PLACE IN RETURN BOX to remove this checkout from your record. To AVOID FINES return on or before date due. MTE DUE MTE DUE MTE DUE 1/98 aICIHCDIbOm.p65-p.14 EFFECTS OF FIELD IMPERFECTIONS ON RADIAL STABILITY IN A THREE-SECTOR CYCLOTRON* By William Stephen Paul Hudec A THESIS Submitted to the College of Science and Arts of Michigan State University of Agriculture and Applied Science in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Physics 1961 Approved *Research Supported in part by U.S. Atomic Energy Commission Contract AT (llOl)-872 ABSTRACT EFFECTS OF FIELD IMPERFECTIONS ON RADIAL STABILITY IN A THREE-SECTOR CYCLOTRON by William S.P. Hudec As the title implies, the main body of this report is primarily concerned with the effects of field imper— fections on radial stability. After a brief discussion of the two fields subjected to investigation in this report, the theory of the effects of field imperfections is presented in terms of the variable y(¢) instead of the usual radial displacement variable x(6). It is then .shown that to a good approximation the predominant first harmonic of x(6) is equal to the first harmonic of y(¢). With this equality established, it is possible to make a direct detailed comparison between the theory which is formulated in terms of y(¢) and computer results which are formulated in terms of the variable x(6). In the final sections, the results of computer pro- grams relative to the effects of certain field bumps are presented and a comparison is made with the theore- tical predictions. EFFECTS OF FIELD IMPERFECTIONS ON RADIAL STABILITY IN A THREE-SECTOR CYCLOTRON* By William Stephen Paul Hudec A THESIS Submitted to the College of Science and Arts of Michigan State University of Agriculture and Applied Science in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Physics 1961 *Research Supported in part by U.S. Atomic Energy Commission Contract AT (llOl)-872 ii ACKNOWLEDGMENTS At this time, I would like to express my sincere appreciation to Dr. M.M. Gordon and Dr. H.G. Blosser who freely gave their time, advice and suggestions during the progress of this report. I am grateful to Dr. M.M. Gordon, Dr. H.G. Blosser and T.I. Arnette for furnishing the necessary computer programs and to the United States Atomic Energy Commis- sion for making this work financially possible. iii TABLE OF CONTENTS Page ACKNOWLEDGMENTS ................................... ii LIST OF TABLES .................................... v LIST OF FIGURES ................................... vii INTRODUCTION AND SUMMARY ................. ...... ... 1 Chapter I. MAGNETIC FIELD INFORMATION . Median Plane Magnetic Field ........... ll . Cyclotron Units ....................... l2 . Field Data on B26.29A and B26Al ....... 13 . Equilibrium Orbit Data ................ 18 5. Fixed Point Data ...................... 21 -: UJFO H II. THEORY OF THE EFFECTS OF FIELD IMPERFECTIONS 1. BaSiC Equations ......COOCCCCOOOOCOCOOC 35 A. Equations for Fixed Point Orbits .. 42 B. Approximate Phase Invariant ....... 43 2. One-Sector Perturbation ............... 44 3. Two-Sector Gradient Perturbation ...... 49 4. Simple Criterion for Stability ........ 55 5. Evaluation of Perturbation Parameters for a Given Bump Field ........... 60 III. COMPARISON OF THEORY WITH COMPUTER RESULTS 1. Orbit Properties in the Absence of a Field-Bump .....OOOOOOOOOOOOOOOOOO 79 2. Flat One-Sector Bump Field ............ 86 A. General Results ................... 86 B. Discussion of the Phase Plots ..... 90 C. Detailed Comparison of Computer Results with the Theory ......... 94 Page 3. Flat Two and Four—Sector Bump Field ... 101 A. General Considerations ............ 101 a. General Considerations for the Two—Sector Flat Bump Field ........................ 102 b. General Considerations for the Flat Four—Sector Field Bump .......................... 105 B. Discussion of Phase Plots ......... 106 a. Flat Two-Sector Bump Field .... 106 b. Flat Four—Sector Bump Field ... 110 4. Two—Sector Gradient Bump Field ........ 112 A. General Considerations ............ 112 B. Discussion of Phase Plots ......... 116 a. Phase Plots for B26Al with 62:0ooooooooooooooooooooooo .116 b. Phase Plot for B26.29A with 62:0......OOOOOOOOOOCOOOOOO 122 0. Phase Plots for B26Al for Other 62 values ......OOOOOOOOOOOOOO 123 APPENDIX Extension of Theory to a Four—Sector Geom- etry ...O0....O0.000000.........OOOOOOOOOO 128 REFERENCES 0.0.0.0....0.0000000000000000.0.00.00... 160 LIST OF TABLES TABLE ' Page 2-1 Correlation between bump field and perturbation parameters for the one— sector perturbation ........................... 70 2-2 Correlation between bump field and perturbation parametersfor the two- sector gradient perturbation .................. 70 3—1 Amplitude and phase of the zero, first, second, third and fourth harmonic of the fixed point orbit for U1 .................. 80 3-2 Specification of o and $2 to obtain "pure one-corner ogening" or "pure two-corner opening" of the stable triangle for fields B26Al and B26.29A ......... 84 3-3 Numerical representation of certain parameters necessary to discuss the phase plOtS O......OCOOCCOOOOOOOOOCOO00.0.00... 86 3~4 Relations between perturbation parameters and the bump field for the flat one-sector bump forB26029A .0................COCOOCCOO... 87 3-5 Relations between perturbation parameters and bump field for the flat one-sector bump for B26Al OO0.0.0.0000........OOOOOOOOOOOO 87 3-6 Values of A and a for E0, U1, U2 and U3 for various bump strengths .................... 96 3-7 Evaluation of perturbation parameters in terms of the bump strength and phase angle for fields B26.29A and B26Al for the two- sector flat bump .............................. 102 3—8 Evaluation of perturbation parameters in terms of the bump strength and phase angle for fields B26.29A and B26Al for the four- sector flat bump .............................. lO2 TABLE 3-9 3-10 3-11 Evaluation of the quantity Ib'/( (46/AO )I for fields B26. 29A and B26A1 ............... Evaluation of the perturbation parameters for fields B26.29A and B26A1 for the two- sector gradient perturbation ............... Evaluation of Ib'/(46/A )I for fields B26. 29A and B26A1 for tfie two- sector gradient bump field-......OOOOOOOOOOOOCCOOOO Page 103 114 114 vi FIGURE 1. vii LIST OF FIGURES Page Average magnetic field Bo’ amplitude of three-sector component, B1 and Spiral angle C1 as a function of r(c.u.) for field B26.29A ......................... 25 Average magnetic field Bo’ amplitude of three—sector component, B1 and spiral angle C1 as a function of r(c.u.) for field B26A1. .............................. 26 Axial oscillation frequency vz as a function of energy for field B26.29A (solid line) and for field B26A1 (broken line) ............................. 27 (Vr-l) as a function of energy for field B26.29A (solid line) and for field B26A1 (brOken line) O0..........OOOOOOOOOOOOOOOOO 28 Square of the mean radius of the equili- brium orbit (R2) as a function of energy for field B26.29A (dots)and for field B26A1 (x's) ............................... 29 x vs. pX phase space diagram showing three unstable fixed points (dots) and approxi— mate stability limit for field B26.29A from 3.29 Mev to 23.45 Mev in steps of 2.24 Mev.0.........OOOOOOOO......IOOOOOOOO 30 x vs. pX phase space diagram showing three unstable fixed points (dots) and approximate stability limit for field B26.29A from 18.97 Mev to 34.65 Mev in steps of 2.24 Mev ..... 31 FIGURE 8. lo. 11. 12a. 12b. 13a. viii Page x vs. pX phase Space diagram showing three unstable fixed points (dots) and approximate stability limit for field B26A1 from 2 Mev to 28 Mev in Steps Of2Mev O......OOOOOOOOOOOOOCOOOO0. 32 x vs. pX phase space diagram showing three unstable fixed points (dots) and approximate stability limit for field B26Al from 22 Mev to 34 Mev in steps Of 2 Mev 0000..0000.....000000..000.000.... 33 Comparison of stable region for field B26.29A (solid line) and field B26A1 (broken line) at 18.97 Mev and 20 Mev respectively; the dots represent the three unstable fixed points ............... 34 Qualitative curves obtained from phase invariant (H) where b and 6' are equal to zeroIO....I......OOOOOOOOOOOOO0.00.0... 71 Qualitative evolution of three-sector phase plots with increasing one-sector perturbation: $1 = $0 0......00....0000.00. 