IEEE TRANSACTIONS ON MAGNETICS, VOL. 30, NO. 4, JULY 1994 2431

Magnet Sorting Algorithms for Insertion Devices for the Advanced Light Source

D. Humphries, E. Hoyer, B. Kincaid, S. Marks, R. Schlueter

Lawrence Berkeley Laboratory, University of California, Berkeley, Califomia 94720

Abstract-Insertion devices for the Advanced Light Source (ALS) incorporate up to 3000 magnet blocks each for pole energization. In order to minimize field errors, these magnets must be measured, sorted and assigned appropriate locations and orientations in the magnetic structures. Sorting must address multiple objectives, including pole excitation and minimization of integrated multipole fields from minor field components in the magnets. This is equivalent to a combinatorial minimization problem with a large configuration space. Multi-stage sorting algorithms use ordering and pairing schemes in conjunction with other combinatorial methods to solve the minimization problem. This paper discusses objective functions, solution algorithms and results of application to magnet block measurement data.

Nd-Fe-B Permanent Magnets

Vanadium

Keeper V Fig. 1. Partial view of lower half of undulator magnetic structure.

I. INTRODUCTION

The goals of magnet sorting algorithms at the Advanced Light Source (ALS) have evolved over a period of time beginning with the first hybrid insertion devices which were designed and built for the Stanford Synchrotron Radiation Laboratory. More recently, the performance requirements for these algorithms have become more stringent and varied. Significant evolution has occurred

Manuscript received September 21, 1993. This work was supported by the Director, Office of Energy Research, Office of Basic Energy Sciences, Materials Sciences Division, of the U.S. Department of Energy under Contract No. DE-AC03-76SF00098.

during the design and production phases of the first three insertion devices for the ALS [ 11.

The first three ALS undulators are hybrid permanent magnet structures which have vanadium permendur poles and six block Neodymium-Iron-Boron (Nd-Fe-B) magnet arrays associated with each pole. Fig. 1 shows a partial view of the lower magnetic structure of a typical ALS undulator. The orientation of the permanent blocks is parallel to the z-axis, i.e. parallel to the direction of the electron beam. The vanadium permendur poles can be seen between the magnet block arrays. The electron trajectory is shown schematically (with exaggerated amplitude) centered above the upper middle block of each magnet array. In the actual device, the structure is mirrored with opposite polarity above the electron beam trajectory.

Magnetic structures for these devices may contain as many as 250 poles in both the lower and upper magnetic structures. Fig. 2 is a photograph of the lower magnetic structure and supporting beam of a 4.5 m long, 5 cm period undulator for the ALS.

A. Direct and Indirect Fields

Permanent magnet materials used for insertion devices are characterized by uniform magnetization along a principal axis. If Mz is the magnetic moment along the magnetization axis, then Mx and My represent the magnetic moments resulting from the orientation or minor component errors of a given permanent magnet block.

In pure permanent magnet structures, each magnet can be modeled as a coil with essentially no nonlinear component. The field distribution at the electron beam trajectory can be calculated by superposition of the fields of block (coil) elements. The fields resulting from the minor component errors superimpose in essentially the same manner, and objective functions can be constructed which represent the combined effects of the minor component errors at the electron trajectory. Sorting algorithms which minimize these objective functions are used to determine block placement in these devices [2].

Unlike pure permanent magnet structures, the field distribution in hybrid structures is dominated by the ferromagnetic pole pair associated with each half period of the device. In this case the poles and the permanent magnet blocks constitute two principal sources of field errors.

Fields from the poles can be thought of as "indirect "

fields in the sense that the pole is first energized by the permanent magnet material and then produces a magnetic flux distribution. This flux distribution is determined by the relationship between the scalar potential of the pole and the scalar potentials of the surrounding poles along with their geometric aspects.

0018-9464/94$04.00 0 1994 IEEE

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Fig. 2. Photograph of the 4.5 m long lower magnetic structure for the ALS U5.0 undulator showing 183 poles and 182 Nd-Fe-B magnet block arrays.

