Practical design approach for trapezoidal modulation of a radio-frequency quadrupole

Transcription

1 PHYSICAL REVIEW ACCELERATORS AND BEAMS 1, (018) Practcal desgn approac for trapezodal modulaton of a rado-frequency quadrupole A. S. Plastun * and P. N. Ostroumov Faclty for Rare Isotope Beams, Mcgan State Unversty, East Lansng, Mcgan 4884, USA (Receved 18 December 017 publsed 19 Marc 018) Trapezodal modulaton of quadrupole electrodes offers addtonal benefts to te concept of a radofrequency quadrupole (RFQ). Because of te sgnfcant ncrease of te effectve sunt mpedance, RFQs wt trapezodal modulaton ave a reduced nterelectrode voltage or resonator lengt as compared to conventonal RFQs wt snusodal modulaton. Ts feature s especally valuable for RFQs operatng n cw mode, snce t reduces te requred rf power. We develop a detaled procedure for te desgn of RFQ electrodes wt trapezodal modulaton. Wt our desgn procedure and by properly coosng te trapezodal cell parameters, we can easly control te peak surface felds n te RFQ to te same level as for snusodal cell modulaton. Te procedure s appled to te desgn of te electrodes for te ReA3 RFQ at Mcgan State Unversty. DOI: /PysRevAccelBeams I. INTRODUCTION A rado-frequency quadrupole (RFQ) s te most essental secton for rado-frequency (rf) on accelerators nowadays. Bot pulsed and contnuous wave (cw) mode on accelerators ave an RFQ secton coverng te energy range of on beams from ð 100Þ kev=u to about ð0. 3Þ MeV=u. Te energy gan rate and ence te effectve sunt mpedance of RFQs decreases wt te beam energy. However, most RFQ applcatons requre g output beam energes and g sunt mpedance [1 4]. For example, n superconductng (SRF) eavy-on lnacs, te RFQ sould provde a g enoug beam energy to enable te effcent use of avalable voltage from SRF cavtes. Hg sunt mpedance of te RFQ reduces te eat load on te resonator and elps to mantan stable and relable operaton n cw mode. Long RFQs wt a reduced voltage can obvously provde te gven beam energy at a reduced eat load. But resonators wt te lengt of a few wavelengts are vulnerable to sgnfcant feld dstorton effects, suc as feld tlt along te resonator and te presence of dpole feld components on te beam axs [5]. Applyng trapezodal modulaton to te RFQ electrodes s an easy and effcent way to ncrease te transt-tme factor of RFQ cells and, terefore, mprove te energy gan and te effectve sunt mpedance of te RFQ. Cylndrcal rods wt trapezodal modulaton ave been proposed for an RFQ for ease of electrode manufacturng [6,7]. Te frst RFQ was bult and commssoned n te Insttute for Hg Energy Pyscs (Protvno) n 1971 [8]. It ad a steplke modulaton of electrodes (see Fg. 1, top left), wc was replaced by trapezodal modulaton wt a smple pecewse lnear profle soon (see Fg. 1, top rgt) [9]. Trapezodal electrodes were easer to manufacture wt a late macne tan te electrodes wt snusodal modulaton. Suc trapezodal electrodes were wdely used by te Insttute for Appled Pyscs (Frankfurt) for four-rod RFQs [10]. However, te accurate calculaton of te feld dstrbuton created by suc electrodes requred te numercal soluton of te Laplace equaton. Los Alamos Natonal Laboratory proposed te wedge-saped vane tps wt a varable radus * Correspondng autor. plastun@frb.msu.edu Publsed by te Amercan Pyscal Socety under te terms of te Creatve Commons Attrbuton 4.0 Internatonal lcense. Furter dstrbuton of ts work must mantan attrbuton to te autor(s) and te publsed artcle s ttle, journal ctaton, and DOI. FIG. 1. RFQ rods and vane tps: top left, cylndrcal rods wt steplke modulaton top rgt, cylndrcal rods wt a lnear slope of trapezodal modulaton bottom left, wedge-saped vane tps wt snusodal modulaton bottom rgt, flat semcrcular vane tps wt snusodal modulaton =18=1(3)=03010(11) Publsed by te Amercan Pyscal Socety

