Volume 21 Received 1 March 2014  First multibend achromat lattice consideration^{a}MAX IV Laboratory, Lund University, PO Box 118, Lund SE221 00, Sweden,^{b}COSYLAB, Teslova ulica 30, Ljubljana SI1000, Slovakia, and ^{c}HAWK University of Applied Sciences and Arts, Hohnsen 4, D31134 Hildesheim, Germany By the beginning of 1990, three thirdgeneration synchrotron light sources had been successfully commissioned in Grenoble, Berkeley and Trieste (ESRF, ALS and ELETTRA). Each of these new machines reached their target specifications without any significant problems. In parallel, already at that time discussions were underway regarding the next generation, the `diffractionlimited light source (DLSR)', which featured subnm rad electron beam emittance, photon beam brilliance exceeding 10^{22} and the potential to emit coherent radiation. Also, at about that time, a first design for a 3 GeV DLSR was developed, based on a modified multiplebend achromat (MBA) design leading to a lattice with normalized emittance of _{x} = 0.5 nm rad. The novel feature of the MBA lattice was the use of seven vertically focusing bend magnets with different bending angles throughout the achromat cell to keep the radiation integrals and resulting beam emittance low. The baseline design called for a 400 m ring circumference with 12 straight sections of 6 m length. The dynamic aperture behaviour of the DLSR lattice was estimated to produce > 5 h beam lifetime at 100 mA stored beam current. Keywords: synchrotron light source; beam optics; lattice; DLSR; MAX IV. 
One of the first proposals for the lattice design of a diffractionlimited light source was produced by the authors in the early 1990s. At that time the new thirdgeneration light sources ALS, ESRF and ELETTRA were commissioned in record times. All three machines reached their target specifications without any significant beamdynamic problems and have operated reliably ever since. The successful commissioning results gave us confidence to expect that it was possible to operate storage rings with horizontal emittance an order of magnitude lower, in the 0.5 nm rad range. With a coupling of 1%, the vertical emittance would be of order 5 pm rad. For this special issue of the Journal of Synchrotron Radiation, we summarize the early design efforts in the 1990s and with hindsight relate those proposals to present storagering lightsource lattice designs (Einfeld & Plesko, 1993a,b; Einfeld et al., 1994a, 1995). Having performed the original calculations over 20 years ago, we are thrilled with the prospect that the new MAX IV facility will be the first synchrotron radiation light source to implement a subnm rad variant of this design and that other proposed machines are following the MBA concept.
One of the most important factors for synchrotron radiation research is the photon beam brilliance which in storage rings is determined by the electron beam emittance and coupling between horizontal and vertical planes. Even in the limit of zero beam emittance, however, the phase space of the radiation emission from an undulator is itself finite due to diffraction effects at the source. For singlemode photon emission, the corresponding diffractionlimited `emittance' of the photon beam is given by
where is the Xray wavelength and E_{} is the photon energy in keV (Coisson, 1988; Wiedemann, 2002). For photon energies of 1, 5 and 10 keV, the corresponding beam emittance should be smaller than 100, 20 and 10 pm rad, respectively. Formula (1) has been much debated and there is still not full consensus that the factor 4 in the denominator is numerically correct (Onuki & Elleaume, 2003). Nevertheless, a light source is referred to as `diffractionlimited' when the electron beam emittance is less than that of the radiated photon beam at the desired Xray wavelength.
By way of review, recall that the horizontal emittance of an electron storage ring beam is determined by a balance between two competing processes: quantum excitation of betatron oscillations from photon emission and longitudinal reacceleration within the RF cavities. The basic formula to calculate storage ring emittance, assuming isomagnetic bend magnets and no insertion devices, is summarized as (Ropert, 1998)
where C_{q} = 3.841 × 10^{13} m is a constant, _{o} is the relativistic Lorentz factor, J_{x} is the Robinson partition number evaluated for the horizontal plane, is the dipole magnet bending radius and <H>_{mag} is the average of H evaluated in the bending magnets,
In (3), , and are the standard positiondependent Twiss parameters in the horizontal plane, and , ' are the horizontal dispersion function and first derivative, respectively. As a general rule for low emittance we aim to minimize the integral quantity <H>_{mag} to maintain sufficient straightsection free space for insertion devices and operate at high enough beam energy to meet the spectral requirements of the user community.
