Quadruple-bend achromatic low emittance lattice studies

Transcription

1 REVIEW OF SCIENTIFIC INSTRUMENTS 78, Quadruple-bend achromatic lo emittance lattice studies M. H. Wang, H. P. Chang, H. C. Chao, P. J. Chou, C. C. Kuo, and H. J. Tsai National Synchrotron Radiation Research Center, Hsinchu 30076, Taian S. Y. Lee, a W. M. Tam, and F. Wang Department of Physics, Indiana University, Bloomington, Indiana Received 16 March 2007; accepted 20 April 2007; published online 30 May 2007 A quadruple-bend achromatic QBA cell, defined as a supercell made of to double-bend cells ith different outer and inner dipole bend angles, is found to provide a factor of 2 in loering the beam emittance relative to the more conventional double-bend achromat. The ratio of bending angles of the inner dipoles to that of the outer dipoles is numerically found to be about for an optimal lo beam emittance in the isomagnetic condition. The QBA lattice provides an advantage over the double-bend achromat or the double-bend nonachromat in performance by providing a small natural beam emittance and some zero-dispersion straight sections. A lattice ith 12 QBA cells and a preliminary dynamic aperture study serves as an example American Institute of Physics. DOI: / I. INTRODUCTION In the last ten years, many lo emittance electron storage rings ere constructed to provide high brilliance x-ray photon sources. 1 Double-bend DB cells ere often used to accommodate many straight sections for insertion devices IDs, such as the igglers and undulators. To operation modes for the DB lattice are either the achromatic condition to achieve the minimum emittance double-bend achromat MEDBA or the nonachromatic condition aiming for the theoretical minimum emittance TME. Theoretically, the natural emittance attained in the nonachromatic mode is three times smaller than that attained in the DBA mode. The horizontal natural emittance of an electron beam in an electron storage ring is determined by the equilibrium beteen quantum fluctuations and radiation damping. The minimum emittances for the DBA and the TME lattices are MEDBA = Jx C q 2 3, 1 TME = MEDBA = C 12 q 2 3, 2 15Jx here C q =55 / 32 3mc = m, J x 1 is the horizontal damping-partition number, and is the bending angle of dipoles in one bending section. Relaxing the achromatic condition, the emittance can generally be reduced by a factor of 3. The bottom plot of Fig. 1 shos the emittance of existing storage rings scaled to 3 GeV beam energy versus the number of pairs of dipole magnets. The top plot of Fig. 1 shos the ratio of the attained emittance to its corresponding theoretical minimum for these storage rings. The diamond symbols in Fig. 1 are associated ith lattices at the achromatic condition, hile the square symbols are results of nonachromatic lattices. We note that the emittances attained in the lattice design are a factor of 2 4 larger than their theoretical minima of Eq. 2, as shon in the top plot of Fig. 1. A simple rule to attain small emittance is to increase the number of cells, so that the bending angle is reduced, but the number of cells and the circumference of the accelerator ill increase. The question is ho to attain a minimum emittance ith a fixed circumference and fixed number of cells in order to accommodate an ever-increasing number of IDs. This article studies the option of the quadruple-bend achromat QBA lattice, hich is defined as a supercell composed of to double-bend cells ith unequal bending strengths for the outer and inner dipoles. We ill compare the emittances attained in the QBA design and the DB A design. There ere to published QBA lattice designs. 