2 A. S. PLASTUN and P. N. OSTROUMOV PHYS. REV. ACCEL. BEAMS 1, (018) of curvature along te electrode and snusodal modulaton (see Fg. 1, bottom left) [11] to create a feld dstrbuton closely descrbed by te so-called two-term potental functon [5,1]. A tree-dmensonal (3D) numercally controlled mllng macne wt a ball-end tool was used to manufacture suc vane tps. Flat semcrcular vanes wt a constant radus of te tp curvature and snusodal modulaton (see Fg. 1, bottom rgt) were proposed at te Insttute for Teoretcal and Expermental Pyscs (Moscow) [13] to be macned by a two-dmensonal (D) mllng macne equpped wt a concave cutter tool. Te wedge-saped and flat semcrcular vane tps bot wt snusodal modulaton and a constant radus of te tp curvature are currently used n most RFQs. Trapezodal modulaton as been recalled only n 000 wt te purpose of mprovng te transt-tme factor of RFQ cells [14]. In 010, te smoot trapezodal vane tp modulaton was mplemented n a cw RFQ [15] for te frst tme. In 017, te upgrade of te ReA3 RFQ [16] was ntated at te Natonal Superconductng Cyclotron Laboratory (NSCL) of Mcgan State Unversty. Te upgrade ncluded te replacement of te modulated electrodes and stems of te four-rod rf structure. Te new RFQ desgn reduces te operatng nterelectrode voltage by 19% n order to lower te rf eat load on te structure. Trapezodal modulaton of te electrodes as been appled n te regular part of te RFQ to keep te beam energy gan. Currently, no RFQ-desgn code capable of desgnng te electrodes wt bot snusodal and trapezodal modulaton s avalable. Ts paper presents a practcal 3D approac, developed for te desgn of suc electrodes. As a result of te new desgn procedure, we obtaned te computer-aded desgn (CAD) model of te new electrodes for te ReA3 RFQ. II. DESIGN REQUIREMENTS Te man purpose for te RFQ desgn s to develop te exact geometry of electrodes wc satsfes te beam dynamcs specfcatons. As a result, a CAD model of electrodes s created. We apply te followng requrements for te CAD model of electrodes: (a) manufactured electrodes ave exactly te same geometry as te CAD model (b) manufactured electrodes provde te desred varaton of te syncronous pase along te RFQ wt a gven accuracy (c) te longtudnal profle of te electrodes s contnuous and smoot (d) te nterelectrode voltage and mdcell aperture radus of te RFQ can be varable along te RFQ electrodes (e) te ntal buncng secton of te RFQ wt a varable syncronous pase and modulaton factor as snusodal electrode modulaton (f) te regular acceleratng secton of te RFQ as trapezodal electrode modulaton (g) te transton from te ntal secton to te regular acceleratng secton of te RFQ as trapezodal modulaton wt a varable modulaton factor and syncronous pase () te peak surface feld n eac RFQ cell s accurately calculated and dstrbuted as unformly as possble along te RFQ. In order to satsfy requrement (a), we use flat semcrcular electrodes (Fg. 1, bottom rgt), snce tey ave a constant radus of tp curvature and can be macned wt bot standard concave and ball-end cutters. More detals on te cutter pat and cutter radus compensaton are presented n te Appendx. For tem (b), we use te accurately smulated felds of te CAD 3D model to desgn te RFQ electrodes modulaton. Te electrodes are desgned n te quaselectrostatc approxmaton. We beleve tat te rf effects on te voltage dstrbuton are small and can be controlled durng te rf desgn of te resonator f needed. Te accurate electrostatc 3D smulatons of te electrodes provde an easy way to control many parameters, suc as te surface electrc feld dstrbuton, te nterelectrode capactance dstrbuton, etc. A combnaton of dfferent software can be used to control te desgn process of te RFQ electrodes, buld te CAD model, and smulate te electrostatc feld as t was done n Refs. [17,18]. Te VBA Macro Edtor of CST STUDIO SUITE [19] as been proposed as a sngle tool to control, buld, and smulate te accurate CAD models of RFQs [0]. Ts approac as been mplemented n te desgn of te trapezodal electrodes wt a constant modulaton factor and syncronous pase [1]. Our desgn approac s based on te CST macro, too, but we are not lmted by te constant