Simple Expression For Minimum Emittance With Linearly Varied Bending Radius In Dipole Magnets

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1 Simpl Eprssion For Minimum Emittanc With Linarly Varid Bnding Radius In Dipol Magnts G.Baranov, E.Lvichv and S.Sinyatkin Budkr Institut of Nuclar Physics, Novosibirsk 60090, Russia W study th thortical minimum mittanc for a non-uniform bnding magnt with th bnding radius linarly rampd from th dipol cntr to its nd. W driv th prssion for th minimum mittanc as a function of th bnding angl and pand it into a powr sris with rspct to a small angl. Th first trm of th pansion givs th TME minimum mittanc whil th high-ordr trms ar rsponsibl for its modification. On th contrary of th vagu and ntangld closd-form solution, th cofficints of th powr sris ar simpl and clarly indicat conditions and limitations for mittanc rduction blow th TME valu. With th hlp of analytical prdictions w dsign a lattic cll with longitudinally varid bnds dmonstrating th mittanc lss than that for th TME structur of th sam bnding angl. I. INTRODUCTION Th quilibrium mittanc in a rlativistic lctron storag ring is dfind by th balanc btwn radiation damping and quantum citation and can b prssd as I 5 Cq, (1) J I whr C q = m, is th rlativistic factor, J is th horizontal damping partition numbr, and two synchrotron radiation intgrals rprsnt damping and citation, rspctivly [1] I M ds H ( s) ds, I 5 ( s). () ( s) M Hr (s) is th curvatur radius and th disprsion action is givn by H ( s), () with th Twiss paramtrs (, rspctivly., ) and th disprsion function and its drivativ (, ), Th minimum mittanc with uniform bnds is achivabl in th TME lattic [ 6] consisting of a bnding magnt with lngth L u (th suffi u indicats th uniform magnt to st th corrsponding valu apart from th varid bnd magnt dnotd blow with th suffi v) and bnding angl, plus numbr of quadrupols to adjust th optics. Th horizontal bta and disprsion hav spcifid minimum at th middl point of th magnt L / 15 and L / 4 (4) 0u u 0u u 1

2 which givs th minimum TME mittanc of u Cq. (5) J 1 15 To gt ovr th TME limit, Wrulich in 199 proposd to us non-uniform magnts with longitudinal variation of th bnding fild [7]. Horizontal bta and disprsion grow from thir minimums at th magnt midpoint toward th magnt nds and caus corrsponding incras of th disprsion action H (s) in th quantum citation intgral Eq. (). According to Wrulich, on can compnsat incras of H (s) by nlarging th bnding radius (or rduction of th magnt fild strngth). As a rsult, additional minimization of I 5 is possibl for th magntic fild high at th dipol cntr and low at its dgs. Th approach has bn intnsivly lucidatd in rcnt yars both analytically and numrically [8 1]. Dtaild study can b found in trio paprs [11 1] whr a minimum mittanc thory was dvlopd for arbitrary dipol bnding profil using vctor and matri form. Eact closd-form prssion for th minimum mittanc with th linarly rampd radius (with th constant fild sgmnt) was drivd and invstigatd. Th prssion (as wll as similar ons from othr rfrncs) is lngthy and cumbrsom and th conclusions following from it ar obscurd by its complity. Blow w also driv closd-form formulas for th minimum mittanc and corrsponding initial bta and disprsion in th magnt with linar ramp of th bnding radius (hyprbolic fild profil). Although ths prssions ar also ntangld, thy ar functions of th bnding angl and for 1thy can b pandd in powr sris with rspct to. Th first trm of th sris is th act TME minimum whil th nt trms plain possibl rduction of th mittanc du to th fild variation. Th trms of th powr sris ar simpl, clar and allow making asy prdictions of th mittanc minimization blow th TME limit. To chck validity of th prdictions w dsign th lattic cll with th varid-bnd magnt and compar it with th sam bnding angl TME cll. II. TASK DEFINITION Numrical amination of a TME-lik non-uniform magnt rvals th fact that to rach th minimum mittanc th curvatur radius tnds to ramp almost linarly from th low valu at th magnt midpoint to th high valu at th magnt dg (s, for instanc, [8, 1]). Hr w study just this cas with th bnding radius and th fild profil dmonstratd in Fig.1. No flat top sgmnt is includd for simplicity. Th magnt cntr and th nd valus ar B c Bma (corrsponding to c min ) and B Bmin (corrsponding to ma ), rspctivly. Th lngth of th varid fild magnt is L v. Du to th symmtry w tak th magnt cntr as th rfrnc point and considr only a half of th magnt as it is shown in Fig.1. Th bnding radius is givn by ( s) k s, (6) c

