EIGENVALUE PROBLEM IN PRECISE MAGNET ALIGNMENT

Transcription

1 EIGENVALUE PROBLEM IN PRECISE MAGNET ALIGNMENT K. Endo and K. Mshma KEK and SKD (Graduate Unversty for Advanced Studes), Tsukuba, Japan Abstract For the algnment of the accelerator components at the desgned coordnates, a matrx connectng observables and coordnates can be treated as an egenvalue problem and has ts own characterstcs dependng on the structure of the survey network. Once ts egenvectors are determned, the msalgnment of the magnets s expressed by ther lnear combnaton. By decomposng the magnet msalgnments nto the egenmodes and evaluatng each contrbuton of the egenmode to the closed orbt dstorton (COD), t s possble to fnd whch mode affects sgnfcantly to the beam n the synchrotron. Ths approach s appled to the synchrotron rngs wth the typcal symmetrc and asymmetrc magnet arrangements to whch several survey networks are adopted and gves the useful nformaton for the magnet precse algnment, to say, the msalgnment modes whch are less sgnfcant to the beam can be safely omtted. In ths work, three knds of survey networks, whch were appled for the real synchrotron, are treated n detals. 1. INTRODUCTION An algnment matrx equaton s obtaned from the trangulaton of the survey networks for the precse algnment of the synchrotron magnets. The horzontal msalgnments, the spatal devatons of magnet from the planned postons n the horzontal plane, are expressed by (x, y) or (r, θ) coordnates wth an orgn at the center of the synchrotron rng. From the survey of the physcal dstances such as the dstance of the adjacent quadrupoles and the perpendcular from the quadrupole to the straght lne stretched between the quadrupoles at both sdes, the devatons of these dstances from the desgn values are expressed approxmately as a lnear functon of the msalgnments of the related quadrupoles. These relatons for all quadrupoles are merged nto a global matrx equaton termed here as an algnment equaton. The matrx depends on both the magnet lattce desgn and the survey network and connects the magnet msalgnment wth the devatons of the surveyed dstances, or the survey varables. As the magnet lattce s fxed, the varous knds of the survey network formed on the targets of quadrupoles or the representatve ponts close to each quadrupole ntroduce the varance n the matrx of the algnment equaton. The survey network or mesh should be determned accordng to the magnet lattce structure and the easness of the survey measurement n the narrow synchrotron tunnel. If the target s placed on each quadrupole magnet, an accurate drlled hole where the target s nserted wth a mnmal clearance s requred. The msalgnment s corrected by movng the

2 quadrupole so that the target sts at the desgned coordnate. If the representatve ponts are used, each quadrupole s algned by referrng these ponts. Large colldng beam synchrotron wth the expermental nsertons s composed of the arced and straght sectons. The arced secton has a regular magnet lattce and quadrupoles are placed equdstant, but the expermental straght secton has dfferent quadrupole arrangements because t accommodates the RF cavtes and mn-beta quadrupole magnet system to squeeze the beam at the collson pont. The synchrotron rng n whch the quadrupoles are not placed on the common crcle s classfed here as an asymmetrc rng (nhomogeneous rng). To the contrary the hypothetcal rng where all quadrupoles st on the common crcle s called here as a symmetrc rng (homogeneous rng).. DERIVATION OF THE ALIGNMENT EQUATION Assumng that ether symmetrc or asymmetrc synchrotron rng has N quadrupoles and ther N survey targets ( = 1,,, N ) are on the desgned orbt, at least two survey varables S and P are necessary to each quadrupole. These varables are expressed as a functon of R and Θ as S = F(R,Θ ) and P = G(R,Θ ), (1) where the subscrpt of R and Θ s consdered for all possble range. For an example, f S s a dstance between the adjacent quadrupole magnets, S = F(R, R +1,Θ ). Dfferentatng the above relatons to the frst order, F F F F s S = R + Θ = r + (θ +1 θ ) () R Θ R Θ and G G G G p P = R + Θ = r + (θ +1 θ ), (3) R Θ R Θ where s and p are the devatons from the desgn value whch are determned from the precse survey, and r and θ the quadrupole dsplacements to the radal and azmuthal drecton, respectvely. Dervng relatons () and (3) for all quadrupoles, p1 a11 a1n c11 c1n r1 : : : : : : p N an1 ann cn1 c NN r N = (4) s1 b11 b1n d11 d1n θ1 : : : : : : s N bn1 bnn dn1 d NN θ N where the N x N matrx s obtaned for the above relatons [1][]. If 3 varables S, P and Q are consdered, the matrx becomes 3N x N and the left column vector has 3N components. In any case the matrx equaton (4) s expressed smply as p = (A)r (5) and solved by the least squares method as

