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Energy resolution in a 500 MeV H⁻ cyclotron Richardson, J. Reginald Jun 30, 1969

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Stf t-ojr ttirt*. &,&o t o wT R I U M FojorTR roI-6934-j 4j 8 100 x.2 57 R6-3r-j  r. r !"#$%MESON FACILITY OF:UNIVERSITY OF ALBERTA SIMON FRASER UNIVERSITY UNIVERSITY OF VICTORIA UNIVERSITY OF BRITISH COLUMBIA*on leave from UCLA TRI-69-6TRIUMFTRI-69-6ENERGY RESOLUTION IN A 500 MeV H" CYCLOTRONJ. Reginald Richardson** on leave from UCLAPostal Address:TRIUMFUniversity of British Columbia Vancouver 8 , B.C.Canada June 1&'&ABSTRACTThis report describes the energy resolution to be expected from a 500 MeV H" cyclotron for the following four cases:(1) Energy resolution of the raw beam, with no selection process in the cyclotron.(2) Energy resolution with low energy defining slits to limit the amplitude of radial osci1lations.(3) Separated turn acceleration and extraction.( 4 ) Selection of energy by determining threepoints on the final orbits (proposed by B.L. White).ENERGY RESOLUTION OF THE RAW BEAMThe energy spread in the extracted beam from the stripping of H" ions is determined primarily by the spread in the amplitude of radial oscillations at the extraction radius, by the energy gain per turn and by the spread in the phase of the ions with respect to the peak in the radio-frequency voltage.Estimates of the spread of the amplitude of the radial oscillations of the beam in TRIUMF can be based on the measured emittance of the ion source which it is proposed to use, and on the measured emittance of the beam from various operating accelerators.For our purposes we define the emittance ase = irtp y = tt nrm y 2'mrwhere <f>m and ym are the maximum values of the angular divergence and displacement of the beam from the equilibrium orbit when the ellipse is upright in phase space. The second equality holds for circular accelerators when the frequency of oscillations with respect to the equilibrium orbit is in units of the ion frequency and r is themean radius of the orbit.Table I gives the emittances as measured from various sources. Since <f> = Py/P> where p^ is the transverse component of momentum, comparison can only be made between two measurements after they have been adjusted to the same forward momentum p. When the energy of the beam differs from 50 MeV (protons) an additional column is shown with the results shown in parentheses for the values extrapolated to 50 MeV inversely proportional to the momentum of the particles. The results from Michigan State shown in Table I are placed in parentheses because of the following: a) the axial emittance varied between the extremesshown, depending upon the length of the axial slot in the ion source and b) the radial emittance was actually measured to be ()ir mm mrad but it is hoped that when the voltage regulation on the M.S.U.Measured Emittances from Various Accelerators-  2  ->CD2: t=CM CM0 • •Lt_ 0 .—x LAID— >OC <Dh - 21 t= t=CM CMl_ Oo OH - CM~o >CD CD+-> 21 fc= t=CD ^ rE O•— LA4-JLT>LU >CD21 t= t=LA LAOCACL > x ^L- c Cl) —^v t=o  o 2 : t= LAc_> J- PA •>C O s— CAC  O LA v—"o  — *—^L- o■M >O  o CD t=•— 21 fc= CAO  I LA •> - DC LA vOo *”.Q . /— ,—^L_ CD t=O  0) 21 OO Oc_> u • •O O ■—C  ZJ LA s— + v—✓o o4- CO+J >O c CD t= t=—  o -X O LAo  — LA vO> - CMo> fc= t=ZD 0) ✓—v ,—vCO X O CAX CA0 1 1LA LA CA------ - ----’cu >c o (DC  CD X t= t=o c 0 0 0 0CD — OL- _ J LA<>< CD fc= t=X CAo r— •ZD O -^ rLA(D(D OO cC CDCD ^ 4-»4-> "D +J TD4-» CD —  CD—  L_ E  L_E  E LU ELU>> E —  Ecn —  E CD ^EL . CD — 'CD — “OC X CDLU < c c- 3 -cyclotron is improved, the radial emittance of the beam will drop to-o -tt mm mrad. The results from the Cyclotron Corporation H- cyclotron at 15 MeV represent 80 per cent of the extracted beam, which in turn represented about 60 per cent of the total internal beam. One would expect, therefore, that the radial emittance of the internal beam was probably about 9tt mm mrad at 15 MeV.It is interesting to compare the emittance of the Cyclotron Corporation (Ehlers) ion source with that of the cyclotron which took the ions from that source and accelerated them to 15 MeV. When one reduces these numbers to a common energy of 50 MeV, one sees that the injection and acceleration process has worsened the emittance by a factor of about four. The fact that the electrostatic mirror used in the injection technique in this case can easily worsen the emittance of the beam