modulaton factor and syncronous pase. Moreover, we use a contnuous and smoot longtudnal profle of te electrode tp, wc supports bot snusodal and trapezodal modulaton of te RFQ cells. III. ELECTRODE PROFILE In order to make a contnuous and smoot electrode profle along te wole RFQ, we developed a profle for a sngle RFQ cell, wc provdes postonal and tangental contnuty wt te adjacent cells. Fgure sows te transverse and longtudnal profles of te RFQ cell. Te radus of te electrode tp curvature R v does not depend on te dstance from te longtudnal axs to te electrode. Te longtudnal profle of te jt cell conssts of two stragt parts of zero slope tangents and a snusodal juncton of te lengt, wc s called te acceleratng gap of te trapezodal RFQ cell. Te total lengt of te jt cell s L j. Trapezodal cells ave <L j, wle snusodal cells ave ¼ L j. Anoter two parameters defne te jt cell cross secton at ts entrance: R j s te average aperture radus at te jt cell entrance, and m j s te modulaton factor at te jt cell entrance. Te cross secton at te end of te jt cell s obvously te same as te entrance cross secton of te (j þ 1)t cell. Te mnmum aperture radus at te jt cell entrance s defned as a j ðzþ ¼ R j m j þ 1 : ð1þ Te maxmum aperture radus at te jt cell entrance s equal to m j a j. Te varaton of te average aperture radus along te jt cell wt te center at z ¼ c j follows te law

3 PRACTICAL DESIGN APPROACH FOR PHYS. REV. ACCEL. BEAMS 1, (018) 8 R j >< R R 0 ðzþ ¼ jþ1 þr j þ R jþ1 R j sn >: R jþ1 z c j L j c j z c j c j þ z c j þ c j þ L j : π z c j For example, f te average aperture radus remans constant along te RFQ, ten R 0 ðzþ ¼R j ¼ R jþ1 ¼ R 0. Te common equatons for te longtudnal profles of orzontal R x ðzþ and vertcal R y ðzþ electrodes are 8 R 0 ðzþ 1 ð 1Þ j mðzþ 1 mðzþþ1 z c j L j >< c j R xy ðzþ ¼ R 0 ðzþ 1 ð 1Þ jþ1 mðzþ 1 mðzþþ1 sn π z c j z c j c j þ ð3þ >: R 0 ðzþ 1 ð 1Þ jþ1 mðzþ 1 mðzþþ1 z c j þ c j þ L j were mðzþ s a contnuous and smoot functon, wc descrbes te varaton of te modulaton factor. Te sgn sould be replaced by þ for te x-plane profle and by for te y-plane profle. Te functon mðzþ s gven by ðþ 8 m j >< m mðzþ ¼ jþ1 þm j þ m jþ1 m j sn >: m jþ1 z c j L j c j z c j c j þ z c j þ c j þ L j : π z c j ð4þ Te snusodal law for te modulaton factor s used wtn te acceleratng gap. However, modulaton can follow any oter smoot functon, wc s symmetrcal about te center of te cell and wc as a zero slope at te gap ends, for example, a polynomal functon gven by mðzþ ¼ m jþ1 þ m j þ m jþ1 m j z cj z cj z c j c j þ g j ð5þ or multarmonc functon gven by mðzþ ¼ m jþ1 þ m j þ m jþ1 m j ð1 pþ sn π z c j p sn 3π z c j z c j c j þ g j ð6þ FIG.. Transverse (top) and longtudnal (bottom) profles of an RFQ cell. were p s te relatve ampltude of te trd armonc of te functon. Fgure 3 sows plots for dfferent modulaton functons and electrode profles for a snusodal alf-cell. Te polynomal functon (5) s very close to te snusodal

4 A. S. PLASTUN and P. N. OSTROUMOV PHYS. REV. ACCEL. BEAMS 1, (018) cell, te electrode profle may become nonmonotonc wtn te cell, because functon (4) canges faster tan (3). Fortunately, te modulaton growt rate (m jþ1 m j )of practcal desgns s usually below te lmt sown n Fg. 4. Te lmtng curve was numercally calculated for te case of R 0 ðzþ ¼R 0. It does not depend on te acceleratng gap lengt, snce bot te modulaton and profle functons ave te same arguments of te sne functon. FIG. 3. Modulaton law functons and electrode profles wtn te acceleratng gap. functon of Eq. (4) terefore, te profles are close, too. Te parameter p of te functon (6) provdes control of te longtudnal profle curvature, wc can be used, for example, to reduce te peak felds on te electrode s surface. Te lnear functon for te modulaton factor s smple but wll lead to an offset of te mnma and maxma of te longtudnal profle as sown n Fg. 3 n te magenta curve. Te mnmum of te profle s located at z=l Te magntude of te offset ncreases wt te