3 h (A T )p = ( A T A)r (6) T T where ( A ) s the transposed matrx of ( A ) and ( AA) s the N x N matrx. The soluton s gven by r = (A T A) 1 h, (7) f the matrx s not sngular. In general, however, the matrx ( A) s sngular. Therefore one must consder to reduce the rank of the matrx by fxng several coordnates, for an example, r 1 =, r = and θ 1 = wthout loosng generalty. The number of coordnates to be fxed depends on the adopted survey network. Another method to solve (6) s to use the generalzed nverse matrx. Even f the matrx s sngular, ts nverse matrx s determned unquely followng the method of the sngular value decomposton. Replacng ( A T A) wth (H) and assumng (H) + the generalzed nverse matrx, the soluton of (6) s smply gven by r = (H) + h. (8) In ths manuscrpt the sngular matrx of (4) s reduced to a lower rank to obtan the non-sngular matrx so as to analyze ts propertes. From the relaton (6) the precse magnet algnment problem can be treated as an egenvalue problem, (H)v = λ v, (9) where λ s an egenvalue belongng to the egenvector v. If all components of egenvector are multpled by the coeffcents and are combned lnearly, the dsplacements of the quadrupole magnets ( r and θ ) are obtaned as a lnear combnaton of the egenvectors, r = c k v k, (1) k where c k s the coeffcent. From (6) and (1), (H)c k v k = h. (11) k The lnear coeffcent of (1) s c k = (v T k h)/ λ k, (1) T where v k s the transpose of the k-th egenvector v k. The j-th component of the dsplacement vector s (v r j = kt h) (v v kj or r j = kt A T p) v kj, (13) k λ k where v kj s the j-th component of the egenvector v k [3]. 3 SURVEY NETWORKS k λ k In many cases of the synchrotron rng, all magnets are algned n the narrow tunnel and the survey network for the measurements of dstances between magnets s restrcted. Nowadays very precse nstruments such as a laser tracker and a mekometer are avalable, but the maxmum length to be measured s lmted to mantan accuracy. Thus the survey network must be establshed consderng the dstrbuton of the survey ponts whch can attan the requred accuracy. From (13), the dsplacement error r due to the survey error p s estmated by

4 a pv vv m km m, kj km rj = vkj = am k λk m k λk Then the standard devaton of m kj km = k, m λk r j s expressed as follows, p. (14) a v v σ r σ j p, (15) where σ p s the expected error for each p, the statstcal error for repeated dstance measurements between the same ponts. Between the standard devatons of the coordnates and dstance measurement errors for the survey network there s a relaton medated by the network structure [4]. The smaller coordnate error s expected f omttng the contrbutons by egenmodes wth small egenvalues. For large synchrotrons there are not so many varatons for the survey networks, because the number of the collmaton axes extended from each pont to another s lmted. The typcal survey networks consdered here for the synchrotron are (Case #1) Short chord and perpendcular, (Case #) Long chord and perpendcular, and (Case #3) Short chord and perpendculars, as shown n Fg.1, where the survey varables are gven by the sold lnes. Dfferences of the numbers of egenmodes and vector elements are compared n Table R-1 θ -1 P R S θ θ +1 Θ -1 Θ +1 R+1-1 R-1 θ -1 S +1-1 P,1 P, P +1 - L -1 + R R R-1 R+1 R+1 R- θ θ +1 R+ θ θ -1 θ +1 θ + Θ -1 Θ θ - Θ - Θ -1 Θ Θ +1 Case#1 Case# Case#3 Fg.1 Survey varables (sold lnes) for the dfferent survey networks. The matrx equatons for 3 cases are derved for both symmetrc and asymmetrc rng, assumng both rngs have the same dameter (48 m) and the same number of the quadrupole magnets (39 quads). The asymmetrc rng s an electron-postron colldng synchrotron called TRISTAN of KEK whch has the long expermental nsertons at both sdes of 4 nteracton ponts. To the contrary, the symmetrc rng s the hypothetcal synchrotron of whch all quadrupole magnets locate on the same crcle at equdstant all around the rng.