2,3 This ork differs from the previous studies in 1 retaining the dispersive straight sections so that the optical matching is easier and 2 providing an optimal length ratio beteen the inner and outer dipoles. There are also multibend achromat lattices proposed to attain lo emittances, as shon in Fig. 1 labeled as nba. These lattices lack the available straight sections for the ever-increasing need for many insertion devices. Our concept can also be considered as a possible upgrade for some existing facilities that presently use the DB A structure. This upgrade is most beneficial for lo to medium energy storage rings that can benefit emittance damping from high field igglers and avelength shifters. We organize this article as follos. Section II revies the dipole and ID contributions to the emittance of DB A designs. Section III introduces the QBA lattice concept and studies the properties of QBA lattices. We compare the QBA lattices ith the existing DB A lattices of the Advanced Photon Source at the Argonne National Laboratory, the National Synchrotron Light Source II, and the Taian Photon Source. In particular, e study a QBA lattice for the Taian Photon Source in depth and compare the effective emittance for the QBA and DB A lattice options. The conclusions are given in Sec. IV /2007/78 5 /055109/8/$ , American Institute of Physics

2 Wang et al. Rev. Sci. Instrum. 78, FIG. 1. Color online Compilation of emittances achieved for some synchrotron light sources Ref. 1. The square symbols are the natural emittances of nonachromatic lattices, and the diamond symbols are emittances of the achromatic lattices. Top: The ratio of the emittances achieved to that of the corresponding theoretical minimum plotted as a function of the number of pairs of dipoles. Bottom: The natural emittance in unit of nanometers for storage rings scaled to 3 GeV plotted as a function of the number of pairs of dipoles. The blue solid and the red dashed lines are the minimum emittance for the DBA lattice and the theoretical minimum emittance TME, respectively. For the multiple-bend lattices, the pairs of dipoles are not equal to the number of cells. For example, a lattice ith 12 7BA is equivalent to 42 pairs of dipoles. The minimum emittance of such multibend lattice is given in Refs. 5 and 6. The red X symbols are results from the QBA lattices to be addressed in Sec. III. The cross symbols in magenta are taken from Ref. 3. II. EMITTANCE OF THE DB A LATTICES The horizontal natural electron beam emittance of a storage ring arises from the balance of quantum fluctuations and radiation damping. The resulting emittance of electron beams is 4 x = C q 2 I 5 I 2 I 4, here is the relativistic energy factor. The radiation integrals are I 2 = 1/ 2 ds, I 4 = D x / 1/ 2 +2K ds, and I 5 = H/ 3 ds, here s is the length coordinate along the accelerator, is the bending radius of dipoles, D x is the dispersion function, K is the focusing function, and the H-function is H = 1 x D x 2 + x D x + x D x 2. Here, D x is the derivative of the dispersion function ith respect to s, x is the betatron amplitude function, and x = 1 2 d x/ds. The minimization of beam emittance requires minimizing the average H-function through the dipoles. For an isomagnetic, separate function accelerator, I 2 =2 /, I 4 0, and I 5 =2 H / 2, and e find x C q 2 H /, here H is the average of the H-function in dipoles. FIG. 2. Color online H/H TME of one superperiod to DB cells for a small emittance DBA lattice red dashed line and a small emittance DB lattice black. Here, H TME = 3 / 3 15 is the maximum H-function for the theoretical minimum emittance TME lattice. The dotted and dot-dashed lines are H in the dipoles. The circumference of the storage ring is m, ith 15 long and short straight sections at and 7 m, respectively. The betatron tunes are and 15.40, respectively. The natural chromaticities are 70 and 33, respectively. The natural emittances are 2.28 nm for the achromatic mode and 0.75 nm for the nonachromatic modes. This lattice as obtained by removing 60 quadrupoles from that of Ref. 8 hile keeping the same circumference. Using a small angle approximation, e find H TME =1/4H TME for the TME lattice, here H TME = 3 / 3 15 is the H-function outside dipoles. For the MEDBA, e find H MEDBA = 3 4 H TME. In this case, the H-function in the dispersion matching section is 3H TME. Figure 2 shos an example of the H-function in a DB lattice in to modes of operation. The dotted lines are the average of the H-function in dipoles. Had e achieved the MEDBA and the TME