modulaton growt rate (m jþ1 m j ). Suc offsets are not acceptable, snce tey cange te cells lengts and even sft cells centers along te z axs. Te advances of te syncronous pase due to te sfts are dffcult to control, wc contradcts tem (b) n te lst of te requrements. Smlar sfts of te longtudnal profle appen wt any oter functon mðzþ wt a nonzero slope at te ends of te acceleratng gap. Moreover, te electrode profle bult wt a nonsmoot functon mðzþ s not smoot, too. It s wort mentonng tat profles (3) wt te modulaton functon (4) are lmted by te modulaton growt rate. If te modulaton factor canges very fast from cell to IV. ACCELERATING GAP Te optmal desgn of a trapezodal cell provdes te same peak surface electrc feld bot nsde and outsde te regon of te acceleratng gap. Fgure 5 sows te feld enancement factor κ ¼ E peak R 0 =U obtaned wt an electrostatc smulaton of te cell wt a lengt of 10R 0 and a radus of te tp curvature of 0.8R 0, were E peak s te peak electrc feld on te electrodes surface and U s te voltage appled to te RFQ electrodes. All results presented below correspond to flat-semcrcular electrodes wt a radus of vane tp curvature of 0.8R 0. Te optmal lengt of te acceleratng gap s about.8r 0. It provdes te maxmum T=κ rato,.e., te gest energy gan rate at a gven magntude of te peak surface feld and average aperture radus. If te gap lengt s smaller tan.8r 0, ten te peak surface feld does not depend on te lengt of te trapezodal cell, because t s reaced n te acceleratng gap regon, wc remans te same at any cell lengt. It can be benefcal for te RFQ desgn to make peak felds ndependent from te cell lengt. Fgure 6 sows te feld enancement factor for a cell wt g ¼.6R 0. It remans constant untl te cell s sort and close to te snusodal one at g=l > 0.7. In ts case, peak felds are drven mostly by te acceleratng gap, not by te transverse profle of electrodes. If one assumes constant R 0, ten g remans FIG. 4. Lmtaton of te modulaton growt rate. FIG. 5. Feld enancement factor of te RFQ cell wt trapezodal modulaton

5 PRACTICAL DESIGN APPROACH FOR PHYS. REV. ACCEL. BEAMS 1, (018) analytcal formula for te transt-tme factor n te wole range of g=l. FIG. 6. Feld enancement factor of te RFQ cell wt trapezodal modulaton at g=r 0 ¼.6. constant, too, wle te L s varable, because te orzontal axs s g=l. It means tat te left part of te plot (g=l < 0.7) corresponds to long trapezodal cells, wle te rgt part of te plot (g=l > 0.7) to sort snusodal cells wt te same snusodal gap lengt. Te feld enancement factor ncreases for sort cells, because peaks felds are not drven only by te transverse profle of te electrode. A small curvature radus of te longtudnal profle may domnate for sort cells. Terefore, Fg. 6 means tat te peak felds of a sort snusodal cell could be larger tan tat of te long cell wt trapezodal modulaton. Te varaton of te transt-tme factor s presented n Fg. 7. Curves n te fgure tend well to te teoretcal lmts: T ¼ 1.0 at g=l ¼ 0 and T ¼ π= at g=l ¼ 1. However, we ave not found a smple and precse FIG. 7. Transt-tme factor of te RFQ cell wt trapezodal modulaton at g=r 0 ¼.6. V. 3D MODEL Te goal of te desgn procedure s to calculate te parameters for eac RFQ cell: average radus of aperture, modulaton factor, gap lengt, and cell lengt. Te modulaton desgn ncludes two steps. Te frst step s an estmate of te output energy, total lengt, and cells parameters. A prmary 3D CAD model of te RFQ electrodes s created as a result of ts step. Ts model s created n CST for te followng electrostatc smulatons. At te second step, te prmary 3D model s modfed cell by cell to provde te desred law of syncronous pase, modulaton, and average radus of aperture along te electrodes wt a gven accuracy. For te ReA3 RFQ, we assumed a constant average aperture radus of 6.56 mm. Te lengt of te acceleratng gap of all trapezodal cells s equal to 15 mm. Te prmary 3D model s desgned usng a smple and fast Fortran code. It usually takes several teratons