5 The egenvectors for largest and smallest egenvalues are shown for the symmetrc 3 cases n Fg. and for the asymmetrc 3 cases n Fg.3. In these fgures the radal and azmuthal dsplacements are gven n seperate both n the order of quadrupole magnet confguraton. Table 1 Dfferences of the numbers of egenmodes and vector elements for 3 survey networks. N s the number of the quadrupole magnets. Case #1 Case # Case #3 Number of egenmodes N-3 N-4 N-3 Number of the egenvector elements of each egenmode N-3 N-4 N # Symmetrc Case# # # Symmetrc Case# # # Symmetrc Case#1.1 # # Symmetrc Case# # Egenvector #779 Egenvector #78.1 # Symmetrc Case# # # Symmetrc Case# # Egenvector #781 Egenvector #78 Egenvector #78 Egenvector # # Symmetrc Case# # # Symmetrc Case# # Egenvector #78 Egenvector #781.1 # Symmetrc Case# # # Symmetrc Case# # Egenvector #78.1 # Symmetrc Case# # Egenvector #781 Egenvector # # Symmetrc Case# # Egenvector #781 Fg. Egenvectors wth largest and smallest egenvalues for the symmetrc cases. has the largest egenvalue and egenvector #781 (#78 for case#) the smallest one. These egenvectors gve the magnet msalgnment patterns to the radal and azmuthal drectons n the order of magnet confguraton. Egenvectors for (a) Upper: Symmetrc case#1, (b) Mddle: Symmetrc case#, and (c) Lower: Symmetrc case#3. In every case the same egenvector s obtaned for the smallest egenvalue. It gves the quadrupole dsplacements formng a snusodal undulaton along the synchrotron rng n both radal and azmuthal drectons. Generally for larger egenvalue the number of the snusodal dsplacement

6 undulatons along the rng ncreases accordng to the magntude of the egenvalue. They gve the characterstc envelope patterns of the snusodal dsplacements for large egenvalues. However, these msalgnment undulatons are obtaned wthout consderng the betatron moton of the partcle. As each egenvector has ts own pattern of the magnet msalgnment, f the betatron oscllaton number becomes smlar to the quadrupole magnet algnment pattern gves the relatve large COD to the beam. Asymmetrc Case#1. # # Asymmetrc Case#. # # Asymmetrc Case# # # Asymmetrc Case#1..6 # # Asymmetrc Case#..6 # # Asymmetrc Case# # # Egenvector #779 Egenvector #78 Egenvector #78.1 # Asymmetrc Case# # Asymmetrc Case# # # # Asymmetrc Case# # Egenvector #781 Egenvector #781 Egenvector #78 Egenvector #78 Egenvector #779 Egenvector # # Asymmetrc Case# # Asymmetrc Case# # # # Asymmetrc Case# # Egenvector #78 Egenvector #781 Egenvector #781 Fg.3 Egenvectors wth largest and smallest egenvalues for the asymmetrc cases. has the largest egenvalue and egenvector #781 (#78 for the case#) the smallest one. These egenvectors gve the magnet msalgnment patterns to the radal and azmuthal drectons n the order of magnet confguraton. Egenvectors for (a) Upper: Asymmetrc case#1, (b) Mddle: Asymmetrc case#, and (c) Lower: Asymmetrc case#3. 4. RMS CLOSED ORBIT DISTORSION (COD) To estmate the rms COD the magntude of the msalgnment s normalzed to N ( x ) =.1mm, (16) N =1