72 Qualitative evolution of three-sector phase plots with increasing one-sector perturbation: $1 = $0 + v. ................ 73 Qualitative evolution of three-sector phase plots with increasing two-sector gradient perturbation: $2 ==¢O. ........... 74 FIGURE 13b. 14. 15. 16a. 16b. 160. 17a. 17b. 18. ix Page Qualitative evolution of three-sector phase plots with increasing two-sector gradient perturbation: $2 = $0 + w/2. ..... 75 Phase plot for the unbumped field B26.29A; the "x's" represent the three unstable fixed points (U1, U2, U3) and the stable equilibrium orbit (E0). ................... 134 Phase plot for the unbumped field B26Al; the "x's" represent the three unstable fixed points (U1, U2, U3) and the stable equilibrium orbit (E0). ................... 135 Phase plot for B26Al with: bl = +.OOl, 6 l = O. .000.0000.00.00.00...0000000....00. 136 Phase plot for B26A1 with: bl = +.OO2, 9 1 = O. a.000000000....00000000000.00...00 137 Phase plot for B26Al with: bl = +.OO4, 61 = O. 0o0..00000....00.00000..0000000000 138 Phase plot for B26.29A with: bl = +.OO4, l = O. 000000..0.......0.0000.000000000.. 139 Phase plot for B26.29A with: b1 = +.oeo, 91 = O. .00000.0.000.000...00..00.000...... 140 Amplitude (A) as a function of bump strength: solid lines represent theore- tical curves; dots represent computer reSUltS.O0......0.0.0.0..........OOOOOOOCO 141 FIGURE Page 19. Plot of aasa function of bump strength: the solid lines repre- sent the theoretical curves; the dots represent the computer results. ...... 142 20a. Phase plot for B26A1 with: b = +.OO4, 2 62:0. 0.........................OCOOOOOOC 143 20b. Phase plot for B26A1 with: b2 = +.020, 62:0. ......O...’..........C............. 144 21a. Phase plot for B26.29A with: b2 = +.O20, 92:0. OO.C.......................O....... 145 21b. Phase plot for B26.29A with: b2 = +.100, 92 = O. 0000..00...0.00.00.00.00.0000000000 146 22. Phase plot for B26Al with: b4 = +.040, 94 = O. 0.00.0........00.0.0.00.00....0.... 147 23a. Phase plot for B26.29A with: b4 = +.100, 64 = 0’ 000000.000.000.00.0...0.0000.00.00. 148 23b. Phase plot for B26.29A with: b4 = +.3oo, 94 = QC 000.000.00...0..0.00.0.0.0.0..0000. 149 24a. Phase plot for B26A1 with: a = +.10, 9 2 = O. 00.000..0..0.0....0...000.00....00. 150 24b. Phase plot for B26A1 with: a = +.30, 62:0. ......OOOOOOOOOOOOOO00.000.00.00... 151 240. Phase plot for B26A1 with: a = +.60, 92:0. ...O......OOOOOOOOOO......OOOOOOOOO 152 24d. Phase plot for B26A1 with: a = -.30, 9 = O. 0000.00000...000.0..0.00000...0000. 153 2 24e. Phase plot for B26A1 with: a = —.60, 92 = O. .000..0000......0.00.00.00.00000000 154 FIGURE Page 25. Phase plot for B26.29A with: a = +.30, 6 2 = O. ......0........0...0.0000...0..0.00 155 26. Phase plot for B26Al with: a = -.30, 92 = -12050.0....00.0....0.00..0.0.0.0.0000 156 27. Phase plot for B26Al with: a = -.30, 62 = ‘16050 00.000.00...00.000.00.00000.0.. 157 28. Evolution of four—sector phase plots with increasing one-sector perturbation: (a) - (C). c1 = cos (d) - (f), bl = do + 45° 158 29. Evolution of four-sector phase plots with increasing two-sector gradient perturbation: (a) - (b). $2 = o0; (c) - (f). $2 = $0 + 45° 159 INTRODUCTION AND SUMMARY In a medium energy three—sector cyclotron the value of Vr is always close to unity. As a result of the vr=3/3 non-linear resonance [1,2]* the radial motion of the particles has stability limits which may be rather stringent. A substantial amount of computer calculation has been carried out on machines of this type which indicate that these stability limits are sufficiently large for machines with weak—Spiral [3,4]. A theoretical analysis of the orbit properties in a three-sector cyclo- tron, including the 3/3 non-linear resonance, has been carried out by Smith and Garren [5], and by Verster and Hagedoorn [6]. The results of this work are in good agreement with data obtained from computer studies [7,8]. Since the radial stability of the particles is rather limited, and since Vr is close to unity, it is clear that the situation could be severely aggravated by the presence of relatively weak imperfection fields ("field bumps"). The present report presents the results of a study made of this problem. These results should be * References will be given in square brackets, and all references will be listed at the end of this report. helpful in establishing limits on the size of imperfec- tion fields of particular types which can be tolerated for a given three-sector geometry. At the same time, these results re-emphasize the importance of using a three- sector geometry with as little spiral as possible. A theoretical treatment of the effects of a field bump on the N/3 resonance was first given by L.J. Laslett and K.R. Symon [9] and by Laslett and S.J. Wolfson [10] at MURA in connection with a scheme for achieving beam injection into a synchrotron. A subsequent analysis was carried out by M.M. Gordon of the effects of a one-sector and two-sector field bump on the 3/3 resonance, which was less quantitative than the present work reported here, but which also included in the calculation the effects on the three outer stable fixed points associated with the edge region of the magnetic field [11]. The purpose of that work was to help clarify and interpret the results of the computer studies on a resonant beam extraction scheme being investigated at this laboratory [12]. Further analyses of this problem have been carried out by T.K. Khoe [l3] and L. Teng [14]. The results of the present study can also be used for calculation of the strength and angu- lar position of a field bump required for effective resonant beam extraction, provided Vr is sufficiently close to unity such that the three unstable fixed point orbits are substantially closer to the equilibrium orbit than are the three outer stable fixed point orbits. The extension of the present results so as to incorporate the three outer stable fixed points into the theory should be fairly straightforward. The two main perturbations of the radial oscillations produced by field imperfections are the one-sector pertur— bation and the two—sector gradient perturbation. The effect of these perturbations is to reduce the size of the stability limits; and when sufficiently strong, to completely destroy the stability. In the absence of any field imperfections, the stability limits are defined by the three unstable fixed point orbits associated with the 3/3 resonance. These perturbations cause one or more of these unstable fixed point orbits and the equili- brium orbit to move toward each other; the new stability limit is then defined by the diSplaced equilibrium orbit and the nearest unstable fixed point orbit. When the field imperfection is strong enough to cause the equili— brium orbit to merge with one of the unstable fixed point orbits, the stability limit goes to zero. An alternative picture of how the loss of stability arises is that the imperfection field produces a shift in the equilibrium: orbit and a change in the linear oscillation frequency about this orbit which tends toward the vr=2/2 stop-band. [A one—sector perturbation is produced mainly by the field imperfection component having frequency n=1, and to a lesser extent by those components having frequencies n=2 and n=4 through coupling with the three—sector structure of the equilibrium orbit and of the alternating-gradient modulation of the linear oscillations. The two—sector gradient perturbation is produced mainly by the imper- fection field harmonic with n=2, and again, to a lesser extent by those harmonics with n=1 and n=5. The one- sector perturbation produced by the harmonics given above can arise not only from the values themselves but also from their first derivatives; in the same way, the two— sector gradient perturbation can result not only from the first derivative of the field harmonics but also from the zeroth and second derivatives. There are two properties of the main three-sector field which determine the sensitivity of the orbits to field imperfections of a given size. These are the values of (Vr'l) and the values of a quantity called AO, which is essentially the (first harmonic) amplitude of the displacement of one of the unstable fixed point orbits from the equilibrium orbit in the absence of any field imperfections. In the