to obtan te desred combnaton of te output energy, RFQ lengt, acceptable ntal syncronous pase, and modulaton factor. A flowcart of te desgn algortm to generate te prmary 3D model for ReA3 RFQ s sown n Fg. 8.A sort descrpton s gven below. () Defne te ntal values of syncronous pase φ, acceleraton effcency A, and transt-tme factor T [5] at te nput normalzed velocty of te reference partcle β ¼ v=c. () Start from te frst cell. () Evaluate te syncronous pase φ and te product AT of te acceleraton effcency A and te transt-tme T factor wt te equatons for adabatc buncng n te RFQ: φ0 W ¼ φ W 0 AT ¼ W sn φ 0 A 0 T 0 W 0 sn φ ð7þ were W s te beam knetc energy. Index 0 means te ntal values for te RFQ or te values for te frst RFQ cell. Nonndexed parameters are te values n an arbtrary cell n te RFQ. (v) Calculate te lengt of te cell usng te average value of β along te cell. (v) If te cell s longer tan 15 mm, te cell s trapezodal wt a 15-mm-long acceleratng gap. (v) Searc for te modulaton factor m, wc provdes AT evaluated from te equatons. (v) If te calculated m >.5, ten apply te lmt m ¼.5 and recalculate te lengt of te cell. (v) Repeat for te followng cells. Te Laplace equaton s solved numercally at eac cell to fnd te modulaton factor. If te current value of m does not provde te requred AT, ten m s canged accordng to te bnary searc (bsecton) algortm and te Laplace equaton s solved agan. Ts process repeats untl te gven accuracy for m s aceved. Te solver for te Laplace equaton uses a

6 A. S. PLASTUN and P. N. OSTROUMOV PHYS. REV. ACCEL. BEAMS 1, (018) FIG. 9. Modelng steps of a 3D model for te RFQ cell. FIG. 10. Closed contour for a 3D model of te RFQ electrode. FIG. 8. model. Flowcart of te desgn algortm of te prmary 3D fnte-dfference metod and a mes wt ponts n a box wt dmensons ð6aþ ð6aþ L, were a s te mnmum aperture radus of te cell. Eac run of te wole algortm to desgn 100 cells takes about 30 mn. Ts tme can be sgnfcantly reduced, f one uses te analytcal formulas for acceleraton effcency [5,1] of snusodal cells nstead of a numercal soluton of te Laplace equaton to searc for te modulaton factor. In ts case, some adjustments of te modulaton factor can be performed n CST durng te second step of te electrodes desgn. Fgure 9 sows an example of te full modelng procedure of a sngle RFQ cell. Te full-lengt electrode may consst of several undreds of cells. Our frst CST VBA macro was developed to create te curve object for te wole electrode profle of te prmary 3D model usng Eqs. (1) (4). Te curve and te lnes lmtng te electrode boundares form te closed planar crcut sown n Fg. 10, wc s extruded nto a sold. Some edges of te sold are rounded. Fgure 11 presents a 3D CAD model of two FIG. 11. A 3D model of te ReA3 RFQ electrodes. adjacent electrodes of te ReA3 RFQ. We defne te geometry of eac cell by varables stored n te parameter lst of CST. Te macro uses te names of tese varables to create te profle curve. Ts approac provdes te nstant modfcaton of te geometry wt no need to execute te macro every tme we cange any geometrcal parameter. We developed a second CST macro code to perform te accurate adjustments of te RFQ cells automatcally. Ts macro canges te values of te varables n te parameter lst of te prmary 3D model. Flowcart of ts macro s presented n FIG. 1. Te man steps of te macro are: () Intal value of te syncronous pase φ 0 s assumed at