7 after gvng the radal and azmuthal msalgnments to every quadrupole magnet. If N =1 ( x ) / N =.1mm s assumed only for the radal dsplacements, an extraordnary azmuthal dsplacement s ntroduced even for the small radal msalgnment. To avod ths problem, the normalzaton over both radal and azmuthal dsplacements s adopted. For the calculaton of COD the beam optcs code s used [5]. An essental feature of the COD calculaton depends on the transfer matrces of all machne elements ncludng quadrupoles, dpoles and drft spaces that the crculatng beam encounters when travelng the synchrotron orbt. The transfer matrx through one perod can be rewrtten by usng the twss parameters and the betatron phase advances. Once the twss parameters are obtaned at a pont, those at other ponts are easly calculated. A matrx to calculate COD at the beam poston montors due to the msalgnments of the quadrupole magnets can be derved by usng the twss parameters. If each egenvector s appled to ths matrx, ts contrbuton to the beam s estmated by usng the rms value of COD. Every quadrupole s shfted to the radal and longtudnal drectons accordng to the assumed msalgnment n the horzontal plane and the drft space lengths are adjusted to make up for the longtudnal shft of quadrupoles before the COD calculaton. By nvestgatng the relatons between the msalgnment modes and the beam orbt deformatons, the underlyng rules for the accelerator algnment ssue wll be found. In the synchrotron the charged partcles perform the betatron oscllaton whch s determned by the magnetc feld gradent and the confguraton of the quadrupole magnets. The COD s defned as the orbt of the gravtatonal center of the crculatng partcles. If the undulaton number of the msalgnment pattern concdes wth that of the betatron oscllaton, partcles suffer the Lorenz force due to the systematcally varyng magnetc feld of the dsplaced quadrupole magnet and wll result n the large orbt dstorton The rms COD s for each egenmode The rms COD s defned here as N k 1 < r >= ( k ) N = 1 r, (17) where r k s the radal COD at the -th beam poston montor and < r k > the correspondng rms COD orgnated from the k-th egenvector. The rms COD may be used to estmate the effect to the beam orbt by each egenmode Symmetrc cases In general, the rms COD responses are dfferent dependng on the survey networks. However, there s no structural dfference between cases #1 and # for egenvalues as shown n Fg.4. The undulaton number of the egenvector (egenmode #687) shown n Fg.5 s very close to the betatron oscllaton frequency. The egenmode #687 of the symmetrc case# has the same undulaton pattern as Fg.5. The largest peak correspondng to the egenmode #178 has the egenvector shown n Fg.6. Ths egenmode has about 4 tmes of the undulaton number of the egenmode #687 whch s close to the horzontal betatron frequency. As almost all dots shown n Fg.7 ft on the phase of the

8 msalgnment pattern of the egenmode #178 whch has the largest rms COD response, the egenmode #178 s expected to have a large contrbuton to COD. The same relaton concernng to the largest rms COD peak holds on other cases. The egenmode #176 of the symmetrc case# have the same undulaton pattern except for the azmuthal part whch has 71 undulatons. rmscod (arbtrary) Egenmode #178 Symmetrc Case#1 Tune hor/ver = 37./39.1 rmscod Egenmode number Egenmode #687 & # rms COD (arbtrary) Egenmode #176 Symmetrc Case# Tune hor/ver = 37./39.1 rms COD Egenmode number Egenmode #687 & # (a) Symmetrc case#1 (b) Symmetrc case# Fg.4 The rms COD responses of the symmetrc cases #1 and #. Egenvector # oscllatons Symmetrc Case#1 37 oscllatons # Azmuthal poston Radal & s Fg.5 Undulaton pattern of the egenvector #687 ( nd peak) of the symmetrc case#1. The survey network of the symmetrc case #3 has the 3 rd peak as shown n Fg.8 whch does not exst n cases #1 and #. The nd peak corresponds to the egenmode #65 havng the undulaton number close to the betatron frequency (Fg.9). But the 1 st and 3 rd peaks have larger undulaton numbers as shown n Fg.1 and Fg.11, respectvely.