case of a one-sector perturbation, the magnitude of the field imperfection which will com- pletely destroy the stability ("critical bump strength”) is essentially proportional to (Vr'l)Ao5 and for a two- sector gradient perturbation, the critical bump strength is essentially proportional to (vr-l). Thus if one or both of the parameters is small, the radial stability will be quite sensitive to small field imperfections of the kind described above. The values of (vr-l) for a three-sector geometry with weak-spiral are consistently larger, though not by a large factor, than those of a comparable three-sector geometry with moderate-Spiral; moreover, the values of A0 in the former case are signi- ficantly greater than those in the latux'case. As a result, a three-sector field with moderate-spiral is very sensitive to relatively small field imperfections; in the case of a tight-spiral design, the situation would be com- pletely intolerable. H.G. Blosser and K. Kosaka [15] carried out a series of computer studies in the Spring of 1961 at this laboratory on the relative merits of two different three-sector fields, one with weak-spiral and one with moderate-Spiral; in the course of these investi- gations, an exploratory study was made of the effects of a flat one—sector bump on stability which brought out the essential features of the results noted above. It was these exploratory computer results which furnished the direct motivation for the work presented here. In Chapter I a discussion is presented of the three— sector fields (B26.29A and B26A1) upon which the computer results of this study given in Chapter III are based. One of these fields (B26.29A) is essentially the same as the weak-spiral three-sector field used in many other studies at this laboratory [16]; the other field (B26A1) of moderate-spiral was so constructed that it has half the flutter of B26.29A and a compensating increase in spiral so as to yield comparable values for vZ. Output data from the Equilibrium Orbit Code [17,18] is presented giving essential properties of the equilibrium orbit and of the linear radial and axial oscillations about this orbit as a function of energy. These data are not signi- ficantly different for the two fields considered. Addi- tional data from the Fixed Point Code [17] is also pre- sented which leads to the evaluation of the radial stability limits as a function of energy in the absence of a field bump; in this case the stability limits for the weak- spiral field (B26.29A) are substantially larger at almost all energies than for the moderate—spiral field (B26A1). The theory of the effects of field imperfections is presented in Chapter 11. Instead of the usual radial displacement variable x=x(9), the theory in section 1 of Chapter II is formulated in terms of the variable y=y(¢) which reduces the linear oscillations to Simple harmonic motion of frequency Vr' It is then Shown that the predominant first harmonic component of x(9) is to a good approximation equal to the first harmonic component of y(¢); as a result, the theory is then formulated "quasi—first entirely in terms of the first harmonic (or harmonic") component of y(¢). The non-linear resonance driving force associated with the 3/3 resonance is intro- duced in a simple, semi—empirical way such that the correct first harmonic component for the unstable fixed point orbits is obtained. The one-sector and two-sector gra- dient perturbation forces are then introduced into the differential equation for y(¢) and the resultant equations for the first harmonic component of the diSplaced fixed point orbits are then derived as well as the equation for the approximate invariant associated with the motion of the phase Space points. In section 2 of this chapter the effect of the one-sector perturbation acting by itself is considered. Explicit equations are obtained for the displacement of the fixed point orbits as a function of the perturbation strength, in the two extreme cases of "pure one-corner Opening” and "pure two-corner opening" of the stability triangle. Equations for the critical bump strengths and values for the required phasescfi‘the perturbation are obtained in each case. In section 3 of Chapter II the same results are derived for a two—sector gradient perturbation acting by itself. In section 4 a Simple, semi-quantitative result is obtained for the critical perturbation strengths when both the one-sector and two-sector gradient perturbations are present sim- ultaneously. This result is based on the calculation of the linear oscillation frequency about the displaced equilibrium orbit. All these theoretical considerations are based on a representation of the perturbations by a strength and phase parameter; in section 5 it is shown how these parameters can be evaluated in terms of the physical properties of a given field bump and explicit formulae are given for carrying this out. In Chapter III results of computer computations relative to the effects of certain field bumps on the fields B26.29A and B26A1 are presented and a comparison is made with the theoretical predictions of Chapter II. The following four types of field bumps are considered: 1. a flat one—sector bump; 2. a flat two-sector bump; 3. a flat four—sector bump; and 4. a two-sector gradient bump. For each type of field bump a series of static (non-accelerated) phase plots are given for different bump strengths which exhibit the changes in fixed point locations and the stability limits. For ease of compari- son all phase plots are at the same energy for each field so chosen such that the stability limits are near their maximum in the absence of any field perturbations. For each of these field bumps the predictions of the theory are in reasonably good quantitative agreement in practi— cally all cases with the computer results as concerns: 1. the corners of the stability triangle which open; 2. the relative displacement of the fixed points; and also 3. the critical bump strengths required. In addition, for the flat one-sector bump a detailed com- parison is carried out between the computed first har- monic components of the fixed point orbits and the theoretical predictions of section 2, Chapter II. In this case the quantitative agreement between theory and computer results is very good. In the Appendix of this work some of the results of Chapter II are extended to include a cyclotron field having four-sector geometry. A note of thanks is in order to Mr. Stan Steinberg for his skillful help with some of the computer calculations. 10 CHAPTER I THREE-SECTOR FIELD INFORMATION Following a brief exposition of the field notation and cyclotron units used in this report, this chapter presents a discussion of the genesis of the two different three-sector fields which are used in the computer studies to be described in Chapter III. One of the fields (B26.29A) is a weak-Spiral geometry closely related to the fields used in other studies at this laboratory; the other field (B26A1) corresponds to a moderate-spiral geometry, which is introduced for comparative purposes. Both fields have been limited to just one harmonic in addition to the average field. Output data from the Equilibrium Orbit (E.O.) Code is presented giving the radial and axial focusing frequencies (vr and vz), the mean radius of the equilibrium orbit (R) and the frac- tional phase slip per turn (A¢/2F), all as a function of energy (E). Further data is presented from the Fixed Point Code giving the (r, pr) coordinates of the three unstable fixed point orbits relative to the E.0. (at 9:0) again as a function of energy. Using this data phase Space diagrams are then given which Show the evolution of the triangular stability region with energy; it is 11 clear from these diagrams that the stability limits for the weak-Spiral field are substantially larger than those for the moderate—spiral field at practically all energies. 