Referring again to equation (2), the horizontal beam emittance is seen to scale with the square of the beam energy, linearly with <H>_{mag}, and is inversely proportional to the partition number J_{x}. For bending magnets with a pure dipole field, J_{x} = 1. Since the Hfunction is determined by the Twiss parameters and dispersion in bending magnets, low emittance can be reached if the beam matrix and (s) have specifically controlled values within the bending magnets. By way of example we can analytically evaluate the horizontal emittance for two well known examples: the first bend magnet of a flatfield doublebend achromat (DBA) lattice (Chasman & Green, 1975) and the centre bend magnet of a flatfield triplebend achromat (TBA) structure (Einfeld & Muelhaupt, 1980).
For the DBA example, the horizontal dispersion function is zero (achromat) at the entrance of the first bend magnet of length L. The emittance is minimized when _{x} has, at the distance S* from the beginning of the magnet, a minimum value _{min} such that the following relations hold,
In this case, the emittance for the bending magnet of deflection angle is
According to (4), the emittance scales with the well known third power law of deflection angle leading to the universal scaling law that a lowemittance lattice requires a large number of bending magnets.
For the second case we evaluate the centre bending magnet of a flatfield TBA lattice. Here the emittance is minimized when both _{x} and _{x} have minima in the middle of the centre bend at position (L/2). The corresponding theoretical minimum emittance (TME) conditions are (Sommer, 1983; Wüstefeld, 1987)
and the resulting TME emittance is
Closer inspection of (4) and (5) reveals that the TME emittance for the TBA magnet is a factor of three less than for the DBA magnet. Taking into account that the structure of a TBA must satisfy `Case 2' in the centre and simultaneously satisfy achromatic `Case 1' at the beginning and end bending magnets, the minimum emittance for a TBA lattice is, in practice,
For the case when the bendingmagnet deflection angle is the same for all three magnets, the advantage of the TBA structure over the DBA structure is only in the range of 23%. The reason for the discrepancy relative to the TBATME lattice is that the outer `achromat' magnets dominate the emittance integral <H>_{mag} producing an overall contribution of 85% to the emittance. Conversely, when we consider the case when the total deflection angle of the full TBA cell is the same as the DBA cell we have deflection angles (TBA) = (2/3)(DBA) and the emittance of the TBA cell is a factor of three less. An even smaller TBA emittance is achieved when the deflection angle of the outer magnets is further reduced and that of the middle magnet increased. For example, taking the bend angle of the outer magnets as half that of the centre dipole, the resulting cell (called TBA_{mod}) has an emittance less than a quarter of the DBA cell,
These examples demonstrate that in order to reach small emittance in an achromat cell the bendingmagnet structure should be built up with bending magnets with different deflection angles: smaller magnets at the ends with larger angles in the centre. An application of this rule using a total of four bending magnets in a flatfield quadruplebend achromatic (QBA) cell design is illustrated by Einfeld & Plesko (1993b).
The most straightforward approach to QBA cell design involves directly inserting a second pair of bending magnets between the two bending magnets of a DBA structure with all magnets of the same deflection angle. The two dipoles in the centre of the cell should yield minimum emittance when the TME conditions leading up to equation (5) are satisfied. Similar to the DBA and TBA designs, the main contribution to emittance is given by the outer magnets in this case (2 × 37.5%). With the optimized TME settings mentioned above, the QBA design with four equal bendmagnet deflection angles yields (QBA) = (DBA)/12.
During the design phase for SPring8, a comparison between DBA, TBA and QBA lattice cell structures was performed under the constraint that the ring circumference and the number of achromat cells was the same for all three cases (Tsumaki et al., 1989a,b). Interestingly, all three lattices resulted in roughly the same beam emittance and any dependency of the emittance upon the deflection angle ( ^{3}) could not be found. The result of this study was to show that the standard QBA lattice described above has no advantage over standard DBA or TBA cell structures. As a result SPring8 chose the DBA cell design to minimize construction risk.