9 oscllatons Symmetrc Case#1 149 oscllatons # Radal poston Radal & s Fg.6 Undulaton pattern of the egenvector #178 (1 st peak) of the symmetrc case#1.1 Egenmode #687 (red and blue dots) s superposed on the egenvector #178 (sold lne) Symmetrc case# A part of radal poston Fg.7 Egenmode #178 (sold lne) has 4 tmes of the undulaton number of the egenmode #687 (red and blue dots) whch s close to the horzontal betatron frequency n the symmetrc case#1. Almost all dots ft on the phase of the msalgnment pattern of the egenmode #178 whch has the largest rms COD response. Only one fourth of the crcumference s shown for the explanaton. #178 rms COD (arbtrary) Egenmode #18 rms COD Egenmode #468 Symmetrc Case#3 Tune hor/ver = 37./39.1 Egenmode # Egenmode number correspondng to egenvalue Fg.8 The rms COD response of the symmetrc case #3.

10 Egenmode #65.1 Egenmode of the nd peak of the rms COD.5 Symmetrc Case# oscllatons #65 36 oscllatons Radal & s Fg.9 Undulaton pattern of the egenvector #65 ( nd peak) of the symmetrc case# Egenmode #18 for the largest peak of the rms COD Symmetrc Case#3 # oscllatons Radal & s Fg.1 Undulaton pattern of the egenvector #18 (1 st peak) of the symmetrc case#3. Egenmode # Egenmode #468 for the 3rd peak of the rms COD #468 Symmetrc Case#3 89 oscllatons 66 oscllatons Radal poston Radal & s Fg.11 Undulaton pattern of the egenvector #468 (3 rd peak) of the symmetrc case#3.

11 4.1. Asymmetrc cases In the asymmetrc cases there appear many peaks n the rms COD response spectra. It seems that the rregularty n dstance between quadrupole magnets plays a role. As shown n Fg.1 the cases #1 and # have almost the same rms COD responses and the peaks around the egenmode #677 correspond to the egenvector close to the betatron frequency as shown n Fg.13. The egenmode for the largest peak (Fg.14) has a smlar egenvector as Fg.6. 5 Egenmode # Egenmode #174 5 rmscod (arbtrary) 4 3 Asymmetrc Case#1 Tune hor/ver = 37./39.1 Egenmode #59 Egenmode # rmscod (arbtrary) 4 3 Asymmetrc Case# Tune hor/ver = 37./39.1 Egenmode # rmscod 1 1 rmscod Egenmode number Egenmode number (a) Asymmetrc case#1 (b) Asymmetrc case# Egenvector #677 Fg.1 The rms COD responses of the asymmetrc cases #1 and #..1 4 oscllatons Asymmetrc Case# oscllatons -.5 # Radal & s Fg.13 Undulaton pattern of the egenvector #677 of the asymmetrc case#1.

12 oscllatons.5 Asymmetrc Case#1 148 oscllatons -.5 # Azmutrhal poston Radal & s Fg.14 Undulaton pattern of the egenvector #176 of the asymmetrc case#1. rmscod (arbtrary) Egenmode #19 #149 rmscod Asymmetrc Case#3 Tune hor/ver = 37./39.1 #454 #63 #67 # Egenmode number Fg.15 The rms COD response of the asymmetrc case #3. Egenvector # oscllatons Asymmetrc Case#3 4 oscllatons # Radal & s Fg.16 Undulaton pattern of the egenvector #69 of the asymmetrc case#3.