conditions, these dotted lines ould have values of 0.75 and 0.25, respectively. In the achromatic mode, the emittance does not reach the MEDBA value. Rather, e find the H-function in the dispersion matching section is larger than 3H TME. Similarly, the emittance does not reach the TME value in the nonachromatic mode. Hoever, the values of the H-function at the ID and dispersion sections are nearly equal to H TME. In this lattice, H /H TME =0.59 instead of 0.25 for the TME. Since the H-function is invariant outside dipoles, the most basic three-bend achromat TBA design can never achieve its intended minimum emittance because the H-functions from both TME and MEDBA cannot be matched in the dispersion matching section. 5 In order to achieve dispersion matching, the outer dipole must be made shorter or have a eaker magnetic field. A. The effective emittance Because the DB A lattice is easily tunable and yields a large number of straight sections, many storage rings use the DB A design. In the DB A design, there are achromatic and nonachromatic modes of operation. Theoretically, the nonachromatic mode can reach an emittance three times

3 Lo emittance lattice Rev. Sci. Instrum. 78, smaller than that at the achromatic condition. Thus, most synchrotron light sources operate in the nonachromatic mode in order to reach a smaller emittance. When the storage ring operates at the nonachromatic mode, the dispersion function is not zero in the ID straight sections, and thus the beam momentum spread can contribute to the horizontal beam idth. We define the one dimensional 1D effective emittance in the ID section as B. Effect of IDs on beam emittance and off-momentum spread We have concluded that the nonachromatic mode has a smaller effective beam emittance. No, e ask another question: to hat degree do IDs in dispersive sections spoil the emittance? Although the formulas to evaluate the effect of IDs on the beam emittances are available in Ref. 7, e rederive these formulas here using the invariant H-function. x,1d = x + H ID 2, 3 here H ID is the H-function at the ID locations, 2 =C q 2 / J E is the square of the rms momentum spread, and J E =2+I 4 /I 2 2 is the longitudinal damping-partition number. More commonly, in the light source community, the effective emittance is defined as x,eff = x x,1d. Interestingly, the 1D effective emittances of the TME and MEDBA lattices are equal. 6 We note that both the angular divergence of the photon beam, 1/, and the angular divergence of the electron beam orbit in undulators, ˆ e K/, here K=eB / 2 mc, are much larger than the betatron divergence x = x x for the small emittance beams, here x = 1+ 2 x / x. Furthermore, hen the linear betatron coupling is minimized, the vertical emittance may be dominated by the vertical dispersion function induced by vertical closed orbit distortion and ske quadrupole errors. Thus, the final photon brilliance may be determined by the 1D emittance! So far, there is no systematic study on this effect. Hoever, e use the slightly rosier to-dimensional 2D effective emittance as the figure of merit. Since it is difficult to design a lattice reaching either the MEDBA or the TME condition, e ask a simple question: are e better off ith the achromatic or the nonachromatic mode of operation? Let us assume e can design a lattice in the achromatic mode ith x,a = f a MEDBA and another nonachromatic mode ith x,na = f na TME, here both f a and f na are typically about 2 4. The effective emittances of these to lattices become x,a = x,1d,na = f a 4 15Jx C q 2 3, f na 12 15Jx C q C q 2 J E, here e have used H ID H TME. Assuming J x 1 and J E 2, e find x,1d,na x,a f na +2 3f a, x,eff,na x,a f na f na +2 3f a. 7 If both f a and f na are 2, the 1D and effective emittance of the nonachromatic lattice are about about 0.66 and 0.47 of the achromatic one, respectively, i.e., the nonachromatic lattice can provide higher beam brilliance. Hoever, the emittance reduction factor is not 1/3. 