3 with th gradint k vm c. (7) L Fig.1 Bnding radius and bnding fild in th magnt half. Blow instad of th orbit lngth s w us as an indpndnt variabl th bnding angl 1 k s ( s) k ln 1, (8) c with th total angl ovr th magnt k 1 ln. (9) c Lt us compar th lngth of th uniform and non-uniform magnts with th sam total bnd and maimum fild B B. Insrting Eq. (7) into Eq. (9) w obtain u c L v 1 Lu, (10) ln( ) whr / c Bc / B. As th factor y ln( ) ln( B c / B ) is crucial in th following thory, Fig. dpicts th longation of th varid magnt with rspct to th constant on. For th sak of brvity, blow w st th horizontal damping partition J 1 and omit th factor C q in mittanc (for mittanc in nanomtrs C q 1468 E ( GV ) ). With this notation th minimum TME mittanc (5) taks th form 1 15 u min.

4 Fig. Elongation of a non-uniform magnt rlativly to th uniform on vs. y ln( B c / B ). Both magnts hav th sam pak fild and bnding angl. III. MINIMUM EMITTANCE To driv analytically th thortical minimum mittanc for non-uniform magnt with th radius (s) dfind by Eq. (6) w follow th convntional procdur. At first, w solv th scond-ordr diffrntial quations for th horizontal btatron motion and for th disprsion function using th initial conditions dtrmind by th symmtry. Thn from th btatron transfr matri w find th horizontal btatron function propagation (which is, actually, almost qual to that for a drift). At th nt stp w assmbl th disprsion action H (s) according to Eq. () and calculat two synchrotron radiation intgrals Eq. (). Finally, w minimiz th mittanc prssion Eq. (1) with rspct to th initial 0 (0) and 0 (0) (taking into account that du to th rflction symmtry ). In procssing w us th bnding angl (s) instad of s as an indpndnt variabl that looks natural for mittanc minimization tchniqu. Th calculations ar tdious and wr carrid out with th hlp of Mathmatica 10.0 computational softwar program [15]. W pand act closd-form solution givn in Appndi A into a powr sris for 1and hav th following prssion for th mittanc 9k 7 k 081k v min (11) Hr a zro-ordr trm obviously rprsnts th TME constant fild cas. Th nt trm provids th major contribution to th mittanc rduction blow th uniform magnt. Rplacing k y ln( B c / B ) w obtain a simpl formula 9 y 7 y 081y v min u min (1)

5 In this approimation th mittanc dcras dpnds only on th logarithmic ratio of th maimum and minimum fild in th longitudinal gradint bnd. Fig. shows th mittanc rduction factor r v / u as a function of k for act formula Eq.(A) and dcomposition Eq. (11) for two bnding angls of 0. 1 and 0.. Fig. Varid fild mittanc rduction factor for two bnding angls. Solid lin corrsponds to th act formula Eq. (A), dashd lin corrsponds to th sris Eq. (11). Eq. (1) cannot giv us dtails on th paramtrs of th uniform TME magnt but th pansions of th initial bta Eq. (A) and disprsion Eq. (A4) c 7k 1859k 16501k ov 1..., (1) c k k k ov 1..., (14) , rspctivly. In- corrspond to th fild and th lngth of th uniform magnt srting y paramtr into Eq. (1) and Eq. (14) yilds B c and L u c 7 y 1859 y y ov 0 u ( B c ) 1..., (15) y y y ov 0 u ( B c ) (16) Fig.4 compars th act solutions Eqs. (A, A4) with th powr sris pansion Eqs. (1, 14) for th total bnding angl of

6 Fig.4 Th initial bta (lft) and disprsion (right) normalizd to th corrsponding uniform magnt as a function of th radius gradint k. Solid and dashd lins show th act solutions from Appndi A and th powr sris pansion rspctivly. Although w plannd to pand Eqs. (A, A, A4) as a powr sris in small, it appard that th ral pansion paramtr is y k / 1. Th dtaild study shows that th numrical cofficints in Eqs. (1 16) dcras ponntially, but th sris convrgnc dpnds on th product of k and for larg bnding angl and/or larg radius gradint convrgnc can b poor. In this cas ithr th pansions should b tndd to th high-ordr trms or th act formulas ar usd. It is worth noting that th transvrs gradint K G / B in th longitudinally uniform TME magnt also can rduc th minimum mittanc blow Eq. (5) 1 K u min, but for 1maks this mchanism inffctiv compard to in Eq. (11). IV. LATTICE CELL DESIGN To validat thortical rsults w attmpt to dsign th minimum mittanc lattic cll with linar ramp of bnding radius in th dipol. For rfrnc w us th uniform dipol with th constant fild B c. Th problm is that in a ral compact lattic it is difficult to satisfy optimal conditions Eq. (4) so th rsulting mittanc dgrads. So a supplmntary issu of th paragraph is how clos w can approach th absolut minimum mittanc for both constant and varid fild TME lattic cll. Solutions providd mor compact lattic cll would b an asst as usual. Also w would lik to mntion that a systmatic dsign and optimization of th lattic with non-uniform dipol is byond th scop of this papr, w purpos only to illustrat analytical rsults of th prvious sctions. 6

