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Nuclear Instruments and Methods in Physics Research A 394 (1997) 295-304 NUCLEARINSTRUMENTS

& METHODSIN PHYSICS

ELSVIER RE!!3H

Study of a novel compact standing wave RF linacR. Zhang*, C. Pellegrini, R. Cooper

Department oJPhwics. UCLA. 405 Hilgard Aw. Los Angeles, CA 90024. cIs.4Received 4 March 1996: received in revised form 19 July 1996

AbstractA novel, compact RF linac structure, the plane wave transformer (PWT), is studied. The PWT provides highaccelerating field and high efficiency, and can be used to accelerate high-brightness beams. PWT linac prototypes witheight cells at S-band have been developed at UCLA and successfully used to accelerate an electron beam by more than10 MeV in 40 cm. In this paper, we describe the principal properties of this structure, the electric parameters obtainedfrom numerical simulation, and measurement results from microwave cold tests. The mechanical design for prototypelinacs is also reported.

1. Introduction

A compact. efficient, high brightness producing, radiofrequency electron linac is desirable for many scientificand industrial applications. Electron accelerators withenergies up to a few MeV are used in radiation chemistrystudies, for instance the radiation-enhanced chemical re-actions of the highly active intermediate chemical statesproduced by the electron beam. Compact RF linacs,combined with compact, multi-terawatt, sub-picosecondlaser systems can be used in the generation of X-rays [ 1 ,which could have a significant impact on a number ofapplications in the areas of X-ray radiology, microscopyand spectroscopy. A compact RF linac that providesa high brightness beam and fits in a small room will alsobe a powerful facility for scientific research in universitylaboratories.

There has been tremendous progress in the develop-ment of RF accelerator structures in the past half cen-tury. Among the widely used RF structures, such as theAlvarez structure [2], the coupled-cavity structure (CCS)[3], and the side-coupled structures (SCS) [4], the SCSlinacs are considered as the most successful ones. How-ever, the non-cylindrical symmetry in the SCS linacmakes it both inconvenient and expensive to fabricateand difficult to assemble and tune. On the other hand, theplane wave transformer (PWT) [S] linac, which promiseshigh gradient and high electric efficiency, is simpler and

* Corresponding author

less expensive to fabricate. These properties make thePWT linac a very promising candidate for use in researchlaboratories and the industrial community.

The PWT linac could be dated back to late seventies[6]. Fig. 1 shows its 3-D schematic view. The structureconsists of a cylindrical tank and an array of disks. Thesedisks are connected together by several metal bars paral-lel to the axis. Because it is separated from the cylindricaltank, the array acts as a center conductor to supporta TEM-like plane wave between the tank and the array.This TEM wave provides the coupling between the indi-vidual cells in the array. Meanwhile, each individual cellsupports a longitudinal electrical field on the structureaxis. In other words, this structure makes use of a planewave to transform the external RF power into a longitu-dinal electric field for acceleration of particles. Thus, it isnamed a plane wave transformer [S]. This feature causesthe PWT structure to have the advantages of high shuntimpedance and strong coupling between individual cells.Furthermore, the PWT configuration provides good vac-uum conductance and ease of fabrication. However, theoperation mode in a PWT structure is a high-orderhybrid one instead of the fundamental TMol mode ope-rated in most RF accelerating structures. This operationwith a high-order mode raises concerns about: the pos-sible excitation of other, undesired, modes either by theexternal source or by the electron beam; the stability ofthe acceleration field; and the wake field effect on beamdynamics. These issues have to be studied in order for thePWT to be operated successfully.In this paper, we describe the principal properties ofthe PWT structures. Then we present the results from

016%9002/97/$17.00 Copyright CI 1997 Published by Elsevier Science B.V. All rights reservedPII SO1 68-9002(97)00684-O

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296 R. Zhang et al. /Nucl. Instr. and Meth. in Phys. Res. A 394 (1997) 295-304

- Support Rods

I_ DisksFig. 1. The 3-D schematic view of a PWT structure with flatdisks.

both the numerical simulation and the microwave coldtests, with a comparison between the two different confi-gurations developed at UCLA. We also discuss the shortrange wake-field and its effect on beam quality. Themechanical design of the UCLA PWT prototypes is alsopresented in detail. In a companion paper we give thenumerical simulation, including the space charge effect,of the beam dynamics in the linac, and the experimentalresults obtained in accelerating a beam.