7 PRACTICAL DESIGN APPROACH FOR PHYS. REV. ACCEL. BEAMS 1, (018) FIG. 13. Tetraedral mes of te full-lengt 3D model of te RFQ electrodes n a longtudnal cross secton. FIG. 1. Flowcart of te desgn algortm of RFQ electrodes modulaton. ntal β 0 n te center of te 1st cell. () Start from te second cell. () Perform te electrostatc smulaton of te prmary 3D model n CST. (v) Perform a 1D smulaton of te reference partcle longtudnal dynamcs. (v) Evaluate te rf pase φ ¼ ωt wen te partcle s n te geometrcal center of te cell, were ω s te crcular frequency of te rf feld and t te tme. We assume tat te electrcal center of te cell concdes wt ts geometrcal center.e., ts rf pase s te syncronous pase. (v) If te rf pase s dfferent from te requred value, ten adjust te lengt of te cell properly. Ts step may also nclude adjustments of te modulaton factor (.e., acceleraton effcency) to follow te adabatc buncng concept more accurately tan provded by te prmary 3D model. If te average aperture radus s varable along te RFQ, t can be corrected at ts step, too. (v) Repeat te electrostatc and reference partcle dynamcs smulatons untl te gven accuracy for te rf pase of te reference partcle n te cell center s aceved. (v) Repeat for te followng cells. We prefer to smulate te full-lengt RFQ electrodes at eac teraton of cell lengt adjustments rater tan sngle cells. It may elp to avod possble errors caused by artfcal longtudnal boundary condtons. Eac run of te CST electrostatc smulaton of te full-lengt RFQ electrodes takes about mn. Fgure 13 sows te tetraedral mes of te CST model. Te smulaton of te longtudnal dynamcs of te reference partcle starts from te begnnng of te RFQ at every teraton of lengt adjustments. Ts s done to avod unexpected advances of te rf pase, nduced n prevous cells by te lengt adjustment process. Tey may be as g as te desred accuracy for te rf pase a fracton of a degree. Te automatzaton of electrodes geometry adjustments by a CST macro sgnfcantly reduces te nconvenence from te long smulaton runtme. Te total tme of te fully automatc adjustment process of te ReA3 RFQ electrodes was about 40 per 100 cells. Obvously, ts tme can be reduced to several ours f one smulates sort parts of te electrodes and takes care of accurate and correct reference partcle dynamcs along tese parts. VI. DESIGN ANALYSIS Te peak surface electrc feld and rf defocusng strengt are te key parameters to be controlled durng te desgn of te RFQ wt trapezodal modulaton. For te gest accuracy, we calculate te peak surface electrc feld magntude for eac RFQ cell usng te model of te sngle RFQ cell (see Fg. 9). Usng te macro, we consequently apply te dmensons of eac RFQ cell to te sngle-cell model, run te electrostatc smulaton, and save te magntude of te peak surface feld. Fgure 14 sows te dstrbuton of te peak feld along te ReA3 RFQ calculated wt te macro. One can see te constant peak feld magntude n te secton wt trapezodal modulaton (z >1 m), snce we keep te lengt of te acceleratng gap constant and sorter tan.8r 0. In oter words, peak felds of tese cells are drven by an acceleratng gap. A smlar macro s used to study te surface feld effects from te random msalgnments of te electrodes. In order to estmate te rf defocusng effect, we calculate te transverse pase advance σ per focusng perod [5] for te partcles at all rf pases n te range ( 180, 180 ). Te transverse pase advance σ for any partcle of te beam can be derved from te elements of te matrx of te focusng perod M:

8 A. S. PLASTUN and P. N. OSTROUMOV PHYS. REV. ACCEL. BEAMS 1, (018) presents te transverse pase advance plot at dfferent locatons along te ReA3 RFQ. Te strongest rf defocusng s aceved at te frst trapezodal cell wt m ¼.5, because ts cell as te largest rato of acceleratng feld magntude to te beam knetc energy. Stablty of te transverse moton of te beam partcles s mantaned wle σ > 0. FIG. 14. Peak surface electrc feld along te ReA3 RFQ. cos σ ¼ M 11 þ M : ð8þ Matrx M for te x plane s gven by x1 x 0 1 M11 M 1 ¼ M 1 M x0 x 0 0 ð9þ were ndex 0 means coordnate and dvergence of a partcle at te entrance of te focusng perod and ndex 1 means coordnate and dvergence of a partcle at te ext of te focusng perod. Assumng x 0 0 ¼ 0, we can get te element M 11 ¼ x 1 =x 0. Assumng x 0 ¼ 0, we get te element M ¼ x 0 1 =x0 0. In bot cases, te coordnate and dvergence of a partcle are taken from te result of te partcle dynamcs smulaton troug te focusng perod. Smlar calculatons can be appled to te y plane. Fgure 15 VII. DISCUSSIONS Te proposed desgn procedure as some