Fields from the permanent magnet blocks can be thought of as "direct" fields, i.e., fields which are not completely determined by the characteristics of the poles. In the pure permanent magnet case, the fields of the device are effectively all direct flux. In the case of the hybrid device the fields are dominated by indirect flux but have an important direct flux component.

The electron beam at the midplane of the undulator sees some combination of direct and indirect fields.

B. Characterization of Error Fields

Field errors can be divided into two distinct types which are associated with indirect fields and direct fields. 1 ) Indirect field errors: These errors are due primarily to defects in the poles of the device except in the case of excitation errors, which may be due to a combination of pole errors and variation in the average MZ value of the permanent magnet blocks associated with a given pole. Indirect field errors may be caused, for example, by variations in the dimensions of the pole [3] or by other characteristics such as differences in chemical composition or inconsistencies during heat treatment of the poles for magnetic properties. Vertical position errors or orientation errors of the pole within the magnetic structure are also causes of indirect or pole associated field errors. 2) Direct field errors: These errors are a result of error characteristics of the permanent magnet blocks in the hybrid magnetic structure. There are two principal sources for these errors within a magnet block. The first is misorientation of the easy axis of the magnet block. The second is what is referred to as "surface fields" of the block.

Misorientation results in minor component magnetic moments (Mx and MY) which cause vertical or horizontal error fields at the midplane of the device and between the poles of the magnetic structure.

Surface fields generally result from irregularities in the magnetization of a block near a particular surface of that block. Depending on the manufacturing process used for a magnet block, there may be some continuous orientation distribution along a given surface. If this distribution is symmetric then the integrated effect of the resulting surface field distribution outside the block will be zero for integration paths parallel to the axis of magnetization of the block. If not, then the surface fields result in net error fields at the midplane of the device. Like misorientation error effects, they occur at the location of the magnet blocks rather than under the poles.

C. Error Effects on Spectrum vs. Beam Dynamics

The error fields discussed above may combine in various ways to affect both the spectral characteristics of the insertion device and the electron beam dynamics [4]. Indirect errors may affect both, but are more closely associated with degradation of the spectral Characteristics of the photon beam. Direct field errors are more closely connected to effects on beam dynamics. This is especially true when the integrated error fields along the length of the magnetic structure become sufficiently large.

D. Role of Block Sorting Algorithms

Through the use of appropriate sorting algorithms, the effects of both types of field errors on spectral characteristics and on beam dynamics can be reduced. Sorting techniques

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can be particularly effective in minimizing the effect of variations in the principal magnetic moment (M,) of the magnet blocks and thus establishing uniform excitation of poles. Effects of direct fields on beam dynamics can be minimized through the use of combinatorial techniques in the later stages of a block sorting process.

11. SORTING FOR UNIFORM POLE EXCITATION

Highly successful sorting techniques for achieving uniform pole excitation have been employed beginning with the early development of Halbach type hybrid insertion devices. Updated versions of these techniques have been applied to the first three insertion devices for the ALS.

A. Principal Sorting Criteriu

In order to establish uniform pole excitation, it is necessary that the average magnetic moment in the direction of magnetization, H z , for the magnet array associated with any pole, be the same as that of any other array.

For the ALS insertion devices, the moment values of the magnet block populations were determined by automated Helmholtz coil measurement [5]. Values of M,, My and MZ were recorded to an electronic file. The data were then arranged in ascending values of a criterion called Ti calculated for each block, where:

r i = M . + M ~ f o r i = l t o n

where n is the total number of blocks being sorted. The r criterion associates a combined minor component magnitude with each block. Blocks with lower r values are preferred for positioning close to the electron beam.

This ordered group was then subdivided into six subgroups with group one having the lowest r values, group two having the next lowest r values and so on. Each of the six groups was assigned a position in the six block magnet arrays as shown by the numbering scheme in Fig. 1. Thus, any block from group one was constrained to a position immediately under, or above, the electron beam.

(1) 2 ' 1 Yi

B. Final Determination of Magnet Arrays

The next stage of the sorting process was to construct the six block magnet arrays. The objective here is to establish uniform averaged MZ values for the final six block arrays. This was done using a non-iterative ordering and pairing technique which is a feature of the sorting code INSORT [6]. The steps of the process as applied to a population of n blocks are as follows:

4) The two groups which did not contain the block selected in step 3) are searched for the block having the greatest magnitude of dMz of the opposite sign of that of the step 3 block.