2. Electrical propertiesThe PWT structure is simulated by using both theSUPERFISH and MAFIA [7] codes. The SUPERFISH

code is very helpful in finding the dimensions of the PWTfor a given operation frequency. However, the connectingmetal bars cannot be included in the simulation due totheir lack of cylindrical symmetry in the structure. There-fore, the 3-D code MAFIA has been used to obtain

a more complete picture of the electromagnetic fielddistribution in the structure and information aboutother, possibly interfering, modes.

Fig. 2 shows arrow plots from MAFIA of the electricfield distribution for two cells (one full cell plus two halfcells) of a PWT structure at different cross sections. Theelectric field distribution of the longitudinal cross sectionalong the axis is shown in Fig. 2(a). A plane wave (TEM-like) pattern is clearly seen in the space between the outertank and the loaded-disk array. The acceleration electricfield distribution in the region close to the axis is a rc-mode. Fig. 2(b) shows the electric field distribution at thetransverse cross section at the center of the full cell. Dueto the transition from a TM-mode field pattern to a TEMfield distribution, the longitudinal electric field variesfrom maximum on the axis to almost zero in the middleof the radial direction. This field pattern does not exist ina pure TMol mode. However, the magnetic field distribu-tion in the transverse cross section, as shown in Fig. 2(c),is very similar to that of a TMol mode. It should be notedthat connecting metal bars produce only a minor per-turbation on the field distribution for this operationmode, which explains why the SUPERFISH code couldgive very good results. In fact, the displacement currenton axis does not return in the region between ,the disks.Instead, it circulates via the wall of the tank. Therefore wename the operation mode as a TMOI_wall ode.

The shape of the disks near to the axis can be used tocontrol the electric field profile on the axis. We haveconsidered two cases: disks loaded with nose cones andflat disks. The disks loaded with nose cones are usedin a PWT to increase the shunt impedance of thestructure by concentrating the electric field near theaxis. This configuration is just suitable for moderateacceleration gradient, due to a high peak surface electricfield on the tips of the nose cones. The linac using diskswith nose cones at UCLA is called PWT2. The electric

b) t YFig. 2. Arrow plots of the electromagnetic field distribution of the operating mode at different cross sections in a two-cell PWT withloaded disks: (a) electric field at the on-axis cross section;(b) electric field at the middle plane of the PWT; (c) magnetic field at the middleplane of the PWT.

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2.q :4I6

9 :

: 4 65,

0.05

rt Z

loaded disk0 O5 10Disk surface location (CT:)

1.25 j . _ . . . I-

5 10 15Disk surface location (cm)

4 rtflat disk Z

Fig. 3. The electric field distribution on the surfaces of two types of PWT disks.

field amplitude on the disk surface in a PWT is shown inFig. 3(a). It is seen that the maximum ratio of the peaksurface field on the tip of a nose cone to the on-axis fieldis about 2.5. This ratio has to be reduced to close to unityin order to achieve a high-gradient operation. This lowpeak surface field can be obtained using flat disks. Thelinac using flat disks at UCLA is called PWT3. Fig. 3(b)shows the relative peak field amplitude along a flat disksurface in the PWT3. The maximum ratio is reduced toabout 1.2 for a disk thickness of 1.2 cm. Thicker disks arefavorable for a low field ratio but unfavorable for goodshunt impedance. The dependence of the field ratio andthe shunt impedance upon the disk thickness is shown inFig. 4. The UCLA PWT3 prototype chose a disk thick-ness of 1.2cm as a trade-off with an effective shuntimpedance of about 57 Mn/m, calculated from MAFIA.The electrical parameters of the PWT prototypes withdifferent types of disks calculated from MAFIA are listedin Table 1. As a comparison, the measured results fromcold tests are also listed.