13 For the asymmetrc case#3 has the rms COD response spectrum dfferent from Fg.1 as shown n Fg.15. The peaks of the rms COD responses for the egenmodes correspondng to the betatron frequency shft to larger egenvalues. The undulaton number of the egenmode #69 (Fg.16) s less than the betatron frequency whch corresponds to the egenmode #63 (Fg.17). Comparng the cases #1, # and #3, there are bg dfferences as shown n Fg.1 and Fg.15. However, t seems lttle effect to COD by omttng the correctons of egenmodes (more than #7) wth small egenvalues. In ths way the beam senstve egenmodes, whch must be corrected, are selected through the nvestgaton of the egenmode problem. Egenvector # oscllatons Asymmetrc Case#3 4 oscllatons # Radal & s Fg.17 Undulaton pattern of the egenvector #63 of the asymmetrc case#3. 4. Solvng the egenmode based algnment problem If several egenmodes are selected as the quadrupole algnment errors, the correspondng egenvectors can be combned lnearly assumng arbtrary coeffcents and transformed to the survey varables such as the perpendcular and short chord lengths n the case#1. Accordng to (13) the dfference of the survey data from the desgn values can be processed to obtan the quadrupole msalgnments. An example for the asymmetrc case#1 s gven n Fg.18, where the 677-th egenmode s assumed as the msalgnment mode and the dfference between the msalgnment and the correspondng egenvector s shown by the blue lne. Most sgnfcant contrbuton to the msalgnment s the 677-th egenmode as shown n Fg.18 by the peak of the green lne. Other egenmodes at the rght sde of the absolute coeffcent (green lne) also contrbute the msalgnment but these egenmodes arse by solvng the sngular matrx problem by the rank reducton method. The survey varables, perpendcular and short chord length errors, are obtaned from the 677-th egenvector as gven n Fg.19. Even f two egenmodes are lnearly combned, the same procedure can be appled to obtan the msalgnment. Combnng the 176-th and the 677-th egenmodes wth the equal weght but the dfferent sgns for the asymmetrc case#1 (Fg.), both egenmodes have sgnfcant contrbutons to the msalgnment and egenmodes at the rght sde of the absolute coeffcent (green lne) also have relatvely large contrbutons whch are also ntroduced through the rank reducton of the sngular matrx. The dfference between the calculated msalgnment and the combned egenvector

14 arses from the exctaton of the lower undulaton frequency modes wth small egenvalues. The egenmodes wth the undulaton frequency much lower than the betatron frequency and wth the small rms COD response do not contrbute to the beam moton and can be safely dsregarded from the msalgnment correcton. If the random msalgnment s assumed to obtan the perpendcular and short chord errors, most egenmodes contrbute equally to the msalgnment as shown n Fg.1 for the asymmetrc case#1. Soluton & Dfference (=Sol-677th egenvector) Asymmetrc Case#1, 677-th Egenvector Sol Sol-677thEV Abs.coeffcent Radal and azmuthal postons of quadrupole magnets or Egenmode number for Absolute coeffcents Fg.18 Msalgnment due to the 677-th egenmode of the asymmetrc case#1. The red lne s the quadrupole msalgnment and the blue lne s the dfference between the calculated msalgnment and the 677-th egenvector. The green lne s the absolute coeffcents combnng all egenvectors to make up the msalgnment of the red lne Abs.coeffcent 677-th Egenvector thEV Asymmetrc Case#1, 677-th Egenvector Perpendcular&ShortChord Perpendcular error Short chord error Radal and azmuthal postons of quadrupole magnets Perpendcular & Short Chord Errors Fg.19 Comparson of the 677-th egenvector and the correspondng survey varables obtaned from ths egenvector for the asymmetrc case#1. The perpendcular and short chord errors (blue lne) gve the soluton (red lne) n Fg.18.

15 Soluton & Dfference(=Sol-176th-677thEV) Asymmetrc Case#1, 176th + 677th Egenvector Soluton Sol-176thEV-677thEV Abs.coeffcent Radal and azmuthal postons of quadrupole magnets or Egenmode number for Absolute coeffcents Abs.coeffcent Fg. Msalgnment due to the combnaton of the 176-th and the 677-th egenmodes. Gven random msalgnment (m) Perpendcular & Shortchord eroor Contrbuton of each egenmode Gven random msalgnment.1 mm (rms) Asymmetrc case#1 Random msalgnment Egenmode Asymmetrc case#1 Soluton - Gven random msalgnment -.6 Random msalgnment Quadrupole magnet.1..5 Perpendcular & Shortchord eroor Asymmetrc case#1 Ramdom msalgnment.15 Perpendcular error Short chord error Soluton Radal msalgnment Azmuthal msalgnment Quadrupole magnet Sol-Gven random msalgnment (m) Soluton (a) Assumed random msalgnment (d) Dfference between soluton and assumed random msalgnment: (d) = (c) - (a) (b) Perpendcular and Shott chord error obtaned from the assumed random msalgnment error (c) Solved by egenvalue method usng errors of (b) (e) Contrbuton of egenmode to the soluton of (c) Random dsplacement error for Asymmetrc case#1 Fg.1 Random msalgnment of.1 mm (rms) s assumed for the asymmetrc case#1. Most egenmodes contrbute equally to the msalgnment and t also gves the smlar dfference between the assumed and calculated msalgnment.