1. Effect of IDs on beam emittance: Nonachromatic lattice In the absence of IDs, the natural emittance is x0 = C q 2 I 50 I 20 I 40, here I i0 s are the standard radiation integrals from the storage ring dipoles. In the presence of IDs, the emittance becomes x = x0 1+I 5 /I I 2 I 4 / I 20 I 40, here I i s are radiation integrals of the IDs. Normally, the change of the H-function due to the IDs is small. The radiation integrals have the folloing properties: I 40 I 20, I 4 I 2, I 4 I 20, and I 2 /I 20 = /, here = C E 4 / 0, = U C E 4 L / 4 2 C E 4 L / 2 2 planer helical undulator, are the single-turn single electron energy losses in the storage ring dipoles and in an undulator or iggler, respectively, C = m/gev 3, 0 and are the bending radii of storage ring dipoles and iggler magnets, L is the length of the undulator or iggler, and E is the beam energy. For a TME lattice, e have H = 1 4 H ID. Here, H ID is the maximum of the H-function at the ID end of dipole magnet. On the other hand, it is not easy to achieve the TME condition, and e find H = f h H ID, here f h 0.56 for the sample lattice shon in Fig. 2. Since the H-function is invariant in the straight section, 6 its value H ID is constant in the entire ID region. For a ell designed lattice ith isomagnetic dipoles, e find I 50 f h H ID 1 3ds = f hh ID 2 0 2, I 5 = H ID W 1 3ds 4 H ID L 3 3 planer H ID L 3 helical,, 11

4 Wang et al. Rev. Sci. Instrum. 78, here 0 and are bending radii of the storage ring dipoles and the iggler magnets, respectively. Here, e assume a sinusoidal undulator ith a planer magnetic field. Thus, e find I 5 I f h = 8B 3 f h B 0 planer 0 f h = B f h B 0 helical, 12 here B and B 0 are the magnetic flux densities of the undulator and the main dipole magnet. Thus, the emittance becomes x x0min = f h f h 1+ U for the planer and helical undulators, respectively. For eaker field undulators B 3 f h /8 B 0, the emittance ill decrease slightly. For strong field igglers, the natural emittance ill increase! To minimize the emittance increase, H ID should be made smaller for strong field IDs. For a given H, e ould prefer a larger H in the dispersion matching section and a smaller H in the ID section. The black curve in Fig. 2 shos that a small improvement can be made by increasing the H-function in the dispersion matching section hile reducing the H-function in the ID straight sections. High field igglers ould preferably be installed in the straight section having a smaller H ID. To minimize emittance degradation in the nonachromatic lattice, one chooses a strong main dipole field by making the dipole length short, or 0 small. The draback is that the value of dispersion function in the dispersion matching section, hich is proportional to 2, becomes smaller, and thus the chromatic sextupole strength becomes larger. Optimization of the dynamic aperture becomes a difficult task in the design of such a machine. Since in high energy photon sources such as the ESRF, APS, and Spring-8, nonzero-dispersion function in the IDs does not cause much emittance degradation, and thus there is a small advantage of the nonachromatic DB mode over the DBA mode of operation, as shon in Eq Effect of IDs on emittance: Achromatic lattice As discussed above, each ID increases radiation poer in a storage ring. If an ID is located in an achromatic straight section, here H=0, the natural beam emittance is reduced by x x0 1+ /, 14 here e assume that the effect of the ID on dispersion function is small so that I 5 0. Since x0dba 3 x0min, the energy loss in the damping IDs must be at least tice that in the dipole to achieve the same emittance as the nonachromat lattice. Hence, in order to maximize the emittance damping effect, one chooses a lo main dipole field option. Thus, the dipole length L is increased, and the dispersion function L in the dispersion matching section is larger. This, in turn, reduces the sextupole strength needed for chromatic correction. The dynamic aperture is still a very difficult problem, but is relatively easier to handle than in the high field dipole option. Hoever, the DBA design requires a larger circumference or, potentially, a smaller fraction of available space for IDs. 3. Effects of IDs on momentum spread