The resonance frequency at n-mode in a PWT struc-ture is exclusively determined by the dimensions of thedisks and the cell length, and is independent of thediameter of the cylindrical tank. This behavior can beunderstood by examining the field pattern as shown inFig. 2(a). Due to the TEM plane-wave field distributionin the region near the inner wall of the tank, the increaseof the diameter of the tank does not perturb the fieldlines. Therefore, the resonance frequency remains unaf-fected. This property might make the PWT more attract-ive to operate at higher frequency, say X-band, where it isusually difficult to fabricate structures because of thesmall physical dimensions and stringent tolerances.However, an increase of the tank diameter will storemore energy in the structure while the RF power dissipa-tion in the outside tank stays almost unchanged, excepta slight increase in the end-flanges of the tank. Thereforethe unloaded-Q increases with the diameter of the tank.In addition, the TEM field dissipates less RF power onthe wall of the tank than does a structure employing

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0.0 0.5 1.0 1.5 2.0Disk thickness (cm)

Fig. 4. Shunt impedance and field ratio versus thickness of theflat disks in an eight-cell PWT.

Table 1The electrical parameters of two PWT prototypes calculatedfrom MAFIA and measured in cold testsParameters PWT2 PWT3

Simulation Cold test Simulation Cold testFrequency 2800

(MHz)Unloaded Q 28 500Effective shunt 62.8

impedance(M n/m)RIQ (a) 1550

Transit time 0.115factor

2856.3 2828 2856.314800 26900 16 100

33.2 51.2 34.6

1592 1586 16040.77 0.75 0.75

Table 2The numerical comparison in half cell electrical parameters andsome of their dimensions for a PWT with nose cones and an SCSParameters PWT scsFrequency (MHz) 2856 2856Unloaded Q 41800 16800Effective shunt impedance 104.8 86.9

(M Rim)RIQ (a) 105 217Transit time factor 0.79 0.79Cavity radius (cm) 6.9 3.9Cavity length (cm) 2.625 2.625

a TM-mode providing the same accelerating gradient.The half cell dimensions and their electric parametersfrom SUPERFISH for a PWT with nose cones andan SCS are compared in Table 2 [S]. Both structureshave the same gap length, the same nose angle, thesame iris radius, and also the same resonance frequency,2856 MHz. It should be noted that a PWT linac has tobe terminated by half cells. The unloaded-Q of a wholePWT linac, which includes the power dissipation ofthe end-walls. is less than that of a half cell. Neverthe-less, the PWT has a higher unloaded-Q than that ofan SCS linac. The PWT also has a high shunt impedance,which is a result of a high unloaded Q. However, theR/Q of the PWT is actually less than that of an SCSlinac.

3. Dispersion curves and other modesThe behavior of the PWT linac can be studied using

a coupled resonator model. For a single cavity mode,there are N structure modes for a chain of N cells. Thedispersion curves of a structure are completely deter-mined when the frequencies of these modes are known inthe interval 0 - 71of phase shifts per cell. This dispersionrelation will become a continuous curve if the number ofcells in the chain is infinite. The dispersion relation can bealso used to determine the coupling coefficients of thestructure if the mode frequencies are found through ex-perimental measurements or numerical simulation.To find these modes and their phase advances per cell,we use MAFIA to simulate this structure with differentcell numbers and different boundary conditions. In thisway, we can make sure that all different modes withdifferent phase advance per cell can be correctly identi-fied. The arrow plots of the MAFIA graphics outputs areused to distinguish the phase advance for different modesby comparing the field patterns with one another. Thedispersion curves of some passbands for a PWT proto-type with loaded disks are shown in Fig. 5. The straightline shown in the figure is for the case of phase velocityequal to the velocity of the light.The dashed line in Fig. 5 is the dispersion curve for thepassband associated with the operation mode. The fre-quency difference between the zero-mode and the K-mode is about 700 MHz. It should be noted that thisdispersion curve does not have symmetry about the n/2mode as does that of a conventionally coupled RF linacstructure with identical cells. This feature is unique toa PWT structure and can be interpreted as arising frommulti-coupling among the different cells. Since the coup-ling is provided by a plane wave running back and forth,all cells in the structure are coupled to each other. If wetreat the PWT structure as a chain of identical cells, itsdispersion relation can, using coupled-mode theory [S].

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5000 i . . . . . . . . . - . * . . : 2866

4000

J: x measurements

TMO2 .;$ i ??

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e-field e-field X b-field (Y=O) t z (z=O) t Y (z=O) t YFig. 7. Arrow plots of the electromagnetic field distribution of the lowest n-mode (TMO1_rod)t different cross sections: (a) electric field atthe on-axis cross section; (b) electric field at the middle plane of the PWT; (c) magnetic field at the middle plane of the PWT.