lmtatons. For example, t assumes tat laws for acceleraton effcency AðWÞ and syncronous pase φðwþ are known n advance. Tese parameters cannot be modfed consstently wt te longtudnal pase-space plot of te wole beam, because only te reference partcle smulaton s performed. Space-carge effects are gnored for te same reason. Also, we assume tat te syncronous pase s equal to te pase of te rf feld wen te reference partcle s at te geometrcal center of te cell [1]. Te value of ts rf pase s easly derved from te smulaton results. VIII. CONCLUSION Beam dynamcs as been smulated wt te TRACK code [], usng a 3D feld of CST model. Te smulaton sows tat te RFQ desgn completely satsfes bot te beam dynamcs and geometrcal requrements. No transverse emttance growt was observed. Detaled smulaton results on te ReA3 RFQ upgrade, wc also nclude te redesgn of te rf structure and ts coolng crcuts, wll be publsed n a future paper. We beleve te presented approac and algortms for te practcal desgn of te RFQ electrodes wll motvate desgners to apply trapezodal modulaton. Te teoretcal lmt for te gan of te sunt mpedance provded by te trapezodal modulaton s about ð4=πþ 1 6%. Ts value comes from te rato of maxmum transt-tme factors for trapezodal and snusodal cells. Practcal values of te gan, averaged over te RFQ lengt, usually do not exceed 40%. Our studes sow tat te properly desgned trapezodal electrodes do not compromse wt ter focusng capablty and peak surface felds as compared to snusodal modulaton. ACKNOWLEDGMENTS Te autors tank Dr. Q. Zao for comments on ReA3 RFQ desgn and Dr. A. C. C. Vllar for contnung support. Ts materal s based upon work supported by te Natonal Scence Foundaton under Grant No. PHY FIG. 15. Transverse pase advance at tree dfferent locatons along te ReA3 RFQ. APPENDIX: CUTTER RADIUS COMPENSATION Mllng of te modulaton on te RFQ electrodes s an mportant process for te performance of te RFQ

9 PRACTICAL DESIGN APPROACH FOR PHYS. REV. ACCEL. BEAMS 1, (018) FIG. 16. A 3D model of te concave cutter and RFQ electrode. FIG. 18. A 3D model of te ball-end cutter and RFQ electrode. FIG. 17. electrode. Te sde vew of te concave cutter and RFQ FIG. 19. electrode. Te sde vew of te ball-end cutter and RFQ Computer numercal control (CNC) mllng macnes can eter calculate te cutter-tool pat from te 3D model of te electrode by temselves, or RFQ desgners provde te data for te cutter. Usually one of two types of cutter are used to macne te modulaton a concave cutter, scematcally sown n Fgs. 16 and 17, or a ball-end cutter, a smplfed model of wc s presented n Fgs. 18 and 19. Durng te mllng process, te cutter sould follow te cutter pat, wc s defned for te geometrcal center of te cutter. Te cutter pat s usually defned as a D or 3D curve. A concave cutter requres only a D mllng macne and D pat. Te 3D pat for a ball-end cutter s muc more complcated and requres a 5D-mllng macne, because te cutter axs s usually kept normal to te mllng surface as sown n Fg. 19. Below, we consder te D pat for te concave cutter. However, te followng results are correct for te ball-end cutter as well. It may seem tat te cutter pat follows te RFQ electrode profle yðzþ translated vertcally by a cutter radus R c as sown n Fg. 0 by a red curve. But t s not true for any R c > 0. Equatons (10) (15) descrbe te cutter s center pat wt cutter radus compensaton (blue curve n Fg. 0). Here α s te tangental angle of te profle, β s te angle of te profle normal, ΔyðzÞ ¼y t y c, and ΔzðzÞ ¼z t z c s te offset of te cutter center from te pont of tangency of te cutter outlne and te electrode tp (see Fg. 0). αðzþ ¼arctan y 0 ðzþ βðzþ ¼ αðzþ π= ð10þ ð11þ ΔyðzÞ ¼ R cjtan βðzþj p ffffffffffffffffffffffffffffffffffffffffffffffffffffffff ð1þ 1 þ jtan βðzþj