16 4.3 Effect of the betatron tunes to rms COD The peaks of the rms COD response shown n Fg.4, Fg.8, Fg.1 and Fg.15 shft, f the horzontal betatron frequency changes. Examples for the asymmetrc and symmetrc case#1 are gven n Fg.. The sharp rms COD response peaks are obtaned for the symmetrc cases whch are better to get the relaton between the betatron tune and egenmode. If the horzontal tune ncreases, the maxmum rms COD response peaks shft towards larger egenmode number whch has smaller egenvalue and the rms COD response peaks n the symmetrc case#1 close to the horzontal betatron tune shft to the smaller egenmode number whch has larger undulaton frequency. 8 Symmetrc case#1 8 Asymmetrc case#1 7 7 Egenmode to rmscod response Peak rmscod response Egenmode close to QH frequency Egenmode to rmscod response Egenmode of max. rmscod response Egenmode close to QH freq Horzontal tune (QH) (a) Symmetrc case# Horzontal tune (QH) (b) Asymmetrc case#1 Fg. Dependence of the rms COD response peaks on the horzontal betatron tune for the symmetrc and asymmetrc case#1. 5. CONCLUSION As the egenvectors of the matrx whch s derved for the survey network correspond to the components of the msalgnment, t s possble to estmate ther effects on COD by treatng each egenvector as a msalgnment mode n the beam optcs code. In ths study the followng results were obtaned. 1) There are some peaks n the rms COD response spectrum correspondng to the egenmodes of whch snusodal undulaton numbers are close to or multple of the horzontal betatron oscllaton frequency. These peaks shft dependng on the horzontal tune.

17 ) For the nhomogeneous quadrupole confguraton, there are many rms COD response peaks, whereas for the homogeneous one only few peaks appear. 3) In practce, the msalgnment modes wth relatvely large rms COD response should be corrected. Consderng the possble extent of the betatron tune varaton, the msalgnment correcton should be made for all egenmodes that the large rms COD response s expected. 4) Lower modes, havng small egenvalue whch have small number of the snusodal undulatons, are nevtably ntroduced through the reducton of the rank when obtanng non-sngular matrx, but these modes can be left uncorrected because ther effects on the rms COD response are very small. 5) The largest peak of the rms COD response corresponds to the undulaton number close to 4 tmes of the horzontal betatron frequency. Ths msalgnment pattern wll ft well to the msalgnment mode of whch undulaton number s close to the horzontal betatron frequency. 6. REFERENCES [1] K. Endo and K. Mshma, Survey Meshes for Synchrotron Magnet Algnment, Proc. EPAC, Venna, pp [] K. Mshma and K. Endo, COD Response to Msalgnment Mode of Synchrotron Quadrupoles, APAC1, Bejng, pp [3] J.H. Wlknson, The Algebrac Problem, Oxford Unv. Press, G. Goertzel and N. Trall, Some Mathematcal Methods of Physcs, McGraw-Hll Book Co. Inc., 196. [4] K. Mshma, Study on the Precse Algnment of Accelerator (n Japanese), thess for dssertaton,. [5] A.S. Kng, M.J. Lee and W.W. Lee, MAGIC, A Computer Code for Desgn Studes of Insertons and Storage Rng, SLAC-183, It was modfed greatly to apply for the COD calculaton due to the quad dsplacements.