is illustrated by the calculations of Banford1 for the mirror used in the polarized proton source for the PLA. It is un­likely that the injection scheme to be used on TRIUMF will be free of these emittance-worsening properties.It should be pointed out that this fact of the growth of the emittance does not contradict Liouvilie's theorem. The actual area occupied by the beam points in phase space does not increase, but the shape of the curves surrounding the separated parts of this area become so distorted and twisted that "all the King's horses and all the King's men couldn't put Humpty-Dumpty together again". In fact, most non­linear effects due to electric and magnetic fields result in an increase in "effective emittance" if we relate this quantity to the minimum size of waist which can be produced in the beam by a linear focusing system of given focal length.In addition to the injection non-linearities, the causes for the worsening of beam emittance include the interaction between the accel­eration process and radial oscillations due to "off-centre" injection, first harmonic in the magnetic field, the gap-crossing resonance, and asymmetry in the dee voltage.- 4 -In the belief that the emittance properties of the injection andacceleration in the TRIUMF cyclotron will be at least slightly betterthan has ever been achieved before, the radial emittance ofo jj  mm mrad has been assigned at 50 MeV. This assignment results inthe radial parameters shown in Table 2, particularly the maximumamplitude of radial oscillation Apm and the maximum amplitude of thecorresponding angle (with respect to the equilibrium orbit) .RmThere is another description of the radial oscillations of the ions about the equilibrium orbit which is frequently useful. It is shown in the accompanying diagram.The smal1 ci rcle (exaggerated in size in the diagram) represents the locus of the instantaneous centres of the ion motion, precessing about the equilibrium orbi t centre wi th a precessional frequency of 1 - w M o Since w M  > 1 for SF cyclo­trons, the direction of precession is contrary to the direc­tion of the ion motion.On this picture must be superimposed the step-wise acceleration caus­ing an increase in the radius of curvature * each turn by 8* . Let r be the radial position of the stripper. Then if the ion hits with its centre of curvature at position (1), its radius of curvature must be * = rg - sin & ; note that the momentum p = (q/c)F *For ions in the neighbourhood of 400-500 MeV, + , = 1.5, so that one turn later (or earlier) the centre of curvature will be near (2),180° away. If the ion hits the stripper under this condition, then * = rs + A r sin '-  5  -TABLE 2 Radial Cyclotron Parametersjp- is the rate of change of the energy of the ion in anequilibrium orbit with its mean radius R. AE = A„ 4I-.Rm dREnergy 30 MeV 50 MeV 200 MeV 500 MeVRad us (in.) 101 129 232 311dEdRMeV' i n ./0.60 0.78 1 .82 5-2APmax ( n.) 0.67 0.51 0.21 0.077e^ (mm mrad) r jj o jj - " 1 . 2irA Rm (mm) 3.6 3.5 3.2 2.5^Rm (mrad) 1.5 1.1 0.6 0.5A Rm (in. ) 0.14 0.14 0.13 0.10AE (MeV) 0.084 0(( 0.23 0.52- 6 -Consider an example where |2AD sin E S = 6ApK->| 2Ap |+- ->| 2Ap |«-Number of 0 2 k 6 8 10turn - 5 - 3 - 1  1 3 5 7 9. p)In this example we see that the jump in radial position from the 0to the 1st turn is 6Ap + Ap = 7Ap since p increased by Ap each turn.In the next turn, however, the jump is -6Ap + Ap = ~5Ap and is back­ward. Thus alternate jumps areAr = Ap + |2O M sin 6 |Ar = Ap - |2A^ sin 0 |and this holds true under the condition |2A^ sin 0| > A p . Note that the difference in radial position for successive even-numbered (or odd-numbered) turns is 2Ap.It is interesting to observe that in our example, if the width of the stripper 6r is less than A p , the stripper can intercept beam from turns differing by as much as 7Ap in radius of curvature, but if the stripper has a width greater than 2 A p , then the spread in the radius of curvature of the intercepted ions is not more than 2Ap for |2O M sin 0 | > A p . However, this is not the whole story, since in actuality there will be present at the same time ions such that either A^ or sin 0 (or both) will force |2A^ sin 0| < A p .We can write down immediately the minimum and maximum values for the radius of curvature under the condition |2A^ sin 0| > A p , namelyp .  = r - I A_. sin 6 I min s 1 R 1p = r  - | A n s i n B | +  2Apmax s 1 R 1and under the condition |2A^ sin B| ^ Ap we have/ 0 /P • = rmi n s NOM sin 0pmax = rs + lA R sin 0 I + A p -The limit Pmgx corresponds to those ions which just miss the stripperwith a radius of curvature rg + SOM sin e| and hit on the next turnwith the additional radius of curvature A p . The maximum value ofp (when the equality holds) becomes p = r + | - A p . max n Kmax s 2Thus the overall result for a wide stripper under the condition 2ARm * Ap ismi n = r - As Rm3p = r + v  Ap max s 2 Kgiving a spread in curvature of A Rm+ l Ap.We also see from this diagram that for a wide stripper the radial spread of ions on the stripper itself is 2Ap under the condition |2O M sin 6 | > A p . On the other hand, if |2AR sin e| ^ A p , then the radial spread on the stripper is |2AR sin e| + A p . Thus for a wide beam 2ARm ^ Ap the radial spread on a wide stripper is 2 A p .Under the condition |2AR sin e| < Ap the radial position of successive turns can be pictured as follows:+  1 2Ap I - > |  2Ap I-*-Number of turn2 A 6 8( )  1 0Thus for a narrow stripper (6r < Ap) the spread in curvature is the same as in the previous case, namely 2AR . However, for a wide stripper (6r > 2Ap) we seep = r - A n minimum s Rp = r + A n + Ap maximum s Rand so the spread in curvature is 2AD + Ap.K- 8 -These results are summarized in Table 3, where the minimum and maximum values of the radius of curvature are given for various condi t ions.TABLE 3Summary - Limits of the Radius of Curvature pnarrow stripperI "  1 8*wide stripper6r > -8*FO ARm > 8* FO ARm < 8*min r -s A Rm rs A Rmmax r +s A Rm r +s A Rmmin r -s A Rm rs A Rmmax r +s 1 8* r +s A RmFor a very narrow stripper ('r << 8*2 the fraction of the internal beam intercepted will be on the average '"38*  However, if |2AR sin ' 4 1 8* there may still be radial structure in the beam, in which case there will be large variations in the stripper current as a function of radius, reflecting a combination of turn separation and radial oscillations.The intensity profile of the extracted beam vs. momentum (or energy) for a narrow stripper (6r < 8*2 can be related to the density dis­tribution of the ions in the precessional circles shown in the first diagram of the report. In particular, the projection of that density distribution on the line connecting the stripper and the centre of the cyclotron will give directly the intensity profile vs. momentum. Under normal conditions the density distribution is peaked as A R -* 0, and so the intensity profile is also sharply peaked with a total width ofOCo I FLO M max jn m0mentum.PThe results obtained so far can be used to estimate the energy resolution of the raw beam as it is extracted from the cyclotron. If a wide stripper is employed and if the radial parameters shown in- 9 -Table 2 are correct, one arrives at an estimate of MeVas the energy resolution for the raw beam at 500 MeV and t o 5 -) MeV as the energy resolution at 200 MeV. On the other hand, with a narrow stripper one predicts ±0.52 MeV at 500 MeV and ±0.23 MeV at 200 MeV. The wide stripper will extract all the beam which comes out to the stripper radius. The narrow stripper may exhibit beam structure, in which case the fraction of beam extracted will vary markedly with small changes in extraction radius.2. ENERGY RESOLUTION WITH LOW ENERGY DEFINING SLITSxA typical phase plot for the radial oscillations about the equilibrium orbit is shown in the adjoining figure, where x is the radial displacement with respect to the equi 1 ibrium orbi t , and <f>R is the corresponding angle. The solid curve denotes the path in phase space for ions which have the maxi­mum amplitude A Rm - other ions will follow other ellipses such as those shown with dashed lines.Now the radial emittance is given by:£R “ rtR A Rbut for a given average radius in a circular accelerator6  r  2 r= constant, so all the ellipses are similar.R- 10 -x Consider the effect of a pair of radial slits, each of width w, spaced apart from each other by a distance corresponding to a quarter wavelength of radial oscillationir RThe effect is to place a barrier pre­venting all the ions but those in the shaded area in the adjacent figurefrom proceeding to higher energy. The slits may consist of very thin mate­rial, in which case the rejected ions are stripped and allowed to spiralout along the hills to a heavy-element target, or the slits may beAs an example, we may consider putting the pair of slits at a radius of 100 inches (30 MeV). In this case the azimuthal distance between slits will be 155 inches. From Table 2 we see that 8* for zero phase (a = 0°) is 0.67 inches while for a = 60° it is 0 .3 b  inches. On the other hand, A^m = 0.14 inches, so we always have the condition | 2Ar sin 0 | 1 8 * From the area of the rectangle in phase space we see that the emittance of the beam which proceeds to higher energy will be:and so the effect at higher energy will be to reduce the amplitude of radial oscillation in the ratioFor w = 0 . 