The beam momentum spread is given by 2 E = E 2 1+I 3 /I 30, 15 E 1+I 2 /I 20 E 0 here I 3 = 1/ 3 ds, and I 4 is assumed to be small. The radiation integrals in the numerator are I 30 =2 / 2 0, = 4 L I 3 and L 3, 2 E = E E E 0 I3 I 30 = , 16 for the planer and helical undulators, respectively. If B 3 B 0 /8 for the planer undulator, the beam momentum spread ill decrease. On the other hand, the high field IDs can increase momentum spread, particularly important for the lo main dipole field design Illustrative example of effective emittance in different modes of operation To minimize emittance, e ant to minimize H. For a small emittance, nonachromatic lattice, the parameter f h = H /H ID is small, but hen f h is small, the IDs in the dispersive straight sections can cause emittance degradation. In this section, e consider an example of the Taian Photon Source. A ne 3 GeV Taian Photon Source TPS has been proposed by National Synchrotron Radiation Research Center NSRRC in Taian. The circumference of the storage ring is 486 m. A ell designed 24-DB A -cell lattice has achieved a natural emittance of 1.7 nm rad for the nonachromatic mode and 5.2 nm rad for the achromatic mode. The

5 Lo emittance lattice Rev. Sci. Instrum. 78, FIG. 3. The optical functions of one superperiod for the TPS double-bend nonachromat lattice. The values of f h in dispersive straight sections are 0.73, 0.60, 0.56, optical functions of the nonachromatic case are shon in Fig. 3. The lattice has a structure of 24 cells of DBA ith a sixfold symmetry. It has 24 straights including 6 long straights 10.9 m and 18 standard straights 5.8 m for injection devices or IDs. The main homogeneous dipole field is about 1.19 T at 3 GeV. From the point of vie of photon beam brightness, it appears that the nonachromatic mode of operation is much more advantageous than the achromatic mode of operation. For the 3 GeV storage ring, high field igglers are needed to produce photon energies larger than about 20 kev. Table I lists parameters of the IDs proposed by the NSRRC user group. The proposed IDs are typical for all high brightness storage rings. The effect of IDs on the beam emittance and momentum spread can be evaluated ith lattice design programs, and can also be evaluated ith Eqs. 13 and 16. We find that the results from the analytic formula agree very ell ith those obtained from lattice design programs. 9 If all proposed IDs ere installed, the total energy lost from all IDs ould be MeV per revolution, and the energy spread ould be %. The ID loaded emittance for the nonachromatic mode ould increase from its original value of TABLE I. Parameters of IDs of TPS. ID name Length m Peak field B z /B x T Number of ID EPU EPU / SW EPU / EPU / IVU SU CU FIG. 4. Color online The effective emittance, defined in Eq. 4 including the effect of all IDs ith f h =0.62 ith a natural emittance at 1.7 nm, plotted as a function of the main dipole field for various lattice types for the Taian Photon Source. The natural emittance of the DBA lattice ithout insertion devices is 5.2 nm. The effective emittance of the QBA lattice is calculated, assuming that high field IDs are installed in achromatic straight sections and medium field IDs are installed in the dispersive straight sections. The natural emittance of the QBA lattice is 3.0 nm. 1.7 to 2.12 nm. On the other hand, the emittance of the achromatic mode decreases from 5.2 to 3.11 nm. The emittance ratio is not 3 but For the nonachromatic case, the presence of the dispersion in straight sections must be taken into account in the calculation of the effective emittance. The 2D effective emittance in straight sections, defined by Eq. 4, becomes 2.72, 2.82, and 2.86 nm rad depending on the actual H ID in each ID. On the other hand, the ID loaded effective emittance for the achromatic lattice is still 3.11 nm rad. The actual gain of the nonachromatic mode of operation is very small! The effect of IDs on emittance depends on the main dipole field B 0. Some accelerator designers employ the high field option to minimize the effect of IDs on the emittance. Figure 4 