1. Median Plane Magnetic Field Notation. In polar coordinates, the median plane magnetic field in a sector—focused cyclotron is specified by: B(r,9) = BO(r) + Bf(r,6), (1.1) where Bo(r) is the average magnetic field and Bf(r,6) is the flutter field which averages to zero over 6. For a three—sector geometry, the flutter field may be expressed as follows: Bf(r,e) = 2 Bn(r> cos 3a [e-cn(r)]. (1.2) n>o where Bn(r) is the amplitude of the nth harmonic and Cn(r) is the Spiral angle for that harmonic. To accomo- date the computer codes used at Michigan State University, this expression is rewritten as follows: Bf(r,9) = Z [Hn(r) cos 3nd + Gn(r) sin 3nd], n>o 'where Hn(r) and Gn(r) the Fourier coefficients of the nth liarmonic, are then given by: Hnesntricos 3ncn(r), Gnsin ancn< “H msswfim aw. ow. mm. mm. mm. am. om. ma. NH. mo. so. 0 ..mNI E) UH bifihnh b h I s b O A.:.ovp t -- H -- - .- -..- -- -- Atv u . -- -- - -- -..rt .mm+ Ammmpwomv.0m+ as. or. mm. mm. mm. am. dA.5.OJVaH1 a “ fl. .1 mSHUmm ofiom A ’ b- P ' F ' .H< "m mssmflm o is. 91. mm. am. ran. a. om. om. fl. mo. so. o 2 47:61. . l I. . J . . a . ...mm + ..0m + ntmN. .+. AVOOH+ Ammmsmmmv.:mma+ fie. or. mm. mm. mm. am. on. ma. ma. mo. :0. .A.5.0V.H. 14 . q i . . - . ] O Hm — -<-q-tnuun--4 t¢--q¢14- L.ON. mSHUmm maom :03. niomo om atom. - ......... >4- iaqtrrntupllrrutunonuiitnohuupruhpoi OO.H .--”1.-«1 ...... A.S.ovm 27 .AooAH oososbv Hammm ofioas soc one Aoeaa oaaomv mocosvopm compwaaflomo Hmflxg “m ossmflm @: om om 0H . o 0 4 1 A>o2v.hmhocm l rofi.+ ..om.+ A .Om.+ wo:.+ L .8... l ros.+ .Aoeflfi cososbvfiabmm RC 9 2 UHoflm now can Aocfla Ufifiomv V ": mpswflm 0: 0% 0m 0H 0 A>ozqiwwhocm .4 q . +VOHOI .fimo.n 29 .Am.xv Habmm efioas hon use Amooev amm.bmm semen sou smsoeo mo coapoczm m we Ammv Danae Esapnfiaasvo esp mo msfipmp Emma one mo chmsvm "m mhsmfim om om on ma 0 A>mzvazmhmcm - 1‘] Jr ..HO.+ fl .1#0.+ * 100.4. 30 \ <-—P-—> "3.29 Mev 10.01 Mev 16.73 Mev 21.21 Mev -.05 ---.08 Figure 6: x vs. pX phase space diagram showing three unstable fixed points (dots) and approximate stability limit for field B26.29A from 3.29 Mev to 23.45 Mev in steps of 2.24 Mev. -.08 I -.O6 31 +.08 +.O6 +.O4 I -.o4 I I I —.02 6-'PX-—9 +.02 +.O4 30.17 Mev 25.69 Mev 21.21 Mev +.O6 +.08 (_.><__.> -.o4 -.O6 -.08 Figure 7: x vs. pX phase space diagram showing three unstable fixed points (dots) and approximate stability limit for field B26.29A from 18.97 Mev to 34.65 Mev in steps of 2.24 Mev. 32 “+.08 -+.O6 -+.04 -+. 02 Mev ‘-002 ‘L.O4 ’—.O6 -.08 -.O6 -.04 -.02 +.O2 +.O4 +.O6 +.08 é—PX -—> Figure 8: x vs. p phase space diagram showing three unstable fixed points (dots) and approximate stability limit for field B26Al from 2 Mev to 28 Mev in steps of 2 Mev. 33 Figure 9: x vs. px phase Space diagram showing three unstable fixed points (dots) and approximate stability limit for field B26Al from 22 Mev to 34 Mev in steps of -+.08 "+.06 ..+.04 '-+.O2 1 1 ’-.O2 --O4 --.O6 --.08 -.lO I +.08 34 ._+.O4 R I I I I I I -.O6 -.O4 —.02 +.O2 +.O4 +.O6 +———PX-——9 Figure 10: Comparison of stable region for field B26.29A (solid line) and field B26Al (broken line) at 18.97 Mev and 20 Mev respectively; the dots represent the three unstable fixed points. --.06 "-.08 —-010 I +.08 35 CHAPTER II THEORY OF THE EFFECTS OF FIELD IMPERFECTIONS 1. Basic Equations. If a charged particle is moving in a magnetic field, and at right angles to the field, the momentum of the particle is given by: p = eB(r:9)P.(I’:9), where p(r,6) is the instantaneous radius of curvature, p and e are the momentum and charge of the particle and B(r,9) is the magnitude of the magnetic field at that point. Substituting the expression for p(r,6) in polar coordinates, the equation for the radial motion becomes: r ) _ r _ engr,92 . . p r2 + r2 r2 + r2 3,( , where the differentiation is with respect to 9 and r is the radial distance to the particle. Let r(e) = re(6)+x(6), where re(6) is the radial distance to the E.O. and x(6) is the radial displacement of a particle relative to this orbit. There exists a function D(9) which defines X(9) as follows: x(6) E D(6) X(9), (2.1) 36 such that with this transformation, the radial equation of motion for oscillations about the E.O. becomes: 36(9) + G(e)x(e) = K(X,e) + 5.K(x,e), (2.2) where K(X,9) represents all the non-linear terms arising from the main field, 6K(X,9) represents all terms arising from a small bump field and G(6) is a periodic function characteristic of the linear oscillations. According to Floquet's theorem[22], when K(X,6) and 6K(X,9) are zero, a solution exists for Eq. (2.2) of the form : x = c exp [ivr¢(9)], (2.3) where ¢(9) = 6 + W(9). 0(9) and w(9) are real periodic functions having the same periodicity as G(9). Moreover, 0(9) and ¢(9) satisfy the equations: 02¢ = 1 (2.4) and, b + G(9)C = vic'3. (2.5) Define y(¢) by: MW 3 C(9)3’(<1>)- (2-5) With this transformation, Eq. (2.2) becomes: 37 d2y<¢>/a¢2 + viy<¢> = c3IK + 6K(X,6)] F(y,¢) + 6F(y,¢). (2.7) where F(y,¢) and 6F(y,o) are the forces associated with the main field and the bump field respectively. It is clear from this equation that the transformation from the variable x(9) to the new variable y(¢) has the effect of converting the linear oscillations to simple harmonic motion; as a result, the non-linear force F(y,¢) and the perturbations produced by the field bump 6F(y,¢) can now be treated as perturbations of a simple harmonic oscillator having frequency vr. Specific formulae for D(9), W(9), 0(9) and G(9) will be given in section 5 below. From the above results, x(e) may be expressed as: X(9) = D(9)C(9) y[9+w(9)]. (2-8) Expanding this equation, the result is: X(9)=D(9)C(9)Y(9)+W(9)D(9)C(9) [dy(9)/d9]+.... Since |w(e)|<< 1 and |D(e)c(e) — 1| << 1, then X(9) = y(9) + [D(9)C(9) - 1] y(9) + r(e) [dy(9)/d9]- The quantities [D(6)C(6)-l] and w(6) contain mainly the harmonic of frequency three with harmonics 6, 9, etc. 38 being small and unimportant for this discussion; in addition, the harmonic of zero frequency in [D(6)C(9)-l] is of second order (w(9) has no zero harmonic by defini- tion). Furthermore, since the second and fourth harmonics of y(¢) and x(6) are usually small compared to the first harmonic, it is therefore possible to equate the first harmonic of x(9) with the first harmonic of y(¢) to a good approximation as follows: Xl(6) = yl(9), (2.8a) " designates the first-harmonic where the subscript "one (or "quasi-first harmonic"). The fact of the near equi- valence of the first harmonic of y(¢) and x(9) allows a simple comparison of the theoretical and computer results to be made; that is, the theory is formulated in terms of the variable yl(¢) while the computer results are in terms of x(6); as a result of the above relation, it is then possible to make a direct comparison between the first harmonic of fixed point orbits in the two cases. It is possible of course, to exactly transform from y(¢) to x(6) and vice versa through the above equations; however, the simplified theory to be presented here deals exclusively with the first harmonic content of y(¢) and Eq. (2.8a) provides the basis for checking the results 39 of this theory. In the differential equation (2.7) for y(¢) the non-linear force F(y,¢) is a complicated function of y(¢) and its derivatives. Although the actual expres- sion can be derived for F(y,¢) from the basic equation of motion [6], the present discussion will be restricted to a simple, semi-empirical expression for F(y,¢). This simplification seems justified by the fact that the present discussion deals only with the first harmonic content of the function y(¢). The present theory is therefore semi—empirical in nature and its principle advantage is its simplicity. In the absence of the field bump, the differential equation for the first harmonic component, or the "quasi- first harmonic" component, of y(¢) in the non-linear case is assumed to be: §(¢) + viy<¢> = F(y.