0 b  in. this means that the energy spread (full width) at 500 MeV will be reduced from ±0.52 MeV to ±0.16 MeV and that at 200 MeV from ±0.23 MeV to ±0.07 MeV.thick and made of a material where the induced radioactivity is low.. _ b [ w )2TM Yt o r  : MO M 2 wO M UVtR B o H O- 1.1 -On the basis of the reduction in phase space the beam intensity wouldbe reduced in the ratio fJL x (phase space density factor). MakingeRa reasonable estimate for the latter factor, one obtains 'v 25 per cent for the surviving beam. There is another reduction factor which enters the picture, however, and that is the phase spread of the beam. If the beam is concentrated in the phase spread a = 0 ± 2.*4°, there will be no reduction in beam. However, if the phase spread is wide there will be certain phases out of the whole phase spread which will be selected by the slits for transmission and will constitute small side-bands at a = ±10°, ±14°, ±17°, ±20°, etc. The beam reduction factor would then be approximately an additional factor of five to give a total of about 5 per cent of the initial beam intensity. In either case, beams of the order of 20 yA could probably be obtained with the above energy resolution.Decreasing the slit width to 0.025 inches would improve the energy resolution at 500 MeV to ±0.10 MeV and at 200 MeV to ±0.0*4 MeV.Beams of intensity 5 to 10 yA would probably be obtainable under these cond i ti o n s .Since the radial emittance e 'v and <j>^ <v -i- for low energies,0O M *w —  = constant independent of radius and the improvement in energy resolution produced by the slits is independent of their radius, provided R < 100 inches. Using a radius less than 100 inches would make more simple the problem of disposing of the unwanted beam. The consideration which favours an energy as high as 30 MeV for the phase space is that most of the factors which tend to worsen the radial emittance, such as electric focusing, first harmonic of the magnetic field, etc., have done their work by this time. Probably a radius of 70 inches, corresponding to an energy of 15 MeV would be closer to the optimum position of the pair of slits. It is also possible thatthree slits (instead of two) would give better energy selectionwithout much further reduction in beam. There is room for furtheroptimization in this scheme.- 1.2 -3. SEPARATED TURN ACCELERATIONThis concept has been discussed by Craddock and Richardson in thetechnical report on magnetic field tolerances.2Since the ion energy is given byE = o 4qVD cos i K 1we can provide the condition that the total spread in final energybe 50 keV by requiring that the average value of cos a not be lessthan 1 - 5.x |°!! « 1 - SlL .5 x 10a 2Thus  - = 2 x 10_l+and  = ±0.014 = ±0.8°.The technical problems are severe but probably they will not be in­surmountable by 1975- They are:1. Average magnetic field controlled to 1/3 x 1052. Frequency of resonator voltage controlled to 1/3 x 1063. Resonator voltage controlled to 1/10^4. Number of circular magnet trim coils increased to 54 andmagnetic field radial gradient tailored to better than1 gauss/ft5- Pulsed injection of ions for 0.2 nanosecondsThe result is an external beam of 500 MeV protons with a full widthin energy of 50 keV and a duty factor of 0.45 per cent. At 200 MeVthe full width in energy would be 20 k e V .The introduction of a certain percentage of third harmonic in the RF voltage gives a flat top to the RF voltage wave and results in a great relaxation of the above technical problems:vrr— = COS <j> - e COS 3<j>- DBecause of the simple structure of the RF electrodes in TRIUMF it/ () /appears feasible3 to introduce the third harmonic electrically, thus obviating the requirement of additional electrodes for this purpose.The addition of third harmonic will increase the phase spread from ±0.8° to ±6.7° and results in relaxation of certain of the technical requirements as follows:K  Average field controlled to 1/A x 1052'. Frequency of resonator voltage controlled to 1/A x 