrotron photons. The procedure of Compton scattering is likewise traced particle-by-particle, and a complete Monte Carlo method is used to simulate the interaction. Since we are only interested in the dynamics of electron beam rather than the temporal and spatial distribution of the scattered
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photons, Compton scattering cross-section is integrated all over the interaction area to determine the interaction chances of electron macro particles with laser pulse in a single turn. Thus, the interaction between the laser and electron beams is therefore considered as one single kick at IP per turn. Furthermore, angular distribution of synchrotron radiation and Compton scattering are taken into consideration to simulate the transverse emittance change of beam in the ring. IBS evaluation is carried out using the method described in Ref. [9]. After the particle goes through SR, CS, IBS, and RF cavity, it is multiplied by a transport matrix to simulate its movement around the ring lattice. 4.1. Simulation of transverse emittance The simulation results of transverse emittance at the stored laser energy of 1 mJ are shown in Fig. 5. As can be seen, emittance grows about four times in horizontal direction. We found that IBS is a dominant factor at such a low energy, and covers up the effects from SR and CS. The laser energy stored in the optical cavity has the potential to increase in the future. Therefore, we simulate horizontal emittance at different laser energy in order to identify the effect of CS to the beam dynamics, and to examine the stability and dynamic features of the electron beam. The result is shown in Fig. 6. We can see that, although the cooling effect of laser becomes stronger as laser energy increases, the emittance change in the horizontal direction is dominated by the effect of IBS. Therefore, in transverse direction, IBS is the dominant factor. Hence, increasing photon yield by means of optimizing our lattice to reduce the effect of IBS can be considered an effective method.
Fig. 5. Transverse emittance change in the pulse mode is presented. The simulation is carried out with electron energy of 50 MeV and laser energy of 1 mJ. IBS, SR and CS are all included in the simulation. As is shown in the figure, transverse emittance grows about four times in the horizontal direction.
Fig. 6. Horizontal emittance growth at different laser energy is presented. Simulation is carried out with electron energy of 50 MeV. We can see that laser cooling can be witnessed as laser energy increases. However, IBS effect dominates the emittance change in the horizontal direction.
4.2. Simulation of relative energy spread and scattered photon intensity We simulate relative energy spread and scattered photon intensity at the stored laser energy of 1 mJ, as shown in Fig. 7. The absolute value of photon intensity is the number of scattered photons integrated over photon energy. In order to identify the contribution of IBS, CS and SR separately, we accomplish simulation in three different ways: with all factors; CS and SR; and SR only. We can see from Fig. 7 that at this energy level IBS and CS have excitation rates of the same order, while SR contributes insignificantly to energy spread. The line corresponding to all factors shows that energy spread grows about 1.6 times, and scattered photon intensity drops to 60% of its value at the end of the pulse due to the increase in transverse and longitudinal beam size. At startup the longitudinal beam size is 3 mm and X-ray intensity is 1.7 1012 photons/s, while at the end of the pulse longitudinal beam size is 9 mm and X-ray intensity is 1.0 1012 photons/s. We also enhanced laser energy to see the effect of CS, as shown in Fig. 8. It is predictable that when laser energy goes up CS becomes the dominant factor in longitudinal direction, because the excitation rate of the energy spread is in proportion with scattering chance. Noticeably, we find that when laser energy stored in the optical cavity is 1000 mJ, energy spread does not follow a linear increase with number of turns. This is because when the energy spread goes too high, the beam exceeds RF bucket, thereby resulting in longitudinal beam instability. This can be witnessed in Fig. 9, in which we witness an abrupt drop of scattered photon intensity in line corresponding to 1000 mJ. We know that energy acceptance is determined by [2]
sRF
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2eV RF pac hE e
(8)
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Fig. 7. Simulation results of the energy spread and scattered photon intensity at laser energy of 1 mJ and electron energy of 50 MeV. The absolute value of the scattered photons intensity is the number of scattered photon integrated over the photon energy. We accomplish the simulation in three different ways: IBS, CS & SR; CS & SR; SR only. As shown in the figure, SR is trivial in the longitudinal direction, while the excitation rates of IBS and CS are of the same order at laser energy 1 mJ. The energy spread grows about 1.6 times. Scattered photon intensity drops to 60% of its value at the end of the pulse.