¢) 5 Kyecos3(¢-¢O): (2.9) where K and do are constants and the subscript "one” on y(¢) will no longer be needed. The constants K and $0, as will be seen later, specify completely the first harmonic component of the three unstable fixed point orbits associated with the main field. These constants could be determined from the theoretical formulae found elsewhere #0 [5,6]; however, we shall instead use the data from the Fixed Point Code for this purpose. The form of the semi- empirical non-linear force given above is the simplest one possible having the required properties. No cubic (frequency shifting) terms have been included in F(y,¢) assumed above since the quantitative theory of the fixed point orbits shows that such terms are canceled for a strictly isochronous field. As a result, the theory presented here will be applicable only in the isochronous portion of the magnetic field; that is, it will not be applicable near the magnet edge if the three outer stable fixed point orbits move in close to the three unstable fixed point orbits. Consider now the effects of a field bump and the resultant perturbations 6F(y,¢) which should be added to the right side of Eq. (2.9). Since v is close to unity, r a first harmonic perturbation force 6Fl(y,¢) given by: 6Fl(y:¢) = ECOS(¢—¢l), where 6 and $1 are constants, acts as a resonant driving force and can lead to trouble even when 6 is quite small. Such a perturbation arises not only from a field bump harmonic with frequency one,but also from those harmonics of frequency two and four. In addition, a second harmonic 41 gradient perturbation force 6F2(y,¢) given by: 6F2(y,¢) = 6'y(¢) cos 2(¢-¢2), where 6' and 62 are constants, acts as an alternating- gradient force which tends to drive the oscillations into the 2/2 stop-band. Such a perturbation can arise not only from a field bump harmonic of frequency two, but also from those harmonics of frequency one and five. Introduction of these two perturbation forces into Eq. (2.9) gives the following differential equation for 37(2): §(¢) + viy(¢) = Ky2 cos3(¢-¢O) + 6 COS(¢-¢l) + 6'y(¢) cos2(¢—d2). (2.10) It is this equation which forms the basis for the analysis to be given here. A field bump will in general produce perturbations of other frequencies and also of higher order in y(¢); however, for VP close to unity and rela- tively small field bumps the two perturbation terms given above will be the most important. Explicit formulae will be given in section 5 below for the parameters 6, 6', $1, and 62 as functions of the pertinent properties of the field bump. 42 A. Equations for Fixed Point Orbits: Since only the first harmonic of y(¢) is considered here, the equa- tion for y(¢) for a fixed point orbit is: y(¢)=A cos(d+d-¢O)=(A/2)exp[i(d+d-¢O)]+c.c., (2.11) where A and d are constants and 60 is defined in Eq. (2.9). Substituting this expression for y(¢) into Eq. (2.10) i¢ and equating the coefficients of e (Harmonic Balance) gives: eAeia = (KA2/8)e-2ia+(6/2)ei(¢o-¢1) + (6'A/4)e-iae2i(¢o-¢2), (2.12) where (vi - 1) 5 2e and where the equation has been multiplied through by a factor ei¢0. Eq. (2.12) is the general equation for determining the constants A and a for the first harmonic of the fixed point orbits. In the absence of a field bump, Eq. (2.12) reduces to: eAei3a = (K/8)A2. Since G, A and K are positive, the acceptable solutions to this equatmm are: A=0 (E.O.); and for the three un- stable fixed point orbits, d = 0, or t 120° and, A A0 = 8e/K, (2.13) 43 where A0 is the amplitude of the first harmonic of all three unstable fixed point orbits. Thus if xl(6) is the first harmonic component of an unstable fixed point orbit in the absence of a field bump, where: xl(6) = lxllcos(9-6i), then from the discussion connected with Eq. (2.8a) above, it follows that: lel = A0 I _ O _ o 91 — $0, ¢O+120 , or $0 120 . As a result, the constants K and 60 can be determined directly from the amplitude and phase of the first har- monic of one of the fixed point orbits in the absence of a field bump. ‘ B. Approximate Phase Invariant: For a non-fixed point orbit, y(o) is sti11 given by Eq. (2-11) but A and a are considered as slowly varying functions of o. The approximate differential equations for A and a are then given by: dA2 _ _gzg 54> — a a dd _ H and 1 aa' — 3 N as. 44 where H is approximately given by: 2 3 H = 6A — (KA /12) cos 3(a) - 6A cos(d-¢O+¢l) - (6'A2/4) cos 2(d-do+o2). (2.14) Using H it is possible to obtain a qualitative under- standing of the phase plots; successive points in the phase plot should lie on a curve given by H equal to a constant. The phase plots obtained from computer results are in terms of (x, pX) rather than the ”quasi-first harmonic" (y, y); therefore, the comparison between com- puter phase plots and the curves obtained from H above can only be qualitative. Fig. 11 is a plot of H for various A and a values with 6 and 6' equal to zero. The fixed points are located at a = 0, or t 120° at a distance AO from the origin. The direction of the flow lines is obtained from A and d, and from H it is possible to determine whether a fixed point is stable or unstable. This figure should be compared with Figs. 14 and 15 of Chapter III. 2. One-Sector Perturbation. The results obtained in this section are very similar to those obtained by L.J. Laslett and K.R. Symon [9] and L.J. Laslett and S.J. WoIfson [10]. Suppose 6' = 0; then 45 Eq. (2112) becomes: eAeia = (KA2/8)e-21a+6/2 ei<¢o'¢1). (2.15) In general Eq. (2.15) is difficult to solve for A and d. The present discussion will restrict itself entirely to the case of a symmetric perturbation where 61:60 or ¢OTV° When 61:60, the situation is referred to as a ”pure one- corner opening" of the stability triangle; when 61:60 i w, the situation is referred to as "pure two-corner opening" of the stability triangle, for reasons which will become apparent later. Actually, both cases will be covered simultaneously by setting o1 = ¢ and considering 6 as 0 either positive or negative. Set 61 = $0 in Eq. (2.15) and substitute for K the value obtained in Eq. (2.13). Upon equating the real and imaginary components of Eq. (2.15) the result is then: (sin a)(l + i—A cos a) = o (2.15a) O A2 5/2e 5 V = A 003 a ' K; cos 2d. (2.15b) From Eq. (2.15a), either sin a = 0 in which case a = 0 or w; or cos a = —AO/2A. The resultant values of A as a function of y for each of these possible choices for a will be discussed in the succeeding paragraphs. With 46 $1 = $0 and 6' = 0 Eq. (2.14) becomes: H/e= A2 -2A 2/3AO cos 3a - 2yA cosd. Figs. 12a and 12b show how the fixed points and invar- iant curves H change as a function of y. With y > 0, the curves illustrate the "pure one-corner opening" of the stable region; with y < 0, the "pure two-corner Opening" of the stable region is observed. If $1 # $0 or $0 plus a multiple of 60°, then a mixture of the one and two-corner cases is generally observed. Consider first the case of a = 0 in Eq. (2.15b) which gives: y = A(1—A/AO). (2.150) When y = 0, A = 0 or A = A0; A = 0 corresponds to the E.0. located at the origin in (y,y) phase Space Space and A = A0 correSponds to the unstable fixed point a distance AO from the origin along the a = 0 axis. As y increases, the unstable fixed point and E.0. move toward each other along the d = 0 axis. When A =A ”/2 y=A 0/4), the two fixed points coincide and the stable region dis- appears; therefore, to have a stable region, y must be less than AO/4. For y < 0, the fixed point along the a=0 axis moves out along this axis away from the origin. 47 Consider next the case where a: w for which Eq. (2.15b) becomes: 'Y = -A(l+A/AO) (2.15d) Since A and A0 are positive by definition, no fixed points exist along the a = w axis for y > 0. When y is zero, A = 0 represents the only valid solution which is the E.0. located at the origin. As y < 0 decreases, the stable E.0. moves out along the a=w axis. It can be shown from the invariant H that when A=AO/2, (y = -3AO/n) the displaced E.0. changes from a stable fixed point to an unstable fixed point. Consider now the effect of the one—sector bump on + the two unstable fixed points at d = - 120°. In this case, cos d: -AO/2A and the resultant formula for y is: y = -AO(1—A2/A§). (2.15e) When y = 0, A = A0 and a = t 120°; this represents the two fixed points at d = t 120°. For y > 0, A is > A0 and - cos a < 1/2. Therefore as y increases the two un- stable fixed points at a: i120° move toward the a: iVrr/2 axes with increasing A. If y<0, A is (A0 and -cosa> 1/2; therefore as y<0 decreases the two unstable fixed points at a: 1'120° move toward the a=w axis with decreasing A 48 (see Figs. 12a and 12b to see how fixed points move as a function of y). However, cos a has a lower limit of -l which occurs when A = AO/2, (y -3AO/4). It has previously been shown that when y = -3AO/4, the stable displaced E.0. moving out along the G=F axis, becomes un- stable. It can be concluded that for negative 7, the E.0. moves out along the a: w axis and the two unstable fixed points at a: 1‘120° move toward that axis. When y = -3AO/4 the three fixed points coincide and only one unstable fixed point remains; when this occurs, the