10 A. S. PLASTUN and P. N. OSTROUMOV PHYS. REV. ACCEL. BEAMS 1, (018) FIG. 0. Te cutter-tool pat and RFQ profle. ΔzðzÞ ¼ ΔyðzÞ tan βðzþ z c ðzþ ¼z þ ΔzðzÞ y c ðzþ ¼yðzÞþΔyðzÞ: ð13þ ð14þ ð15þ [1] P. N. Ostroumov, B. Mustapa, A. Barckowsk, C. Dckerson, A. A. Kolomets, S. A. Kondrasev, Y. Luo, D. Paskvan, A. Perry, D. Scrage, S. I. Saramentov, R. Sommer, W. Toter, and G. Znkann, Development and beam test of a contnuous wave rado frequency quadrupole accelerator, Pys. Rev. Accel. Beams 15, (01). [] M. Comunan, A. Psent, E. Fagott, and P. A. Posocco, Beam dynamcs of te IFMIF-EVEDA RFQ, n Proceedngs of te 11t European Partcle Accelerator Conference, Genoa, 008 (EPS-AG, Genoa, Italy, 008), p [3] M. Vretenar, A. Dallocco, V. A. Dmov, M. Garlascé, A. Grudev, A. M. Lombard, S. Matot, E. Montesnos, and M. Tmmns, A compact g-frequency RFQ for medcal applcatons, n Proceedngs of te 014 Lnear Accelerator Conference, Geneva, Swtzerland (JACoW, Geneva, 014), p [4] A. S. Plastun, B. Mustapa, A. Nassr, P. N. Ostroumov, L. Fallace, S. V. Kutsaev, and E. A. Savn, Beam dynamcs studes for a compact carbon on lnac for terapy, n Proceedngs of te 016 Lnear Accelerator Conference, East Lansng, MI (JACoW, East Lansng, 017), p [5] T. P. Wangler, RF Lnear Accelerators (Wley-VCH, Wenem, 008). [6] I. M. Kapcnsky and V. A. Teplyakov, A lnear on accelerator wt spatally unform ard focusng, Prb. Tek. Eksp., 19 (1970). [7] I. M. Kapcnsky and N. V. Lazarev, Te lnear accelerator structures wt space-unform quadrupole focusng, n Proceedngs of te 1979 Partcle Accelerator Conference, San Francsco, CA (IEEE, New York, 1979), p [8] A. P. Maltsev, V. B. Stepanov, and V. A. Teplyakov, Insttute for Hg Energy Pyscs, Serpukov Report No , 1971 (n Russan). [9] B. M. Gorskov et al., Insttute for Hg Energy Pyscs, Serpukov Report No , 1976 (n Russan). [10] A. Scempp, P. Junor, H. Klen, M. Daene, M. Ferc, K. Langben, and N. Zoubek, Status of te Frankfurt zeromode proton RFQ, n Proceedngs of te 1983 Partcle Accelerator Conference, Santa Fe, NM (IEEE, New York, 1983), p [11] C. W. Fuller, S. W. Wllams, and J. M. Potter, Mecancal desgn consderatons n FMIT RFQ development, n Proceedngs of te 1979 Lnear Accelerator Conference, Montauk, NY, edted by R. L. Wtkover (Brookaven Natonal Laboratory, Upton, NY, 1979), p [1] I. M. Kapcnsky, Teory of Resonance Lnear Accelerators (Harwood, Cur, Swtzerland, 1985). [13] R. M. Vengrov, E. N. Danltsev, I. M. Kapcnskj, A. M. Kozodaev, V. V. Kusn, N. V. Lazarev, A. A. Nktn, and I. V. Cuvlo, Te pulsed proton prototype of a g current on lnac, n Proceedngs of te 1981 Lnear Accelerator Conference, Santa Fe, NM, edted by R. A. Jameson and Louse S. Taylor (Los Alamos Natonal Laboratory, Los Alamos, NM, 1981), p. 9. [14] O. K. Belyaev, O. V. Ersov, I. G. Maltsev, V. B. Stepanov, S. A. Strekalovskk, V. A. Teplyakov, and A. V. Zerebtsov, IHEP experence on creaton and operaton of RFQs, n Proceedngs of te 0t Internatonal Lnac Conference, LINAC-000, Monterey, CA, 000 edted by A. W. Cao (SLAC, Menlo Park, CA, 000), p

11 PRACTICAL DESIGN APPROACH FOR PHYS. REV. ACCEL. BEAMS 1, (018) [15] B. Mustapa, A. A. Kolomets, and P. N. Ostroumov, Full 3d modelng of a rado-frequency quadrupole, n Proceedngs of te 5t Internatonal Lnear Accelerator Conference, LINAC-010, Tsukuba, Japan (KEK, Tsukuba, Japan, 010), p. 54. [16] D. Letner et al., Commssonng results of te RΕA RFQ at MSU, n Proceedngs of te 4t Partcle Accelerator Conference, PAC-011, New York, 011 (IEEE, New York, 011), p [17] S. Jolly, M. Easton, S. Lawre, A. Letcford, J. Pozmsk, and P. Savage, Novel ntegrated desgn framework for rado frequency quadrupoles, Nucl. Instrum. Metods Pys. Res., Sect. A 735, 40 (014). [18] B. Mustapa, A. A. Kolomets, and P. N. Ostroumov, Full tree-dmensonal approac to te desgn and smulaton of a rado-frequency quadrupole, Pys. Rev. Accel. Beams 16, (013). [19] CST Studo Sute, ttp:// [0] A. A. Kolomets (prvate communcaton). [1] C. L, Y. He, and Z. Wang, Optmzaton desgn of te RFQ trapezodal electrode, n Proceedngs of te 015 Internatonal Conference on Heavy Ion Accelerator Tecnology, Yokoama, Japan (JACoW, Yokoama, 015), p. 03. [] Te beam dynamcs code TRACK, ttp://