7 According to Eq. (1) th only paramtr y ln( B c / B ) dfins mittanc rduction blow th uniform TME magnt. Th maimum fild Bc is constraind by availabl tchnology. For instanc, a suprconducting dipol with th fild ~10 T is rfrrd in [16] and it sms to b clos to th practical limit for rasonabl vrtical aprtur. W can incras y by dcrasing th dg fild B but according to Fig. it rsults in lngthning of th varid fild magnt with rspct to th uniform on. Emittanc rduction by an ordr of magnitud with y = 4 (s Fig.A1) longats th magnt by factor ~14 (s Fig.). Th lattr invitably implis difficultis to th magnt dsign and causs incras in th storag ring circumfrnc. Fig.5 shows th mittanc rduction factor for th linar radius ramp magnt with rspct to th uniform on vrsus th corrsponding lngth incras. Almost linar curv in Fig.5 indicats that for th magnt with linar radius ramp th mittanc rduction factor roughly corrsponds to that for th orbit longation. Fig.5 Non-uniform magnt mittanc rduction factor vs. th lngth incras (th maimum fild of th uniform and non-uniform magnts is assumd th sam). W start lattic dsign with a homognous fild TME cll providing 1nm minimum mittanc at E GV. W st 10 T maimum fild for both magnts. High fild givs th lngth of th varying fild dipol convnint for production and savs th machin siz. A sgmntd magnt with a high fild compact cor and low fild wings can also b considrd. According to Eq. (5), th optimal bnding angl for th minimum TME mittanc is and th uniform magnt lngth is Lu m. Eqs. (4) giv th initial horizontal bta and disprsion at th dipol midpoint. Bc For th non-uniform dipol w spcify th linar radius ramp according to Eq. (6) with 10 T and with th doubld lngth of th uniform magnt. Fig.5 rsults th minimum mittanc 0.5 nm with th initial bta and disprsion dfind by Eq. (A) and Eq. (A4), rspctivly. Th thortical valus ar listd in th scond column of Tabl 1.. 7

8 To simulat th minimum mittanc w split th dipol with th hyprbolic fild profil into a larg numbr of slics and rquir MAD8 [17] quippd with a simpl optimizr to find th initial bta and disprsion providing th minimum mittanc. Th optimization rsults ar listd in th third column of Tabl 1 and corrspond wll to th thortical prdiction. Tabl 1 Uniform and varid fild magnts paramtrs ( E GV, ) Uniform Varid (th) Varid (op) L, m , nm , m , m At th nt stp w dsign th simplst TME cll whr th cntral dipol (uniform or non-uniform) is surroundd by two quadrupol doublts. MAD8 with th built-in optimizr was run to find priodic solutions with minimal mittanc. A constraint of th maimum quadrupol gradint of 40 T/m is imposd on th optimization procss. Th rsultd optical functions ar plottd in Fig.6 whil som rlvant paramtrs of th clls (markd as Dtund ) ar listd in summarizing Tabl. Fig.6 Uniform (lft) and non-uniform magnt lattic cll. Th numbrs from Tabl clarly show that with th simplst and compact lattic cll w ar vry far from th optimum conditions and th minimum mittanc. For th uniform dipol w hav.75 nm instad of 1 nm and for th varid fild dipol w hav.14 nm instad of 0.5 nm. Th rason is in poor convrgnc of th priodical solutions for th horizontal bta and disprsion: thr ar no control knobs to tun both of thm simultanously. And th situation is vn wors for th non-uniform magnt bcaus (s Tabl 1) th initial bta incrass with rspct to th uniform magnt whil th initial disprsion, on th contrary, dcrass. To improv th priodic optics convrgnc w, according to th rcommndation in [14], introduc a low fild ngativ curvatur magnt (anti-bnd) into th lattic cll. Varid ralizations of th anti-bnd ar possibl. W transfr two quadrupol doublts btwn th dipols in Fig.6 to th quadrupol triplt in Fig.7 and insrt th anti-bnd into th middl dfocusing quadrupol. 8