stable region has disappeared. Therefore y: -3AO/4 is the critical value below which no stable region exists in the case of ”pure two-corner opening". For 7 less than this value, the one unstable fixed point continues to move out along the a = w axis. It is now possible to sum up the results for the one-sector perturbation. AS y > 0 increases, the stable E.0. and the unstable fixed point along the a = 0 axis move toward each other along that axis; at the same time, the two unstable fixed points at d: t 120° move toward the a = t w/2 axis with increasing A. When y equals the critical value Yo given by: Yo = Ao/u’ AO/e. the stable displaced E.0. and the unstable fixed point for which, A 49 coincide and the stable region disappears. As y < 0 decreases, the unstable fixed point along the a = 0 axis moves out along this axis; at the same time, the stable displaced E.O. moves out along the a = W axis and the two unstable fixed points at d = f 120° move toward that axis. When y equals the critical value Yc given by: for which, A = AO/2, the displaced E.0. and the two unstable fixed points coincide on the d = w axis leaving only one unstable fixed point and no stable region; therefore as far as stability is concerned, the critical values for y are - 3AO/4 and AO/4 for two and one-corner opening reSpec- tively. 3. Two-Sector Gradient Perturbation. For the case of the two—sector gradient perturbation, 6:0 and Eq. (2.12) becomes: eAeia = (KA2/8)e-21a + (6'A/4)e—iae21(¢O—¢2). (2.16) Note that unlike the one-sector perturbation case discussed above, A=0 is always a solution; that is the E.0. is never SO displaced; however, this fixed point changes from stable to unstable when [bIlg 46 which correSponds to the 2/2 stop-band. In general Eq. (2.16) is difficult to solve except when d2 = d or dO t w/2. As in the case of the one— o sector perturbation, the two cases correSpond to the "pure one—corner opening" and "pure two-corner opening" of the stable triangular region. For cases other than these two, a mixture of one and two-corner opening is observed. In the present discussion, d2 = dO and 6' is considered either positive or negative to cover both cases. Taking the real and imaginary components of Eq. (2.16) gives: (sin a) (2 cos d + A/AO) = 0 (2.16a) 6'/4e E y' = cos 2 a - A/AO cos a, (2.16b) ‘where K = 8e/AO from Eq. (2.13). These equations then specify the dependence of the parameters A and a on the strength of the perturbation. The movement of the fixed points for a two—sector gradient perturbation is easier to visualize with the aid of a diagram. Therefore Special attention should be paid while reading the following analysis, to Figs. 13a and 13b which indicate the movement of the fixed points 51 as a function of y'. Again as in the case of the one- sector perturbation, the sequence of figures was obtained from the invariant H given by: 3 H/€ = A2 - §%—- cos 3d - 5'A2COS 2d. 0 From Eq. (2.16a), d = 0 or v, or cos a : -A/2AO. For a = 0, Eq. (2.16b) gives: 6'/4e 2 y' = 1 - A/AO (2.16c) The above equation states that as y' > 0 increases the E.0. remains fixed at the origin while the unstable fixed point along the a = 0 axis moves in along this axis toward the E.0. at the origin. When y' = +1, the two fixed points coincide at the origin leaving an unstable fixed point there; at this point, the stable region has disappeared. The effect has been to convert the E.0. from a stable to an unstable fixed point. The particular value y' = +1 correSponds to that value at which the frequency of the radial oscillation about the E.0. reaches the stop-band value vr = 2/2; it is clear then that the conversion of the E.0. from a stable to an unstable fixed point is due to this merging with one of the three unstable fixed point orbits. For a = 0 and y' < 0, the unstable fixed point along the d = 0 axis will move out along this 52 aXis away from the E.0. located at the origin. Consider next the case where d = w; in this case, Eq. (2.16b) becomes: y' = 1 + A/AO. (2.16d) From the above equation, Since A is positive, y' is never less than unity; for y' < + 1 no fixed points exist along the a = 7 axis. As y' increases above unity, a stable fixed point moves out along the d = w axis leaving the unstable E.0. at the origin. When +1< y'< +3, the stable fixed point continues to move out along this axis. An analysis of the invariant H will Show that when y' = +3 and A = 2A0, the stable fixed point becomes an unstable fixed point. For y' > +3 and A > 2AO this unstable fixed point continues to move out along the d = W axis. Consider next the case where cos a = -A/2AO; in this case Eq. (2.16b) becomes: v' = (A/AO)2 - 1. (2.16e) This equation represents the two unstable fixed points at a = t 120° and their motion as a function of y'. When y' = 0 (zero perturbation), A = A0 and a = t 120°. As y' > 0 increases, A > A0 increases and -cos a > 1/2. There- fore, as y' increases the two unstable fixed points at 53 a = t 120° move toward the d = v axis with increasing A; however, cosa has a lower limit which occurs when A = 2A0 and y' = +3. It has previously been shown that when y! =+39(A=2Ao)’ the stable fixed point becomes unstable. It can be concluded that when y' = +3 the stable fixed point moving out along the a = F axis and the two unstable fixed points moving toward the d==w axis coincide on that axis and an unstable fixed point remains. When y' = +3, the stable region has disappeared. For y' > +3 the unstable fixed point continues to move out along the a = w axis. As y' < 0 decreases, - cos a < 1/2, the two unstable fixed points at d = t 120° move toward the a = I v/2 axis with decreasing A. When y'= -1, the two unstable fixed points are at the origin where they merge with the E.0. leaving an unstable fixed point there. It is now possible to sum up the results of a two- sector gradient perturbation. As y' increases from 0 to +1, the E.0. stays fixed at the origin and the unstable fixed point along the d = 0 axis moves toward the E.0. along that axis; when W' = +1, the two fixed points coin- cide leaving an unstable E.0. at the origin and no stable region. At the same time, the two fixed points at a = +120° move toward the a = y axis with increasing A. AS y' in- creases from +1 to +3, a stable fixed point emerges from 54 the origin leaving an unstable E.0. at the origin. For a particular y' = +2, a closed triangular stable region bounded by the three unstable fixed poinusis again formed. For y' > +2 but less than +3, there is then two—corner opening on the opposite side of this triangular region. When y' = +3, the stable fixed point and the two unstable fixed points coincide on the d = 7 axis leaving an unstable fixed point and no stable region. For y'>+3 this unstable fixed point continues to move out along the d = w axis. This sequence of events is illustrated in Fig. 13a. Consider next the case where y' is negative (two- corner opening). As y' decreases from 0 to -l, the unstable fixed point along the a = 0 axis moves out along this axis away from the origin. At the same time, the two unstable fixed points at d = + 1200 move in toward the a = f W/E axes with decreasing A. When y' = -1, the two unstable fixed points originally at a = + 120° and the stable E.0. coincide at the origin leaving an unstable fixed point and no stable region. As was noted previously, Figs.l3a and 13b are plots Showing the movement of the fixed points as a function of y'; the one and two-corner Opening effect of the stable triangle is again illustrated. When y' = + 1, an unstable 55 E.0. is situated at the origin in (y,y) phase space. It is for this value of y' that the two—sector gradient perturbation has driven the linear oscillations into the 2/2 stop—band. As far as stability is concerned, the critical value of y' is then: I'Y' = 'i' 10 4. Simple Criterion for Stability. In the previous sections of this chapter, the effects of a field bump have been analyzed in terms of the be- havior of the fixed point orbits and the characteristics of the phase space diagrams. The problem of stability or instability produced by a field bump can however be attacked from another point of view. The field bump produces a shift in the position of the ”equilibrium orbit” and a change in the linear oscillation frequency about this orbit. If v; is the frequency of the linear