A quite different behaviour is found when the lattice design permits a modified QBA structure. We first proposed a modified QBA cell design during the planning phase for the synchrotron light source LISA (Einfeld et al., 1992) with details described by Einfeld & Plesko (1993b). As shown in Fig. 1, the modified QBA structure contains two internal `unit cells' each with the deflection angle accompanied on each side by a matching section with deflection angle /2. Similar to lattice designs used for particle colliders, the matching sections force the dispersion to zero in the straight sections and permit optimum matching of the functions in the insertiondevice straight sections. In order to achieve the desired Twiss parameters throughout the unit cells, focusing and a defocusing fields are needed. This was achieved in part by integrating vertical focusing into the dipoles (combined function magnets) along with standard standalone quadrupole magnets. The use of combinedfunction bending magnets increases the partition number J_{x} and yields an overall decrease in emittance of the QBA lattice by 3040%. The novel introduction of internal unit cells composed of combinedfunction bending magnets provided the following advantages: (i) reduction in the total number of magnets per achromat; (ii) increase in horizontal partition number J_{x} (emittance reduction); (iii) creation of compact cells increasing straightsection space for insertion devices.
 Figure 1 A modified QBA achromat's inner structure. The deflection angle of the magnets in the unit cell is and in the matching section it is /2. 
In terms of emittance, for the modified QBA lattice, _{x} is again dominated by the bending magnets in the internal unit cells which gave an overall contribution of 2 × 42.1%. By switching from a standard QBA lattice to the modified QBA lattice the emittance is reduced by an additional factor of 2.53. The net result relative to the flatfield DBA lattice is (mod QBA) = (DBA)/30.36.
The extended concept of inserting multiple unit cells within two outer `matching' cells at the ends of the achromat can be successfully generalized to obtain what can be called a multiplebend achromat (MBA). An early example of a lowemittance MBA lattice was proposed for the 3 GeV synchrotron light source ROSY (Einfeld et al., 1994b). The ROSY design featured a small circumference of 148.1 m and relatively low emittance for such a small ring, 28.5 nm rad. In this case we designed the MBA cells with five bending magnets in an achromat configuration (5MBA). The deflection angle of the dipoles in the unit cells was 20° and in the matching sections 15°. As shown in the next section, the exercise of progressively reducing the bend angle of the individual dipole magnets while simultaneously increasing the number of unit cells within the achromat eventually leads to a lowemittance diffractionlimited light source.
By fixing the electron beam energy at 3 GeV and scaling down the bend angle of the ROSY QBA cell to 5° per bend, we soon realised that it was possible to construct a diffractionlimited storagering light source with horizontal emittance _{x} < 0.5 nm rad. Assuming a modest coupling factor of 3%, the vertical emittance is _{y} = 12 pm rad. With five 5° unit cells per achromatic, the circumference of such a machine is approximately 400 m (see Fig. 2). The lattice for this layout has a 7MBA structure and was given the acronym `DIFL'. Although early results from the DIFL lattice investigation were published by Einfeld & Plesko (1993a) and Einfeld et al. (1994a, 1995), an outline of the design is repeated here to provide insight into the design process and to update the lattice calculations.
 Figure 2 Layout for the diffractionlimited light source utilizing the MBA structure. The bending magnets are in the unit cell with a deflection of 5° and within the matching section with an angle of 2.5°. This is the magnetic structure which has been used for the lattice DIFL which is described by Einfeld & Plesko (1993a), Einfeld et al. (1994a) and Einfeld et al. (1995). 