shos the effective emittance, including the effect of IDs ith f h =0.62, as a function of the main dipole magnetic field B 0 for various designs of the Taian Photon Source. Here, the natural emittances for 24 cell DBA and DB nonachromat are 5.2 and 1.7 nm, respectively. The TPS design corresponds to an effective DBA emittance of 3.11 nm and an averaged DB nonachromatic emittance of 2.8 nm at B 0 =1.19 T shon in Fig. 4. In the folloing section, e examine a hybrid option of a storage ring design to minimize emittance. III. THE MEQBA LATTICE In an effort to design a MEQBA lattice for the Taian Photon Source, e combine to DB cells into one QBA supercell hile retaining all long straight sections. Let the lengths of outer and inner dipoles be L 1 and L 2, respectively. Based on the small angle approximation, the MEQBA lattice requires a necessary condition L 2 /L 1 =3 1/ to match the dispersion function. The minimum emittance for the

6 Wang et al. Rev. Sci. Instrum. 78, FIG. 5. H/H TME of one super QBA cell to-db cells, here H TME = 3 2 / 3 15 is the maximum H-function for the theoretical minimum emittance TME lattice based on the middle dipole. The parameters used in this lattice are L 1 =1.8 m and L 2 =3.0 m. The dotted line is the H in dipoles, derived from the emittance of 1.02 nm at 3 GeV. FIG. 6. Optical function of a single QBA cell. The resulting emittance is about 3.0 nm, or about a factor of 3 larger than the minimum theoretical emittance. MEQBA is given by Eq. 1, here = 1 is the bending angle of the outer dipoles. Hoever, the distribution of quadrupoles may not yield the minimum emittance condition, thus the condition for an optimal L 2 /L 1 may vary. The QBA lattice has an advantage over the the nonachromatic DB lattice in having some zero-dispersion straight sections and has an emittance much smaller than that of a DBA lattice. For example, the natural emittance can reach an emittance of 1.0 nm for a lattice ith 16 QBA supercells 64 dipoles at 3 GeV or 2.8 nm for a lattice ith 20 supercells 80 dipoles at 7 GeV. The natural emittances of some examples, scaled to a 3 GeV beam energy, are shon as red X symbols in Fig. 1. A lattice ith P QBA cells has P zero-dispersion straight sections and P dispersive straight sections. Each QBA cell can be individually tuned to match its ID requirements; i.e., a fe QBA cells can form a superperiod ith varying ID lengths in the dispersion-free straight sections. Figure 5 shos the H-function of a lattice ith 16 QBA cells, a circumference of m, and an emittance of 1.02 nm at the 3 GeV beam energy. The lengths of the straight sections are 10.0 and 5.77 m, respectively. A. A QBA lattice example To provide a concrete example, e study a QBA lattice ith a circumference of 486 m and 12 QBA cells for the TPS design. We choose L 2 /L 1 =1.5 and L 1 +L 2 =2.5 m, i.e., 1 =6, 2 =9, and B 0 =1.048 T. The length of L 1 +L 2 is chosen to loer the dipole radiation poer loss, optimize emittance reduction from damping igglers and avelength shifters, and allo for lo field, lo poer undulators in the dispersive straight sections. To maximize the number of insertion devices, e have achieved and m straight sections. The ratio of straight sections to the total circumference is 33%. The lattice design is thus very compact. 10 There are ten families of quadrupoles ith reflection symmetry in a superperiod. The betatron tunes are x =26.28 and y =12.25, and the natural chromaticities are 64 and 30, respectively. Figure 6 shos the optical functions. The emittance of this lattice is 3.0 nm rad. The IDs in the dispersive straight sections should be emittance neutral; i.e., they must have lo field or lo total poer. We can estimate these effects as follos. Using f h = H H ID H H TME = 3 15 x C q 2 2 3, 17 here 2 is the bending angle of the long dipole of the QBA lattice, e find f h =0.68. All planer undulators ith a peak field near or loer than 0.84 T does not cause a substantial emittance groth; i.e., they are emittance neutral. About 16 of the planned undulators at the TPS fall into this category. rf cavities and other emittance