Nuclear Instruments and Methods in Physics Research B56/.57 (1991) 1130-1132 North-Holland
Orbit analysis of the superconducting
storage ring Super-ALIS
K. Yamada, J. Nakata and T. Hosokawa
Nippon TeIegrapk and Teiephone Corporation, 3-1, Morinosato
Kanagawa Pref., 243-01 Japan
To examine the effects of the fringing field and nonlinear fields of the superconducting bending magnets of a compact storage ring, we have developed an orbit analysis code which integrates the equation of motion numerically to derive the electron trajectory. The field distribution of the bending magnets is estimated by the three-dimensional finite-element method. As a result of the analysis, we derived stable system conditions.
1. Introduction QD-mag.
Synchrotron radiation (SR) is thought to be a promising X-ray source for lithography. However, to realize SR lithography, it is necessary to optimize SR sources for factory use in terms of cost and size. A superconducting storage ring (Super-ALIS) has been developed for this purpose [l]. Adopting superconducting bending magnets and a low energy injection scheme has effectively reduced system dimensions and total cost. There are several problems linked to superconducting rings. For instance, the fringing field length of superconducting magnets is long while the size of the good field region inside the bending magnets is relatively small. The orbit radius is also too small to neglect higher order terms in the equation of motion. The existing orbit analysis codes which assume isomagnetic bending field and use linear approximation are not suitable to examine these problems. In this paper, we describe an orbit analysis code developed for the design of Super-ALIS and we examine these problems with this code.
2. Superconducting storage ring (Super-ALE) Fig. 1 shows the Super-ALE layout. It has a racetrack shape with two 180° superconducting bending magnets. A low-energy (15 MeV) injection scheme is adopted to reduce the size and cost of the injector linac. With this scheme, electrons are directly injected into a storage ring and accelerated to the final energy (600 MeV). A high-energy injection system is also installed to avoid the risks of low-energy injection.
RF Cavity 1-1 QF-msg.\ _QD-mag.
Fig. 1. Layout of Super-ALIS.
decrease the magnetomotive force, reduce the maximum magnetic field strength at the superconducting coil and the force required to support the coil, and shietd the leak field. At first, the coil arrangement and the shape of the iron yoke are optimized by two-dimensional analysis supposing a 360” (cylindrical) bending magnet. Then the field distribution of the 180” bending magnets (especially in the fringing region) is estimated by the thr~-dimensional finite-element method. Results of the analysis indicate that the field distribution changes dynamically during acceleration due to the saturation of the iron yoke. Particularly serious problems are that the fringing field distribution at injection differs from that at storage and that field uniformity worsens in the middle of acceleration.
4. Orbit analysis of Super-ALIS 4.1. Procedure of orbit anatjxis
Iron-yoke-type superconducting bending magnets are chosen to secure adequate field uniformity at injection, Ol68-583X/91/$03.50
We performed the orbit analysis of Super-ALIS in the following sequence. At the beginning, the lattice arrangement was designed with an existing orbit analy-
0 1991 - Elsevier Science Publishers B.V. (North-Holland)
M. Nokajima et al. / Orbit analysis sis code assuming an isomagnetic guide field. We define an electron orbit in this analysis as the mechanical orbit (which consists of straight lines and circular arcs). Three-dimensional magnetic field analysis was performed to derive the field distribution of the superconducting bending magnets along this mechanical orbit. With our orbit analysis code, a closed orbit was calculated based on this field distribution. We call it the design orbit. After the design orbit was defined, we examined closed orbit distortion (COD) due to the deviation of the fringing field and compared the lattice parameters for the design orbit with those for the mechanical orbit. We also examined the effect of multipole components at the bending magnets. 4.2. Orbit analysis method
Let us define the s-axis along the mechanical orbit. We define x as the deviation from the mechanical orbit in the horizontal plane and y in the vertical plane. The Frenet-Seret equations of motion along the s-axis are X”
hx) = $g’/’
x” + y”
(1 + hx)‘,
optimized, and horizontal steering stalled near the fringing region.
5.2. CO1) due to deviation of the fringing field If the fringing field deviates from the designed field, the electron orbit deviates from the designed orbit. As a result, the circumference deviates from the designed circumference. According to the principle of phase stability, the electron energy changes automatically to satisfy the synchronized condition between the circumference and the rf frequency. This condition is responsible for making COD much worse, especially in the case of racetrack type storage rings. Therefore, we cannot neglect this effect in calculating horizontal COD in small storage rings. Iron-yoke-type superconducting bending magnets cause the fringing field to change dyna~cally during acceleration and it is anticipated that the fringing field will differ from the three-dimensional magnetic field analysis. Orbit analysis has shown that these differences of fringing field cause fatal COD without any correction but can be corrected satisfactorily by exciting the steering magnets placed near the fringing region. 5.3. Lattice parameters around the design orbit
((1 +hx)B,-x’B,}, g=
where h( = l/p) denotes the curvature, 3 the field strength, e the electron charge, p the electron momentum, and ’ means differentiation with regard to s. The electron trajectory is derived by integrating this equation of motion with the fourth order Runge-Kutta method. In this calculation, we can use any field distribution derived from either three-dimensional magnetic field analysis or magnetic field measurement. The field distribution in each section of the bending magnets is decomposed into multipole components. In each step of integration, the magnetic field (S,, B,,, B,) is calculated from these multipole components and substituted in eq. (1). Between each section, multipole components are interpolated.