As mentioned above, the emittance of an MBA achromatic cell is mainly determined by the bending angles of the internal unit cells. The DIFL lattice optimization was therefore first carried out using a hypothetical ring consisting of 72 5° unit cells (no matching cells). For purposes of this paper, all calculations have been performed with the lattice design code OPA (Streun, 2010). The layout of a single unit cell is presented in Fig. 3 with magnet field strengths listed in the caption. Unitcell tuning is only possible by changing the strength of the QF magnet and field gradient in the bending magnet. The resulting beam emittance is plotted as a function of QF magnet strength in Fig. 4 and the horizontal chromaticity in Fig. 5. As seen from Fig. 4, the emittance goes through a minimum near K_{QF} = 2 m^{2} and then increases rapidly. Similarly the natural horizontal chromaticity increases rapidly at higher values of K_{QF}. Accordingly we set QF to achieve a minimum emittance value of 0.5 nm rad. Compared with the TME expression shown in equation (5), for a TME machine operating at 3 GeV with 5° bend magnets and J_{x} = 1.29, the minimum emittance is 0.15 nm rad. The 7MBA unitcell design therefore has a higher emittance by a factor of only 3.3.
 Figure 3 The arrangements of the magnets within a unit cell of the MBA and the corresponding machine functions. The parameters of the magnets are as follows. Bending: L = 0.931482 m, = 10.674 m, B = 0.93749 T, k = 0.900 m^{2}. QF: L = 0.35 m, k = 1.992 m^{2}, g = 19.92 T m^{1}, gL = 6.972 T. Sh: L = 0.1 m, m = 53.347 m^{3}, Sh*L = 5.335 m^{2}. Sv: L = 0.2 m, m = 42.730 m^{3}, Sv*L = 8.546 m^{2}. 
 Figure 4 Emittance of the unit cell as a function of the strength of the focusing quadrupole. 
 Figure 5 Horizontal chromaticity of the unit cell of the lattice DIFL. 
To optimize the dynamic aperture the bendingmagnet field gradient was set to maximize the ratio of _{x} and _{y} at the chromatic sextupole locations. The resulting dynamic aperture of a single unit cell is shown in Fig. 6 with both onenergy and offenergy beam conditions, p/p = 0% and ±3%. The barelattice dynamic aperture is in the range x = 3020 mm and y = 1416 mm, or, in terms of acceptance, A_{x} = 31 mm mrad and A_{y} = 83 mm mrad. This result is particularly large when evaluated relative to beam size: 300 to 180 times in the horizontal plane and ±2400 times in the vertical dimension.
 Figure 6 Dynamic aperture of the unit cell for the energy deviations p/p = 3% (green line), p/p = 0% (blue line), p/p = 3% (red line). 
Surprisingly, tracking studies showed the tune shift with the momentum with up to p/p up to ±3% changes by only 0.025. With a peak dispersion value of 0.075 m, an energy acceptance specification of 3% and a ratio of 2 for the functions from the arcs to the straights, an aperture of only ±3.2 mm is needed to capture Touschekscattered particles. Hence the dynamic aperture should be sufficient for both injection and to yield an electron beam lifetime of the order of hours. Based on the simple unitcell concept, in principle there were no `showstoppers' to go forward with the 7MBA DIFL lattice design.
The Twiss parameters within a full achromat of the DIFL lattice including matching cells are plotted in Fig. 7. In order to optimized the dynamic aperture, tune shift with amplitude and tune shift with momentum deviation, additional sextupole families SS1 and SS2 were introduced. Sextupole families SV1 and SH1 have the same settings for all unit cells. The main characteristics of the _{x} = 0.5 nm rad DIFL lattice are given in Table 1. As expected, the emittance remains at _{x} 0.5 nm rad even with the addition of the outer matching cell dipole magnets. Assuming 1% coupling, the r.m.s. beam size is _{x} = 52 µm and _{y} = 3.8 µm in the straight sections.

 Figure 7 Machine function of the chosen lattice DIFL for the proposed diffractionlimited light source. 