neutral devices can be installed in dispersive straight sections. High field damping IDs, installed in the dispersive-free straight sections, can further decrease the emittance. Figure 4 shos an effective emittance for the QBA lattice. Note that if all or 60% of the IDs are installed in the achromatic straight sections of a QBA lattice, the effective emittance is smaller than that of the DB or the DBA lattices, particularly ith the lo dipole field option Fig. 4. Although the QBA emittance 3.0 nm is larger than the emittance of the nonachromatic DB lattice 1.7 nm, the effective emittance of the QBA lattice in the achromatic straight section is smaller after including the insertion devices. This effect is shon as red lines in Fig. 4. Control of the dynamic aperture is a difficult problem in all lo emittance lattices. We chose to use eight-families of sextupoles to correct the chromaticity and to improve the dynamic aperture. The lattice contains to-families of chromatic sextupoles located in the dispersion-function matching section and six families of harmonic geometric sextupoles elsehere. To families of harmonic sextupoles are located in zero-dispersion regions. These families can be used to correct geometric aberration ithout affecting the chroma-

7 Lo emittance lattice Rev. Sci. Instrum. 78, FIG. 7. Color online Dynamic aperture in units of millimeters based on tracking calculations ith no random errors. The rms beam sizes at this location are x 0.19 mm and z mm. ticities. At zero chromaticity in both planes, the dynamic aperture is shon in Fig. 7 for on-momentum and ±3% offmomentum particles. B. The effect of dipole length ratio Although the theoretical necessary matching condition is L 2 /L 1 =1.44 based on the small angle approximation 5, the actual optimal matching condition may vary. The left plot in Fig. 8 shos the emittance versus L 2 /L 1 for lattices ith 12 QBA cells and L 1 +L 2 =2.0 m straight sections: 9.6 and 6.0 m, 16 QBA cells ith L 1 +L 2 =4.8 m straight sections: 10.0 and 5.77 m, 20 QBA cells ith L 1 +L 2 =7.9 m straight sections: 10.4 and 6.0 m, and 24 QBA cells ith L 1 +L 2 =4.8 m straight sections: 7.8 and 5.6 m. The circumference of these accelerators are m for the TPS, m for the NSLS-II, and 1104 m for the APS. The right plot of Fig. FIG. 8. Color online Left: The emittance of QBA lattices vs L 2 /L 1 hile keeping L 1 +L 2 =const. For the 3 GeV storage rings, e choose lattices ith 12 and 16 super QBA cells. For the 7 GeV ring, e choose lattices ith 20 and 24 super QBA cells for a possible retrofitting of the APS. Right: Emittances normalized to 3 GeV and 16 QBA cells. Note that it is harder to reach a small emittance for lattices ith short high-field dipoles. 8 shos the emittance normalized by 2 3 to 3 GeV and 16 QBA cells. A small spread in the emittance normalized by this scaling la indicates the effect of dipole length. It is harder to reach a small emittance ith short, high field dipoles. We also note that there is a broad minimum emittance at L 2 /L For a properly matched lattice, the H-function at the dispersive straight section, H DID, increases ith the ratio L 2 /L 1 parameter. As shon in Fig. 5, e find f h =0.46. The ID used in this dispersive straight section must be a lo field undulator in order to minimize emittance degradation see Sec. II B 1. We note that three DB cells can also be formed into an achromatic 6BA ith one achromatic and to nonachromatic straight sections. For example, a 24-cell DB lattice can be converted to an 8-cell 6BA lattice ith 8 dispersion-free and 16 dispersive straight sections. The resulting emittance ould be smaller than the QBA lattice. Hoever, the number of achromatic straight sections is also reduced by 2/3. Finally, e also note that the DB lattice can operate in a mode ith alternate zero-dispersion and dispersive straights, as shon in Fig. 8 ith L 2 /L 1 =1. A strong iggler installed in the zero or small dispersion straight section can help to reduce emittance. Hoever, the natural emittance is not as good as that achieved ith an optimal L 2 /L 1 ratio because of the dispersion