105Y. Resonator voltage controlled to 1/10*+Af Number of trim coils and tolerances on the magnetic radial gradient could be relaxed to normal, but at the expense of requirements l'and 2 'S ' Pulsed length of the ions would be about 1.5 nanoseconds6'. Implicit in y  is the requirement that the fraction of third harmonic e be controlled to 1/1031'. The phase change , of the third harmonic with respect to the fundamental as in the expression7^ —  = cos <f> - e cos (34>_<5)Dshould be held to 6 ^ ±0.15°.Again the result is an external beam of 500 MeV protons with a full width in energy of 50 keV but now with a duty factor of 3-6 per cent.A. ENERGY SELECTION IN THE FINAL ORBITBruce White4 has suggested the possibility of making the energy selection in the final orbit by the use of two slits and a stripper, with the beams stripped by the two slits also being led outside for/ (8 /use as external beams. This has the advantage of not increasing the internal radioactivity of the cyclotron.For our purposes of discussion we can consider that the final orbit of the ion is circular, although actually it has a six­fold scallop due to the azimuthal variation of the magnetic field. In the adjoin­ing diagram the radius of curvature of the orbit is determined by the three points of the first slit, the second slit and the stripper. The two slits are spaced one quarter of a wavelength of radial oscilla­tion apart 60°). The arcdistance from the second slit,to the stripper is chosen to be the same.Orbits such as the following must be prevented:Particle goes through first slit while at (1) on precessional circle, makes one complete turn to (2), then goes inside second slit at (3) and hits stripper while centre is at (A). Preventing this motion requires the radial thickness of the second slit on the inner sideto be t2 > A^. On the other hand, in order that the des ? redparticles will have missed t2 on the previous turn, we must have t2 < A p , thus requiring A. < Ap.A sufficient condition for the elimination of all motions of this type is 2 A R < A p .- 15 -Referring to Table 2 we see that at 500 MeVAp> v = 0.077 in.maxso Ap = 0.060 in. for a = 38°.We can therefore requireFO M < 0.060 in.or O A < 0.030 in.From our earlier considerations we can see that the requirement would be met by a pair of slits of width 0.0A0 in. placed at low energies. Also we can set tj = t2 = 0.0A0 in. without interfering with the desired beam.An important question connected with this technique is the minimum width of slits which it is profitable to use in the final orbit. This question can be turned around into the form: "What is the radial spread of a monoenergetic beam of particles in the final orbit?" The most important contribution to the radial spread will be due to the fact that the axial magnetic field is a function of the axial distance from the magnetic median surface.,2Thus B = B (0) - —Z Z onF E l + a2^l 9x2 8y2.Bz may be either a maximum or a minimum on the magnetic median surface (m.m.s) in SF cyclotrons.The maximum value of the parenthesized expression occurs in TRIUMF at large radii and at certain azimuths where it is 10 gauss/in2 . Thus 0.3 in. away from the m.m.s.Bz - Bz (0) = 10_4 Bz (0)and so the radius of curvature would differ by the same fractional amount, giving 10-4 x 300 = 0.030 inches.On the other hand, if we use the average value of the second/ (' /derivative of the field, we obtain a change in radius of only 0.001 inch.In order to make a better estimate of this effect, the general orbit code GOBLIN was used to track 450 MeV ions through two turns of unaccelerated motion; the magnetic field expansion and equations of motion were correct to second order. One ion was started off on the magnetic median surface and gave the equilibrium orbit. Another was started off parallel to the m.m.s. but with an axial displacement of 0.5 inch. A third was started off on the m.m.s. but with an appropriate axial component of momentum. One triad of ions was started off just before an axially defocusing spiral edge of the magnetic field, and another was started off just before a focusing edge. The results indicate that the spread in radial position for a monoenergetic beam over an azimuthal path of 120° with the spread in initial conditions mentioned above would be 0.005 inch. It would therefore be very difficult to design a system of slits giving an energy resolution better than that corres­ponding to this spread in radius.The radius of curvature of a circle is determined by three points as indicated in the above figure. The sagitta is given bys = p 1 - COS yQwhere y  is the azimuthal angle between the two slits (and also between the second slit and the stripper). Define £ = pe, then approximatelye2 £2 5 = p T “ WThus a slop in the sagitta As - ”g~ ^ -" would lead to an uncertainty in the radius of curvature of8* 7T s* 10 8In our case, with As = 0  0 0 )  in. and a final radius of ) 0 0  in., we have- 17 -^  = 8 x 3 x 10~ 3 = 8 x 10-5 9 300 x 1 o x iuor Ap = 0.024 in., corresponding to an energy spread of(5 MeV/in.) (0 . 