9 Fig.7 TME clls with th ngativ fild in th cntral triplt quadrupol. Rathr low fild ( 61 mt with th full angl of 4. mrad in th uniform magnt and 1 mt with th full angl of 8.1 mrad in th non-uniform magnt) is nough to match th clls proprly and obtain th thortical minimum mittanc from Tabl 1. Th rsults ar dnotd in Tabl as Tund. To produc th rquird anti-bnd, th dfocusing quadrupol should b displacd horizontally by 1.5 mm. Tabl Rsults of th low mittanc clls dsign. Paramtr Uniform magnt Varid radius magnt Dtund Tund Dtund Tund Lngth L, m Emittanc, nm Initial bta 0, 10 - m Initial disp. 0, 10-4 m Enrgy sprad, Mom.compaction, Enrgy loss, 10-1 MV I 1, 10-4 m I, 10 - m I, 10 - m I 4, 10-4 m I 5, 10-6 m V. CONCLUSION Simpl prssions for th minimum mittanc and corrsponding initial horizontal bta and disprsion in th TME cll magnt with linar radius ramp ar prsntd. Powr sris pansion with rspct to th bnding angl dirctly givs th uniform magnt paramtrs as a zro-ordr trm whil th fild variation trms ar suprimposd ovr thm. Th logarithmic ratio of th pak cntral fild to th dg on is th only paramtr dfining th mittanc rduction factor blow th uniform magnt. Th minimum mittanc lattic clls dsignd confirm th thortical prdictions. 9

10 ACKNOWLEGEMENT Th authors thank A.Bogomyagkov and K.Zolotarv for hlpful discussions. This work was supportd by Russian Foundation for Basic Rsarch undr Grant No and by Russian Scintific Foundation Gran No APPENDIX A: CLOSED-FORM MINIMUM EMITTANCE W hav obtaind and compard th closd-form prssions for minimum mittanc and for rlvant initial horizontal bta and disprsion both with and without th focusing trm 1/ ( s). Evn for rathr strong pak fild B c Bma th diffrnc is ngligibl, so hr w ignor th focusing trm. For brvity w introduc th paramtr (s also Eq. (10)) k B y ln ln c B c, (A1) whr k (s) is th bnding radius gradint Eq. (7) and is th total bnding angl Eq. (9), and intrmdiary functions 4 y y y y 4y 8y 5 y 1y A ( y), B( y) y y y. Thn th minimum mittanc for th magnt with linar radius ramp is givn by y 15 A( y) B( y) 0v y y y 1, (A) with th following optimal bta and disprsion at th magnt midpoint 1/ c 15 A( y) 0v, (A) y 15 y 1 B( y) y y c 1 4 y 0v. (A4) y 4 y 1 In th uniform magnt limit as th radius gradint approachs 0, th abov quations convrg to th corrsponding Eqs. (4, 5) 1/ 0v c k 0 0u, 4 0v c k 0 0u, v k 0 0u. Not that th uniform TME bnd paramtrs corrspond to th pak fild of th varid bnd B, so th lngth of th uniform magnt is L. c u c 10

11 Fig.A1 plottd Eqs. (A-A4) as a function of y normalizd to th rlvant uniform magnt TME (i.., ratios / and th sam for th optimal bta and th disprsion). r ov 0u Fig.A1 Rlativ chang in th minimum mittanc (uppr plot), initial bta (lowr plot, lft) and initial disprsion (lowr plot, right) as a function of y ln( B c / B ). REFERENCES [1] R.H.Hlm, M.J.L, P.L.Morton and M.Sands, IEEE Trans. Nucl. Sci., 0, p.900 (197). [] M. Sommr, Intrnal Rport No. DCI/NI/0/81, [] L. Tng, FNAL Rport No. FNAL/TM-169, 1984; ANL Rport No. LS-17, [4] D. Trbojvic and E. Courant, in Proc. of th 4 th EPAC London, 1994, p [5] S.Y. L, Phys. Rv. E 54, 1940 (1996). [6] H. Tanaka and A. Ando, Nucl. Instrum. Mthods Phys. Rs., Sct. A 69, 1 (1996). [7] R. Nagaoka and A. F. Wrulich, Nucl. Instrum. Mthods Phys. Rs., Sct. A 575, 9 (007). 11

12 [8] J. Guo and T. Raubnhimr, in Proc.of th 8 th EPAC, Paris, 00, p [9] Y. Papaphilippou and P. Ellaum, in Proc.1st PAC, Knovill, 005, p [10] A. Strun, PSI Intrnal Rport SLS-TME-TA (007). [11] C.X.Wang, PRST AB 1, (009). [1] C.X.Wang, Y.Wang, Y.Png, PRST AB 14, (011). [1] C.X.Wang, W.W.Ho, PRST AB 15, (01) [14] A. Strun, Longitudinal gradint supr-bnds and anti-bnds for compact low mittanc light sourc lattics, Low mittanc rings workshop IV, slid 18, Frascati, Sp , 014. [15] [16] E.I. Antokhin t al. NIM A 575 (007) 1-6. [17] 1

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