mismatch. IV. DISCUSSION The lo emittance QBA-lattice concept has been studied for a fe lo emittance storage rings and compared ith DBA and nonachromatic DB lattices. We find that the emittance of the QBA lattice is smaller than that of the corresponding DBA lattice by nearly a factor of 2, and yet the QBA lattice retains the same flexibility as that of the DB A lattice. The effective emittance of the QBA lattice is also smaller than that of the nonachromatic DB lattice, hile keeping some zero-dispersion sections for high field damping igglers. We presented simple analytic formulas to calculate the effect of IDs on emittance in the dispersive straight sections. The formulas sho that hen B 3 /8 f h B 0, the emittance ill increase, and hen B 3 /8 B 0, the momentum spread ill increase. The f h parameter is typically in DB or QBA lattices. The QBA lattice obeys a simple emittance scaling la of 2 3, as shon in Fig. 8. Applying the scaling la at 7 GeV, a QBA lattice ill reach an emittance of 1 nm ith 28 QBA cells and 2.8 nm ith 20 QBA cells. It is knon that the dynamic aperture is an important and difficult subject in high brilliance storage rings. Based on actual lattices 12, 16, 20, and 24 QBA-cell storage rings studied in this article, e note that the lo emittance QBA lattice has moderate natural chromaticities, C x,z / x,z 2. A good dynamic aperture should be attainable for an operational robust lattice design. We found that e can attain a very good dynamic aperture ith eight families of sextupoles in a 486 m 12-cell QBA lattice at the NSRRC. Issues on the chromatic correction are the betatron phase advance, the magnitude of dispersion

8 Wang et al. Rev. Sci. Instrum. 78, functions, and the ratio of betatron amplitudes at the sextupoles. These properties can be addressed in the final lattice design as they depend on specific requirements of the light source. ACKNOWLEDGMENTS This ork is supported in part by grants from the U.S. Department of Energy under Contract No. DE- FG0292ER40747 and the National Science Foundation Contract No. NSF PHY The authors thank Dr. W. P. Jones for reading this manuscript. One of the authors S.Y.L thanks colleagues at the NSRRC for their hospitality. 1 G. Mülhaupt, Proceedings of the EPAC, 1990 unpublished, p.65;m. Böge et al., Proceedings of the EPAC, 1998 unpublished, p. 623; M. Böge, Proceedings of the EPAC, 2002 unpublished, p. 39; L. Dallin, I. Blomqvist, D. Loe, M. Silzer, and M. de Jong, Proceedings of the PAC, 2003 unpublished, p. 220; M.-P. Level et al., ibid. unpublished, p. 229; R. P. Walker, ibid. unpublished, p. 232; R. Hettel et al., ibid. p.235;d. Einfeld et al., ibid. unpublished, p. 238; B. Podobedov et al., ibid. unpublished, p. 241; H. Ohkuma et al., ibid. unpublished, p. 883; G. LeBlanc et al., ibid. unpublished, p. 2321; M. Eriksson et al., Proceedings of the EPAC, 2004 unpublished, p. 2392; J. B. Murphy et al., ibid. unpublished, p. 2457; A. Jackson, Proceedings of the PAC, 2005 unpublished, p. 102; Z. Zhao et al., ibid. unpublished, p. 214; R. O. Hettel, ibid. unpublished, p. 505; G. Vignola et al., ibid. unpublished, p.587; V. M. Tsakanov, ibid. unpublished, p. 629; E. S. Kim, Proceedings of the APAC, 2004 unpublished, p. 85; C. C. Kuo et al., Proceedings of the PAC, 2005, p. 2989; D. Einfeld et al., ibid. unpublished, p K. Tsumaki, R. Nagaoka, K. Yoshida, H. Tanaka, and M. Hara, Proceedings of PAC, 1989 unpublished, p D. Einfeld and M. Plesko, Nucl. Instrum. Methods Phys. Res. A 335, M. Sands, Report No. SLAC-PUB-121, 1979 unpublished. 5 S. Y. Lee, Phys. Rev. E 54, S. Y. Lee, Accelerator Physics World Scientific, Singapore, See e.g., H. Wiedemann, Particle Accelerator Physics Springer, Ne York, 2007 ; A. Ropert, Proceedings of the CERN Accelerator School, 1996 unpublished, Paper No. CERN S. Krinsky, J. Bengtsson, and S. L. Kramer, Proceedings of EPAC, 2006 unpublished, p H. C. Chao, C. C. Kuo, and S. Y. Lee, Proceedings of the Asian Particle Accelerator Conference, 2007 unpublished. 10 D. Einfeld private communications ; see the Proceedings of the CANDLE Workshop, 2006 unpublished.