\ I>. -

b-field y e-field X(x=0) t Z Y (z=O) t Y

Fig. 8. Arrow plots of the field distribution of a dipole HEM mode (i R) at different cross sections for a two-cell PWT with loaded disks:(a) electric field at the on-axis cross section; (b) magnetic field at the on-axis cross section normal to (a); (c) magnetic field at the middleplane of the PWT; (d) electric field at the middle plane of the PWT.

velocity of light, could be excited by an off-axis electronbeam. Off-axis electrons will be kicked further from theaxis by the magnetic field of these modes. If the electronshappen to be trapped in the deceleration phase, thenthese modes will have positive feedback and will grow inmagnitude. For multiple-bunch acceleration, the excita-tion of these HEM modes will increase the energy spread,cause emittance growth and even induce the beambreak-up (BBU) instability. The effects of these HEMmodes on the accelerated beam remains to be studied.It should be pointed out that the cavity mode asso-ciated with the lowest passband is also a hybrid TMolmode. The field distributions of the x-mode at differentcross sections are shown in Fig. 7. As is seen in Fig. 7(c),the connecting metal bars greatly perturb its field distri-bution. Some return current in fact flows along the bars.As a result, this mode has a much lower unloaded Q-value. Not only are their frequencies much lower thanthat of the operation mode, their phase velocities are also

much different from the speed of light. These modesshould have little effect on the performance of the struc-ture. Fig. 8 shows the field distribution of the HEM modeat different cross sections.

4. The cold test resultsTo test what was found in the numerical simulation,a cold test bench stand with a HP network analyzer was

set up to measure the parameters of the PWT prototypes.Both the PWT2 and PWT3 prototypes used in the coldtests are built using copper plated external cylinders andaluminum disks.The on-axis electric field distribution is measuredby a bead-pull perturbation technique. The diameterof the copper bead we used is 1.5 mm, which inducesa maximum frequency shift of about 1 MHz by on-axis perturbation. To reduce the system error in the

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(a) Z-axis cm)1 . 0 - aPERnsH

0.8go 0.6w 0.4

0.2nn.

(h) O5 1 0 15 20 25

Axis position (cm)Fig. 9. The on-axis field distribution in one half of an eight-cellPWT from SUPERFISH and bead-pull measurement. (a)A PWT with loaded disks. (b) A PWT with flat disks.

measurements, the room temperature has to be regulatedwith very small fluctuation. The relative electric fieldprofile on the axis of the two PWT prototypes frommicrowave measurements and simulations are shown inFig. 9. It is seen that the measurement field profiles ofboth PWT2 and PWT3 agree well with those of thesimulation.

For the measurement of the unloaded Q-value, thedisks are made from oxygen-free high conductivity cop-per (OFHC). The measured unloaded Q-value is about14000 for PWTZ, compared with the computation valueof about 28000. The measured Q-value for PWT3 is16000, which is greater than that of the PWT2, becauseall the joints in PWT3 were brazed. The discrepancybetween the measured values and the computed ones canbe attributed to many factors, such as disk-surface pol-ishing, the copper plating of the external cylinder, whichhas a lower conductivity, the slots for RF coupling andvacuum conductance, both contributing to additionalpower losses. The characteristic impedance (R/Q), cal-culated from the measured field distribution, agrees wellwith that computed from numerical simulation. Themeasured R/Q is about 1590 n in the PWT2 and 1600 Rin the PWT3 (SUPERFISH convention).To obtain the dispersion curves from measurement,one must identify different modes and their phase ad-vance per cell. One method to make this identification isto measure the phase variation by pulling a metal beadalong the axis. From the cycles in phase variation, we canfind the phase advance of that mode. This technique is

effective in finding modes which have both high Qevalues and strong electric field distribution on the axisbut fails to identify the majority of higher-order modes.In general, higher-order modes either have low Q. valuesor have little electric field on the axis. The conventionalway to measure the phase advance is not applicable tothe measurement of the PWT due to the lack of a metalboundary between cells. What makes the measurementmore difficult is that the resonance frequencies of somemodes are very close to each other and their frequencyspectra become overlapped due to their low Q. values. Asa result, it is extremely difficult to find the phase advanceof these modes in a PWT structure. The solution to thisproblem is to start from a test module with just a few cellsso that few modes will exist. Then we gradually increasethe cell numbers to find more modes. Additionally, weput the coupling probes at locations where the electro-magnetic field amplitude is relatively high to enhance thesensitivity and to keep the perturbation to a minimum.These results of the measurements are shown in Fig. 5 bycrosses.