5 ) The blocks selected in steps 3 and 4 (above) are paired and their dMZ values are averaged. They are also re- moved from the'data set and stored along with their aver- age dMz value and group affiliations.

6) Steps 3-5 (above) are repeated d 3 times which results in n/3 paired blocks and n/3 single blocks.

7) The d 3 pairs are monotonically ordered by their average dMZ values and the single blocks are monotonically or- dered by their dMz values.

8) The ordered pairs and single blocks of step 7) are then paired by opposite extremes. The average dMz value of each of the resulting triplets is then calculated.

9) Steps 3-8 (above) are repeated for blocks in groups 4,5, and 6.

10) The group 1-2-3 triplets and group 4-5-6 triplets are sep- arately ordered monotonically by their average dMZ Val- ues and paired by extremes to form the final six block ar- rays.

The above algorithm results in very rapid reduction in deviation from global average of Mz values for typical magnet block populations used in hybrid insertion devices. Fig. 3 shows the reduction in deviation of a block population with an initial deviation of 4 % for single blocks. The final maximum deviation from global average at the six block array stage is less than 0.02% which is significantly below measurable levels.

- Max Dev from Global Avg

The global average of MZ for the entire block population

(dMz) is calculated for each block. The six groups, described in Section 11-A. above, are then monotonically ordered in ascending values of dMz. the goup. Groups one, two and three are searched for the block having the greatest absolute value of dMz.

is and the deviation from that global average Fig. 3. Redudion of variation in average principal magnetic moment of six block magnet arrays. "Original Data" is for the total measured population. "Final Data" is after extreme blocks were removed to form

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III. SORTING T o CONTROL INTEGRATED FIELDS REFERENCES

Analysis of magnetic measurements of the first three insertion devices for the ALS indicate very good spectral performance [4]. This implies that pole excitation and indirect fields were well controlled by sorting and precision fabrication techniques.

Integral measurements, on the other hand, indicated a systematic accumulation of direct field errors along the length of the first two devices. Integrated Hall probe measurements taken at the midplane revealed a net negative field on side of the beam trajectory and a net positive field on the opposite side of the trajectory. Examination of block measurement data indicated that this was caused by systematic, non- canceling orientation of the minor field components of the blocks. The result was a transverse gradient in the integrated By fields or, in terms of beam dynamics a pseudo quadrupole field. This effect was corrected by tuning devices installed at the ends of the two insertion devices.

These effects can be minimized in future devices by utilizing appropriate extensions to the above described block sorting algorithm. A large parameter space for optimization remains after the sorting algorithm described in Section I1 is applied. Parameters such as block rotation and position shifts within six block arrays as well as relocation of arrays in the magnetic structure can be used in conjunction with measured information about block minor components and surface fields. minor component information for individual blocks is readily available from systematic Helmholtz measurements. Surface fields require special measurement techniques which are currently under development. Iterative optimization techniques such as synthetic annealing [7] can be used to minimize field integral objective functions after parameter effects have been quantified.

IV. CONCLUSIONS

Block sorting algorithms are a critical element in the fabrication of modern insertion devices. They have demonstrated valuable optimizing aspects for the spectral quality of the first ALS undulators. Extended algorithms currently in progress show promise in meeting increasingly stringent beam dynamics requirements.

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[2] M. E. Couprie, C. Bazin, M. Billardon, "Optimization of the Permanent Magnet Optical Klystron for the SUPER- ACO Storage Ring Free Electron Laser," Nuclear Instruments and Methods in Physics Research, A278, pp.

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[5] S. Marks, J. Carrieri, C. Cook, W. Hassenzahl, E. Hoyer, and D. Plate, " A L S insertion device block measurement and inspection," proceedings at the IEEE PAC, May 1991.

[6] D. Humphries, "INSORT, A Sorting Code for Hybrid Permanent Magnet Structures," LBL Engineering Note M7427, Sept. 1993.

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