5. Resultsof the analysis
By pursuing the small oscillation of an electron around the design orbit, we can derive a transfer matrix around it. Once a transfer matrix is obtained, tune or Twiss parameters can easily be calculated. The deviation of the design orbit from the mechanical orbit causes a deviation in lattice parameters partly because electrons get into the bending magnet obliquely along the design orbit and are affected by the edge focusing effect and partly because the orbit radius (p) of the design orbit is greater than that of the mechanical orbit at the fringing region. (Note that the focusing force at the bending section is proportional to l/p’). However, these lattice parameters can be corrected nearly to the original value by adjusting the quadrupole magnets. 5.4. Chromaticity
in small storage rings
Analytical formulae maticity with separated
to calculate the natural chrofunction magnets are :
<,=GkLds 5.1. Design orbit We tried to bring the design orbit close to the mechanical orbit because it is easier to design bending magnets with a good field region along a circular mechanical orbit. For this purpose, the shape of the iron yoke and coil arrangement at the fringing region were XIV. ACCELERATOR TECHNOLOGY
M. Nakajima et al. / Orbit analysis
1132 Table 1 The chrornaricity of Super-ALIS
5.5. Effects of multipole components
Analytical formula Total
- 1.61 -2.82
-1.63 - 2.91
- 1.53 - 0.64
0.49 - 2.30
- 0.59 0.03
a Quadrupole effect.
b Fringing effect. ’ Other terms in eq. (2). where p and y are Twiss parameters and q is the dispersion function. In this formula, PK represents the contribution from the quadrupole magnets while #h’v: means the frin~ng effect (because h’ appears in the fringing sections). However, in large storage rings with a large bending radius (p), only the effect of the quadrupole magnets is dominant and taken into account. It is possible that the other terms in eq. (2) are not negligibly small in small storage rings and even the validity of these formulae is unknown, Meanwhile, because chromaticity is generated by (less than) second order terms (regarding x and/or y) in the equation of motion, exact chromaticity can be computed by the orbit analysis method with higher order terms. Table 1 shows the natural chromaticity of SuperALIS calculated from our orbit analysis code and from eq. (2). The results are in good agreement with each other. In small storage rings, the fringing effect is important in calculating vertical chromaticity, but it cancels other effects in horizontal plane. The fringing effect is interpreted as follows. An energy deviation of an electron causes COD which introduces an energy dependent edge focusing effect in the fringing sections and thus chromaticity.
Table 2 Deviation of lattice parameters due to multipole components
v.x “v .$, 4-v
1.60 0.66 - 2.2 - 2.2
1.61 0.63 - 2.4 - 0.2
1.61 0.57 -1.9 - 2.0
Field analysis reveals that field unifor~ty is comparatively good at injection because the field distribution is determined by the shape of the iron yoke. On the contrary, large quadrupole and sextupole components are observed during acceleration due to the saturation of the iron yoke. At final energy, field uniformity is good at the center of the bending magnet (because the optimization is performed for this purpose) but it is not so good in the fringing region (mostly due to the quadrupole component). Among the multipole components, the quadrupole component produces a tune shift while the sextupole component changes chromaticity. We examined the effects of multipole components at each excitation level (table 2). These effects can be corrected by the quadrupole or sextupole magnets along the straight sections. However, the sextupole component affects the dynamic aperture as well. With multipole components and chromaticity correction sextupoles taken into account, we got a sufficiently wide dynamic aperture in comparison with the size of the electron beam. Although quadrupole and sextupole components turned out to be allowable, quadrupole and sextupole correction coils were installed at the surface of the bending magnets because of the uncertainty of the three-Dimensions magnetic field analysis. 6. Summary Orbit analysis based on a three-dimensional magnetic field calculation has been performed on a superconducting storage ring. The change in the fringing field and poor field uniformity during acceleration were shown to be serious problems. The solutions to these problems were presented and we confirmed stable system conditions. We have succeeded in accelerating and storing more than 100 mA at the final operating energy.
References [l] T. Hosokawa, T. Kitayama, T. Hayasaka, S. Ido, Y. Uno, A. Shibayama, J. Nakata, K. Nishimura and M. Nakajima, Rev. Sci. Instr. 60 (1989) 1779.  M. Bassetti, LEP Note 504 (1984).