The dynamic aperture of the DIFL lattice is plotted in Fig. 8, along with the tune shift as a function of momentum deviation in Figs. 9 and 10. For energy deviations p/p = ±3%, the horizontal dynamic aperture remains in the range ±10 mm and in the vertical ±8 mm. In terms of acceptance, we have A_{x} = 17.7 mm mrad and A_{y} = 21.7 mm mrad. Comparing with the lattice design composed of unit cells only, the dynamic acceptance is reduced by factors of 1.75 and 3.82 in the horizontal and vertical planes, respectively.
 Figure 8 Dynamic aperture of the lattice DIFL for the energy deviations p/p = 3% (red line), p/p = 0% (black line), p/p = 3% (blue line). 
 Figure 9 Change of the tunes for energy deviations of up to ±3%. 
 Figure 10 Tune diagram for the lattice DIFL with the movement of the working point for energy deviations up to ±3%. The resonance lines are the following: blue = third order, brown = fourth order, green = seventh order. 
To complete the conceptual design we assumed a standard 500 MHz RF acceleration system with a factor of three overvoltage and 1% coupling and calculate an electron beam lifetime of 5 h. The electron beam energy spread is 0.85 × 10^{3}, the natural chromaticities are manageable (_{x} = 79.4_{y} = 38.5) and the damping times are of order 10 ms. The betatron tunes are Q_{x} = 37.82 and Q_{y} = 13.20, well away from dangerous resonances.
As a final example we now examine the lattice design for MAX IV (Tavares et al., 2014). MAX IV has roughly the same magnet structure as the 7MBA DIFL with the unitcell dipole bending angles reduced to 3° and the matching cell dipoles 1.5°. The unitcell emittance is plotted as a function of QF strength in Fig. 11. From this plot it is clear that the minimum emittance for the case with QF = 5.7 m^{2} is around 0.12 nm rad. Referring again to the expression for TME [equation (5)], the emittance of a 3 GeV TME storage ring with a dipole deflection angles of 3° and J_{x} = 2.07 is 0.02 nm rad, or a factor of six below the MAX IV value.
 Figure 11 Emittance of the unit cell of MAX IV as a function of the strength of the focusing quadrupole. 
Detailed tracking studies of MAX IV show that the dynamic aperture is too small with a beam emittance of 0.12 nm rad so the QF strength was reduced to 4 m^{2} yielding _{x} = 0.33 nm rad for the unit cell (factor of 16.5 below the TME value). The bore radius of the MAX IV magnets is 12.5 mm (Johansson et al., 2014) producing pole tip fields of 0.5 T in the quadrupoles and 0.33 T in the 207 m^{3} sextupole magnets. The integrated strengths of the dipole, quadrupole and sextupole magnets are roughly the same as the unit cell of the DIFL. Given the small magnet radii at MAX IV, the field strength can be much higher which leads to a compact lattice. In this case, the overall length of each cell is reduced by 7.3 m. The resulting ring circumference with 12 achromat cells is therefore reduced by 84 m relative to the DIFL design. For the final MAX IV machine configuration, 20 achromat cells were used to produce an overall ring circumference of 528 m and horizontal emittance _{x} = 0.33 nm rad.
It is well known that the emittance of a storage ring synchrotron light source scales with the third power of the bendingmagnet deflection angles. In order to keep the accelerator compact and overall size reasonable, we have systematically demonstrated how a lattice with many bending magnets integrated into each achromat cell leads to an optimal design. In general, the MBA lattice provides a means to obtain lower emittance and higher dynamic apertures than with classical DBA lattices. The advantages of the combinedfunction MBA lattice include the small contribution of the outer bending magnets to emittance, a large horizontal partition number, a low number of quadrupole magnets, short unitcell lengths and relatively simple chromatic correction. The historical investigation of the sevendipole MBA revealed that the emittance of a diffractionlimited light source with values below 0.5 nm rad was possible. Further advances are of course possible by permitting finite dispersion in the straight sections and with the introduction of sophisticated sextupole magnet arrangements. Nevertheless, the original design proposed over 20 years ago has proved its validity and opened up new R&D paths that are fully blossoming today.
The authors would like to thank Jeff Corbett, SLAC, for the review of the manuscript and the introduction of many text improvements.
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