Fig. 8. We show energy spread in this figure. Simulation is carried out at the electron energy of 50 MeV with all effects turned on. We can see that energy spread grows in direct proportion with scattering chance. We observe that with the line corresponding to laser pulse of 1000 mJ, energy spread does not follow the linear increase at around 100,000 turns. This indicates that the beam has exceeded the RF bucket.
We have to either raise RF voltage or reduce momentum compaction factor in order to enlarge energy acceptance. We tried both ways, and the results show that the abrupt drop is eliminated by both means. Yet it is quite difficult to raise RF voltage in such a compact ring design; thus, we have to reduce momentum compaction factor in the lattice design. 4.3. Summary of simulation results In this study, we found that SR does not contribute much to the emittance change either in the transverse or in the longitudinal direction. IBS is fixed when the ring lattice and beam energy are both fixed, while CS may become
Fig. 9. Scattered photon intensity in different laser energy with electron energy of 50 MeV. The absolute value of the scattered photon intensity is the number of scattered photons integrated over photon energy. The intensity decreases quite rapidly to about 60% of its peak at the end of the photon pulse. As we can see from the line corresponding to 1000 mJ, there is an abrupt drop, which indicates that the longitudinal instability has occurred.
strong with the increase of laser energy. However, IBS covers up the effects from CS and SR, and is dominant in the transverse direction. Due to the high energy of scattered photon and small collision angle, emittance growth due to CS is quite large. In the longitudinal direction, CS has an excitation rate with the same order as that of IBS. These results are in accordance with the theoretical analysis in part 3. As mentioned above, when the laser energy goes up to 1000 mJ, it is high enough to cause the energy spread to increase rapidly that longitudinal instability is achieved even in the pulse mode.
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5. Summary We present a lattice that provides an electron beam with a transverse size of electron beam of 32 mm, and X-ray beam intensity of 1.7 1012 photons/s. Since dynamic aperture is intolerable due to large natural chromaticities caused by low beta insertion, we placed groups of harmonic sextupoles to enlarge DA. In calculation and simulation, we know that in the transverse direction emittance grows mainly due to IBS, while in the longitudinal direction CS and IBS have excitation rates of the same order. Moreover, CS has the potential of becoming stronger when laser energy stored in optical storage system goes higher. Both longitudinal size and longitudinal stability require for small momentum compaction factor, which adds to the difficulty of lattice design. Meanwhile, IBS can also be diminished with better lattice design. Therefore, finding a better lattice with smaller momentum compaction factor and less IBS can be the next step of this scheme.
colleagues, Yingchao Du and Dao Xiang, for helpful discussions. This work was supported by the Chinese National Foundation of Natural Sciences under Contract no. 10735050. Reference [1] [2] [3] [4] [5] [6]
[7] [8] [9] [10]
Acknowledgments [11]
The authors are greatly indebted to Professor Peter Gladkikh for many helpful discussions and advice on lattice design. The authors would also like to thank their
[12] [13]
Z. Huang, R.D. Ruth, Phys. Rev. Lett. 80 (1998) 976. Peter Gladkikh, Phys. Rev. Spec. Top. Accel. Beams 8 (2005) 050702. E. Bulyak, et al., Nucl. Instr. and Meth. A 487 (2002) 241. J. Urakawa, et al., Nucl. Instr. and Meth. A 532 (2004) 388. R.J. Loewen, Ph.D. Thesis, Stanford, 2003. Shane Koscielniak, Automated self-convolution applied to synchrotron radiation spectra for use in longitudinal particle tracking, /http://www.triumf.ca/people/koscielniak/synkro-aps.pdfS. IanC. Hsu, Energy measurement of relativistic electron beams by laser Compton scattering, Phys. Rev. E. 54 (5) (1996). J.D. Bjorken, S.K. Mtingwa, Intrabeam Scattering Part. Accel. 13 (1983) 115. Sergei Nagaitsev, Phys. Rev. Spec. Top. Accel. Beams 8 (2005) 064403. Marco Venturini, Study of intrabeam scattering in low-energy electron rings, In: Proceedings of the 2001 Particle Accelerator, p. 2961. Andreas Streun, Practical guidelines for lattice design, SLS-TMETA-1999-0014. H. Wiedemann, Synchrotron Radiation, Springer, Berlin, 2003. J. Yang, M. Washio, A. Endo, T. Hori, Nucl. Instr. Meth. Phys. Res. A 428 (1999) 556.