5. WakefieldsAs is well known, when a bunched electron beam with

high peak current traverses an RF accelerating structure,strong wakefields will be excited in the structure [9]. Thelongitudinal wake will induce an energy spread in thebunch, while the transverse wake leads to kicking thebeam further away from the axis and induces emittancegrowth. However, the longitudinal wake field effects maybe compensated, at least partially, by the RF field whenthe appropriate linac injection phase of the beam ischosen.

The wakefields of the PWT were calculated by usingthe 2-D numerical code ABC1 [lo], which treats cylin-drically symmetric structures, in order to save comput-ing time and to reduce memory requirements. Althoughthe support rods are not included in the simulation,there is little error induced by this approximation asfar as the short range wake field is concerned, becausethese support rods are far from the beam axis [9].We compared the results from ABC1 with that from the3-D code T3 [7]. The discrepancy between the twocodes is less than 1.2%. Fig. 10 shows the wake poten-tials in an eight-cell PWT2 for a beam bunch length (1 a)of 1 mm. Fig. IO(a) shows the monopole (longitudinal)wake potential W,,(s), and Fig. 10(b) shows the di-pole wakes WT (s) (both transverse and longitudinal). Themaximum energy loss (monopole wake) is about48 keV/nC. The maximum transverse kick is about3.5 kV/(mm nC).

The total energy loss per cell, k,,,, and integratedtransverse kick per cell, Akick,for different bunch lengthis shown in Fig. 11. The definition of k,,, and Akick

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302 R. Zhang et al. / Nucl. Instr. and Meth. in Phys. Rex A 394 (1997) 295-304I . . . . . . . . . . . . :...:...

I \ 108 >denstiy

~ transverse b)

\ /\ /- longitudinal \ ,, , ,

-1.00 0.002 0.004 0.006 0.008 0.01 0.000 0.002 0.004 0.006 0.008 0.010s (m)

Longitudinal: Max = -48 V/PCS (ml

Transverse: Max = 3.5 V/mm/PCLongitudinal: Max = -1.1 V/mm*/pC

Fig. 10. Wake field potentials for an eight-cell PWT structure with loaded disks and bunch length 0: = 1mm: (a) longitudinal wake;(b) dipole wake.

6;....:.. . .: ....:....:. . ...0.2F.--.. .-. --0.18 R._. cmTransverse wake a-0.16 g

;-0.14 fl-o.12 3

-Longitudinal wake Y. !3, . . . . I . . . . I . . . . s . . . . I . . * . .O.OB3Bunch Length (lo / mm)Fig. 11. The variation of integrated wakefield per cell versus the bunch length of a beam.

are given by1 cok,,,=Q _-m

sP(S) W,,(s) ds,

(2)

where Q is the total charge of the bunch, p(s) the chargedensity. Compared with the SLAC disk-loaded S-bandstructure, the PWT2 has slight higher wakefield poten-tials. It should be noted that the cell length and the irisdiameter in the SLAC structure are 3.5 and 2.3 cm, res-pectively, compared to 5.2 and 1.6 cm in a PWT struc-ture. Since the longitudinal wake field scales down as 1.5power of the iris radius, the wakefield in the PWT2 is

essentially of the same order as that in the SLAC struc-ture. The wakefields in the PWT3 are basically the sameas that in the PWT2.

6. Mechanical design of the PWTThe UCLA RF linac system, consisting of an RF

photoinjector and a compact linac, is dedicated to study-ing the production of high quality electron beams, devel-oping novel RF structures, providing a high brightnessbeam for other particle beam physics experiments, suchas free electron lasers (FELs), plasma-based focusing lensand plasma-based accelerators [ll]. These experimentsrequires a nominal beam energy of about 15 MeV witha normalized emittance of less than 5 x 10m6 m-rad. Thebeam energy exiting from the RF gun is about 4 MeV.

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Therefore the PWT linac should provide an energy gainof more than 11 MeV. Additionally, the RF photoinjec-tor and the compact linac share an XK-5 SLAC-typeklystron. The RF pulse duration from the klystron isabout 3 us. Therefore, a moderate unloaded merit factor,QO, or the linac, matching with that of the RF gun cavity(Q. = 10 k), is desirable for the RF system.