024 in.) = 120 keV or ±60 keV full width.In estimating the intensity of this energy selected beam we must start with the 20 pA estimated as coming through the low energy slits, since these are necessary for the satisfactory operation of the final orbit selection scheme. The phase spread is reduced by the 0.003 in. slits from ±2.4° to ±0.4°, and the phase spacereduction is 0.0032= 10- 2 , giving an overall reduction of beam0.03intensity by a factor of 600. Thus one would expect an energy resolved beam of approximately 0.03 pA from this technique. As indicated above, the microscopic duty factor would be of the order of 0.2 per cent.4. SUMMARY AND RECOMMENDATIONSThe following table summarizes the results we have obtained on these three methods of improving the energy resolution of the proton beam:Full Width Energy SpreadSpread in PhaseEstimated 1 ntens i tyDutyFactor(500 MeV)Raw Beam +600 keV ±30° 200 pA 15%-520 ±50° (3rd) 200 27Low Energy Slits0.040 in. ±160 ±2.4° 20 1 .30.025 in. ±100 ±2.0° 5 1.0±15° (3rd) 40 8Separated Turn ±25 ±0.8° 10 0.5Acceleration ±25 ±6.7° (3rd) 50 3.6Final Orbi t ±60 ±0.4° 0.03 0.2Selection ±4.0° (3rd) 0.3 2(3rd) indicates that the proper mixture of third harmonic of the RF voltage is used./ (7 /Since each of the above methods has its own advantage, it is recommended that the cyclotron be designed so that each may be incorporated in its operation.ACKNOWLEDGEMENTSI wish to extend my thanks to Dr. M.K. Craddock for several helpful discussions and to Dr. George Mackenzie and Mr. Michael Linton for their computations with the general orbit code GOBLIN.REFERENCES1. A.P. Banford, PLA, Acc. Phys. 29 (unpublished report)2. M.K. Craddock and J.Reginald Richardson, TRI-67-23. K.L. Erdman, private communication4. B.L. White, private communication- 19 -APPENDIXThe Use of Third Harmonic in the RF VoltageLet us assume that a fractional amplitude e of third harmonic is added out of phase with the first harmonic as indicated by:V ,rj— = cos <f> - e cos 3<f>.DThis can be expanded for <f» << 1X  = i- i i + £ ^ _  fj _ H i  + 8J± Vn 2 t T  2 T t- 1 - e - (1 - 9e) + ijy (1 - 81 e) - ...Note that the dependence on phase can be cancelled out to second order by letting the admixture of first harmonic be e = 1/9.Then the dependence becomesJ L = , . e . £ LVD 3In order to achieve a certain uniformity of energy gain with e = 0 it may be necessary to restrict the ion phase position to ±<t>0.The corresponding restriction for e = 1/9 would thus be±4> = ±(/3/2<t>0) V 2 and the ratio of the allowable spread in phase ise a I4* o/3?2 1/2«  1Thus the addition of an admixture of third harmonic of e = 1/9 would increase the allowable phase spread from ±1° to ±8.4°.The situation which occurs for e > 1/9 is illustrated by Figure 1 for the case where e = 1/6. The curve of the total RF voltage now-  2 0  -exhibits a small dip at the centre. <}> is the phase angle whered v ^= 0 and <j>3 is defined by v(<f>3) = v(0) = 1 - e. Finally, <j>0 isdefined byin energy gain per turn with the pure fundamental as would the phase spread ±<f>3 with the admixture e of third harmonic.±(j>0 is the phase angle spread which would have the same variation<j> 3The ratio —  gives a measure of the improvement in the phase spreadt gacceptable in the acceleration process where the limitation is due to posts, slits or other considerations which limit the variation in energy gain per turn. This ratio is shown in Figure 2 as a function of <f> Q . If the acceleration phase spread is limited by electric focusing, the slope of the v vs. <|> curve becomes of greater importance, and Figure 2 does not apply. This situation will be discussed by Craddock et al in another paper.v(<J>m ) (1 - e) _ 1 _ cos v = vv  (4>m ) D 'o00o(4oOCVIOO) knOOCDO0000o(4o o CMd CMoFIGUREIMPROVEMENT RATIO *3/*0FIGURE 2

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