oscil- lations about the diSplaced E.0.,then a Simple criterion for stability is that v; be real. Moreover, when v; becomes imaginary (stop—band), the stability region van- ishes. The present section considers then a semi- quantitative calculation of v;. The principle advantage of the results obtained here over those obtained in the 56 previous sections is that no restriction is made on the values d1 and d2 in obtaining these results. The discussion begins with the differential equation for y(d) given by Eq. (2.10). A particular periodic solution of this differential equation is ye(d) which represents the equation for the displaced equilibrium orbit, such that ye(d) is zero in the absence of a bump field. The variable n(d) is defined by: y(d) = yew) + n(). so that n(d) is then the coordinate describing the motion about the displaced equilibrium orbit. The equation for the linear oscillations about the displaced E.0. is there- fore obtained by inserting the above definition into the differential equation for y(d) and maintaining only the linear terms in n(d); the resultant equation for n(d) is then: fi<¢> + vin<¢> = n(¢)[6' cos 2(¢—¢2> + 2Kye(¢) cos 3(d-do)]. The right—hand side of this equation is an alternating- gradient force predominantly of frequency two, and Since vr is close to one, the effect of this force is to drive the oscillation frequency v; into the 2/2 stop-band. 57 The first problem to be dealt with is the evalua- tion of ye(d). It is assumed here that 6 and 6' are sufficiently small such that a perturbation expansion of ye(d) is possible in powers of 6 and 6'. Actually, only the first order effect of the field bump will be considered here. The resultant equation for y(d) is then: N 6/26 COS (¢'¢l)3 c< m A ‘9 II IR R) A C where 26 = (V? - 1) When this y?(¢) is inserted into the differential equation above for n(d) the result is: fi+ Vin = [6' cos 2(d-d2) + K(6/2e) cos(2d-3do+dl)]n, where a term of frequency four has been discarded since its effect is negligible. This differential equation is rewritten in the following form: .0 2 ' n + Vrn = [Acos 2(d—¢2)]n, where dé is irrelevant to the discussion and A is given by: A2 = (5')2 + (46/AO)2 + 26'(46/AO) cos(3dO-dl-2d2L (2.17) where the substitution K =Eie/AO has been made in accor- dance with the results of Eq. (2.13). Since v; and Vr 58 are both close to unity, an appropriate equation for the value of v; is then: 2 2 2 (vg-l) = (vr-l) -|A/4I , which is correct to second order in A. Since the above calculation gives A correct to first order in 6 and 6', this result is then correct to second order in both 6 and 6'. According to this equation, v; will be real and a stability region will exist when |A|< 4e; moreover, v; will be imaginary (in the stop—band) and no stability region will exist when IA‘> 4e. Inserting the above value for A2, the simple criterion for sta- bility is then given by: (6:)2 + (As/A0)2 +26'(46/AO) cos(3dO-dl-2d2) < (4e)2, (2.18) and naturally, the more strongly this inequality holds, the larger the available stability region will be. If 6' = 0, the above inequality reduces to simply: I6/2el 5 lyl < 1/2 AO. (2.182) In section 2 above it was shown that conditions for stability with one-corner opening and two-corner Opening are respectively: 59 [WI 3 1/4 Ao’ and . lyl g 3/4 A0, so that the results given by Eq. (2.18a) above is Just the average of these two extreme situations. Now if 6 = 0, then the condition for stability given by Eq. (2.18) is: lfl/4el E Iy'l < 1, (2.18b) which agrees with the results obtained in section 3 above for both one and two-corner openings. The above condition (2.18) for stability shows that when both a one-sector and two-sector gradient perturbation are present, their effects add "vectorially". Thus if both perturbations are so phased as to produce one-corner opening at the same corner (d1 = do, d2 = do), or two-corner Opening of the same two corners (d1 = dO + n, d2 = dO + w/2), then their individual effects reinforce each other such that the stability condition (2.18) becomes: 6' + (46/AO) < 42. (2.18c) On the other hand, if one of these perturbations is phased to produce one-corner opening at a specific cor- ner while the other is phased to produce two-corner 60 opening of the opposite two corners, then their indi- vidual effects tend to cancel each other such that the stability condition becomes: 6' — 46/AO| < 46. (2.18d) 5. Evaluation of Perturbation Parameters for a Given Bump Field. The theory presented above contains certain parame- ters (6, 6', d1, d2) characterizing the effect of the field bump. In the present section, a reasonably accurate procedure will be given for evaluating these parameters in terms of the properties of the field bump. Unlike the case of the non-linear force F(ywd), for which a semi- empirical representation was employed, the force 6F(yvd) in Eq. (2.1) associated with the field bump will be evaluated in a realistic fashion. Assume that b(r,6) is the magnetic field associated with the field bump; the expression for 6K(r,6) is then: 6K(r,6) = -(eR/p) rb(r,6), (2.19) where R is the mean radius of the E.0. and p is the momentum of the particle; and p E eRBO(R). Introducing the variable x(6) through the defining equation: 61 r(e) = re(6) + x(6), where re(6) is the equation for the E.0., then the equation for 6K in terms of x(9), expanded to first order in this variable, is: 6K(x,6) = _[rbB:,: ]e - x[%—- Tar— rb(r,6)]e, (2.19a) o where the subscript "e” implies that the quantity in the bracket is to be evaluated at r = re (the E.0.). The equation for the E.0. is of the following form: re(9) = R[1 + §(6)]: where §(9) is the periodic function, having zero average, which Specifies deviation of this orbit from a circle of radius R. The expression for 6K(x,6) is now expanded to first order in €(6) with the following result: 6K(x,6) = —R[b(R,6) + €(9) B'(R,e)] - X[B'(R,9) + 6(9)B"(R,9)], where the dimensionless quantities B, 6', and 6” are defined by the following equations fi(R,9) = b(R,9)/BO(R) B'(R.6) = Ejfl 53g [rb(r,e) 62 2 2"(R:9) = 30%,) 3:2 [rb(r,6)]. The next step is to transform from the variable x(6) to the variable x(e) using Eq. (2.1) which is: X(9) = D(9)X(9). II? where, D(e) 1 + 1/2 e(e), which is correct to the 1st order in the quantity 5(9). The resultant expression for 6K(x,6) is then: 6K(x,e) = -R[B +ed'] - X(6) [2' + 6(2" + 1/22')], which is then correct to first order in 5(6). The next step is to carry out the transformation from the variables (X,6) to the variables y and d through the defining equations: X(9) = 0(9) y(d) and, 6F(y,¢) = C3[6K(X,9)], where the functions 0(9) and d are defined through Eqs. (2.4) and (2.5). The quantity 6F(y,d) is then the force term due to the bump field which must be inserted on the right-hand side of the differential equation (2.7). The functions w(6) and d(9) are defined as follows: 0(9) = l + w(6); d(e) = 9 + w(9), (2.20) 63 and as will be shown later, the magnitude of w(9) and w(9) are small compared to unity. Using these relations, the resultant expression for 6F(y,d) is then given by: 6F(y,d) = -:‘R[B-W(dB/d9) + 3wt3 +£3'] - be'-v(d27d9) + 4WB' + 6(2" + 1/22')], where this expression has been evaluated correct to first order only in the quantities w, w and 2. Note that all quantities in this experssion, which were formerly functions of 9 are now functions of the new variable d. In order to cast the form of 6F(y,d) given above into an expression more suitable to calculations, it is necessary to develop suitable expressions for the func- tions 6(9), w(6) and W(6). Assume that the main magnetic field has the form: B(r,6) = Bo(r) + Bl(r) cos 3(6—Cl(r)), as discussed in Chapter I; this is actually not a very restrictive assumption since the higher harmonics in the magnetic field play an insignificant role in these phenomena. The equation for 5(6) can then be written: 5(9) = 61 COS 3(9-Cl): (2'21) 64 where C1 = §l(R) and £1 is given by: 61 g f/(8-22), (2.22) where f = B1(R)/BO(R and p is the momentum in mc units. 