The UCLA PWT linac prototypes has seven full cellswith half-cell termination at both ends. The total lengthof the linac is about 42 cm [12]. The linac length isa compromise between the acceleration gradient and thefrequency separation, which is discussed in Section 4. Thelinac structures are assembled using flanges for the con-venience of study. The inside joining-slit between anend-flange and the cylinder tank is one quarter ofwavelength away from the inner surface of the end-flange,

where the current is at the minimum. The cylindricaltanks are manufactured from stainless steal with OFHCcopper plated on the inner wall.

The schematic drawing of the PWT2 is shown inFig. 12. These loaded disks are soldered to four watertubes which provide temperature control to the linac tostabilize the resonance frequency. Both the disks andwater tubes are made from OFHC copper.There is nowater flow inside the disks. There is a water reservoir ineach of the end-flanges. The four water tubes are connec-ted at one end to a water reservoir but at the other endpass through the end flange, with the joints being vacuumsealed by viton gaskets, and return the water to thetemperature-controlled bath. The four small ports ateach end of the PWT2 tank, as shown in Fig. 12, areinherited from an earlier design [13]. In the current

Fig. 12. The cross section schematic drawing of the PWT2 linac mechanical design.

Flange DIska \,a-....- --

Fig. 13. The cross section schematic drawing of the PWT3 mechanical design.

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304 R. Zhang et al. /Nucl. Instr. and Meth. in Ph_vs. Rex A 394 (1997) 295-304design, two ports are used: one is for housing an RFmonitor; another for an ion gage as a vacuum pressuremonitor. The remaining ports are idle.

Since the flat disks used in PWT3 are much thicker,they have internal water channels to provide better tem-perature control. The disk array is separated into twohalves. Each half consists of four disks and one end-flange. The two halves join tightly together in a slip fit atthe middle of the linac structure. Water does not flowthrough this joint, but flows through the inner channelsin the disks of each half and forms a close loop with theoutside water bath. All other joints in the structure arebrazed. The viton gaskets present in PWT2 are elimi-nated. Therefore, the PWT3 could be operated at a muchlower vacuum pressure, as is necessary for operating athigh acceleration gradient.

The RF power coupling in both linacs is achieved bycutting a slot in the tank. To minimize the effect of thiscoupling slot on the beam dynamics, the vacuum slotsare cut just opposite against the coupling slot. The watertubes, which also serve to connect the disks, are locatedat places where the RF field is at a minimum. The finetuning of PWT2 was accomplished by slightly changingthe dimensions of the end cells. This tuning for PWT3 isachieved by slightly changing the length of the middlecell. Tuning from cell to cell is not necessary in a PWTstructure because of the strong coupling between cells.The fine tuning of the linac resonance frequency isachieved by adjusting the temperature of the coolingwater, which is stabilized by a constant-temperaturewater bath. No water cooling is provided to the outsidetank because its low RF power dissipation and the insen-sitivity of the resonance frequency to the tank diameter.Both PWT2 and PWT3 are successfully operated underhigh RF power to accelerate beams.

7. SummaryThe PWT linac has many advantages over other

known structure, such as high gradient, high efficiency,and low cost. Because the separation of the cylindricaltube and the central disk array, this structure is easy tomanufacture. These advantages make it attractive for use

in medical and industrial application. However, theoperation with a higher-order mode (TMoI_& mightmake it difficult to build a long structure with muchmore than eight cells because of the proximity of unde-sired modes. For acceleration of a multi-bunch electronbeam, higher-order mode damping becomes impor-tant to preserve a high quality beam and increase theBBU instability threshold. These issues remain to bestudied.

AcknowledgementsThe authors wish to thank J. Billen, C. Joshi, R. Miller,

J. Rosenzweig, S. Schriber, D. Swenson and P. Wilson formany useful discussions. We also thank J. Weaver, B.Rome and M. Woodle for their help in the mechanicaldesign and the brazing work for the PWT3. The helpfrom S. Hanna and A. Menegat in the microwavemeasurements at SLAC is also appreciated. The work issupported by the US DOE under Grant DE-FG03-92ER40693.

ReferencesCl1VIr31c41

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