9 ) To evaluate w( ) and w(9), it is necessary to have an expression for G(6) appearing in Eq. (2.5); actually, only the oscillatory (zero-average) part of this function is needed here. If 0(9) is defined as G(9) = G0 + g(9) where g(9) averages to zero, then the value of g(9) cor- rect to first-order in the flutter is given by: 2(9) = [(R/BO)(dBl/dR)+(3/2)f+(3/2)§l(l+3p2+2pq)]cos3(9-Cl) + 3f(R6Cl/dR) sin 3(e—ql), (2.23) where all functions of r are evaluated at R. Let g1 and A be defined by the equation: g(6) = gl Sin 3(6—A), (2.24) where g1 and A may then be calculated from the above equation for g(9). The differential equations for w(6) and d(9) follow from those of C(9) and d given in Eqs. (2.4) and (2.5); when these are evaluated correct to first order in the flutter field, the result is: 65 ’w' + 4v§w = -g(e) d = -2w(9). Solutions of these equations are then: w(9) = wl sin 3(e—x) v(e) =(2/3)chos 3(e—x), where, wl = gl/(9swvi). (2.25) Consider now the representation of the field bump in a Fourier series as follows: b(r,9) = an(r) cos n(9-6n), (2.26) where bn(r) is the amplitude of the nth harmonic and an is its phase. In the present discussion, an is treated as a constant; that is, the bump is considered to have negligible "spiral”. If it should be desirable to consider bumps having considerable spiral, then the subsequent analysis should be modified accordingly. The bump force 6F(y,d) is likewise resolved into components as follows: 6F(y,d) = 25Fn(y,¢), (2.27) where 6Fn is the perturbation produced by the nth harmonic 66 of the bump field. The following set of parameters are now defined in connection with Eq. (2.27) above which characterize the pertinent properties of the bump field: B = bn(R)/BO(R) n 2;, =§jfl§fi [rbn(r)] R 32 1’1 = BJR) 3R2 [I’bn(l”)]. With these definitions, the expression for 6Fn becomes: 5Fn(y,¢) = -R[Bn cosn(d-9n) + (2n/3)wlfo"n sin n(d-6n)cos3(d-A) + 3wlf5n cos n(d—6n) Sin 3(d-A) + 61o; cos n(o-9n) cos 3(d-Cl)] - y(o)[og_ cos n(¢-en) + (2 /3)W18A, sin n(d-9n) cos 3(d-A) + 4wlBAH cos n(d-9n) sin 3(d-A) + 61(sg_ + 1/225) cos n(o-en) cos 3(9-Cl)], where the quantities WI, 51’ A, and Cl have been defined above in the Eqs. (2.25), (2.22), (2.24) and (2.21). It 67 is clear from this formula, that a one-sector perturba- tion of the form considered in the previous sections can arise not only from the n=l component of the bump field but also from n=2 and n=4 components as a result of the coupling of these harmonics with the three-sector struc- ture of the Floquet functions and the E.0. In addition, it is clear that a two-sector gradient perturbation can arise not only from the n=2 component of the bump field but also from the n=1 and n=5 components. Note that higher harmonics in the bump field can give rise to such perturbations only through interference with the 6, 9,... frequency components of the E.0. and alternating—gradient motion. For the n=l component of the field bump, the following expression for 6Fl(y,d) is obtained: 6Fl(y,¢) = -Rol cos (o-91)—[y(5/3)w18i Sin(2¢-3A+91) +1/251(31 + l/2Bi) cos(2d—3c1+el)], where all terms not having the desired periodicity have been dropped. In the same way the expressions for 6F2(y,d), 6F4(y,d) and 6F5(y,d) are: 68 6F2(y,o) -R[(5/6)wlsg sin(s-3x+2cg> + l/2gleé cos(d-3Cl+292)] — y[6§ COS 2(d-92)] 6F4(y,c) = -R[(-l/6)wls4 sin(¢-464+3A) + l/2ngh cos(d-464+3Cl)] 6F5(y,¢) = y[(l/3)w16é sin(2¢-595+3A) - 1/2&1(og+1/2og) cos(2o-565+3C1)]. In the theory presented in the previous sections, 6F(y,d) was represented as follows: 6F(y,d) = 6cos(d-dl) + 6'y cos 2(d—d2). The parameters 6, d1, 6', and d2 can now be directly correlated with the bump field characteristics as eval- uated above. For convenience, table 2-1 and table 2-2 are given showing the contribution of each harmonic in the bump field to these parameters. If the bump field contains more than one harmonic, then the entries in a given column should be added algebraically to get the net result in each case. It is evident from the results in tables 2-1 and 69 2-2 that a one-sector field bump (n=l) will produce not only a one-sector perturbation but also a two—sector gradient perturbation; moreover, this is true even if bl(r) is constant, that is a flat bump. In like manner, a two-sector bump (n=2) produces not only a two—sector gradient perturbation but also a one—sector perturbation; moreover, this is true even for a pure two-sector gradient bump where b2(R) E 0, dbe/dR % 0. However, as will be seen later, for a flat one-sector bump, the resultant two-sector gradient perturbation is negligibly small; likewise, for a two-sector gradient field bump the result- ant one—sector perturbation is smaller, though not neg— ligible. 70 Zommoflfiflmomimvsom -Amomémvaoomaaam Zommsmvmoo Imam +mov$m -Amsmémvcam warm. m Aao-HUchHmAmmm+=anwm -Agmnaovmoommfism I Afio-fiwmvmooAfiom +mmvawm -AKmIHovsam «mfiz_m I H mom can we- mom moo on. m mom cam .o mom moo .o c .COHmeLSdeQ podfiomhw 90poom I025. mflp ..HOQ mhmmewth COHDOQMFHQQEH USN Ufimflh QESQ. mflp CTQBUTQ. COHuswfimcapHOO ”WIN Emu/5H. Aaom-sosvoam enawmw -Asosugmvnoo smfizmm ANQNIHumVCHm wmfiwmm IAKmImmmvmoo mmfizmm I Ho can Ham- Afiwm-sosvmoo quamm_m -Asos- 0; d1 = 91, b1 < O. (3.4) cul— From Eq. (2.180), the stability condition gives the critical value of 6 for ”pure one—corner Opening” as: oC/AO = 5/2. (3.5) For ”pure two—corner Opening", the critical value is three times as great and is given as: °c/AO = 3e/2. (3.6) 89 If the field bump is phased so that a mixture of one and two—corner opening is obtained, the critical value of 6 will lie between these two limits. Combining Eqs. (3.3) and (3.5) gives the critical value of bl for which the stable region disappears in the case of ”pure one-corner Opening" as: = eAOBO/2R. (3.7) Below are the critical values of bl for both fields for both one and two-corner Opening. One-corner Opening: B26.29A: (b +.0117 159 gauss B26A1: (b +.0027 38.8 gauss Two—corner Opening: B26.29A: (b +.O351 = 478 gauss “ B26A1: (b +.0081 116 gauss. F-J V O H It is evident from Eq. (3.7) that the critical bump strength is much greater for B26.29A than for B26A1 because the value of A0 is substantially higher for B26.29A. This result will be generally true over the entire machine since the stability limits for the weak- Spiral field B26.29A are much greater than those for the moderate-spiral field B26A1. 90 B. Discussion of the Phase Plots: The field bumps to be investigated in this section have been Specialized to the case where 81 = O and b1 > 0. Since 6 is positive by definition, the following results are obtained from the formulae of tables 3-4 and 3-5 for the case where 91 = 0 and b1 > 0. B26.29A, B26Alzdl = 180° (3.8) B26Al: o = .1992bl (3.9a) B26.29A:6= .l954bl. (3.9b) It was stated in the previous section that if dl=167.5°, "pure one-corner opening'l is obtained at U1 for field B26Al. Since d1 = 180° when 6 = 0 and b1 > O, a 1 mixture of one and two-corner opening is expected at U1 and U3 (see table 3-2) with the one-corner opening at Ul dominating over the two-corner opening at U1 and U3. Figs. 16a, 16b and 16c are phase plots for B26A1 with 81 = O and bl = +.OOl, +.OO2 and +.OO4 respectively. On all of the phase plots to be presented in this chapter, the new location of the four fixed points will be desig- nated by ”circles" (O) and the position of the four fixed points before any bump field was added will be designated by "x's". Figs. 16a, 16b and 160 Show that as bl increases, 91 the two fixed points U1 and E0 move toward each other. As evidenced by Figs. 16b and 160, the two fixed points U1 and E0 coincide and the stable region disappears at a value of +002 < b < +.OO4. According to Eq. (3.7), 1 the critical value of bl which corresponds to the dis- appearance of the stable region for "pure one-corner Opening" is given by: (b = eAOBO/2R = +.OO27 = 38.8 gauss. 1)c Since the case being considered here is not "pure one- corner Opening", the critical value of bl for which the stable region disappears should be somewhat greater than b = +.OO27 but substantially less than three times this 1 value (i.e. b1 = +.0081). Using the results of Figs. 14, 16a and 16b, a semi-quantitative extrapolation based on the theory of Chapter II gives the value of bl = +.OO3l as the critical value for which the stable region Should disappear in this case which is in agreement with the results obtained above. At the same time as U1 and E0 move toward each other, the two fixed points U2 and U3 move away from the initial position of E0 and qualitatively move toward the a = + 90° axes as measured from the axis Joining the initial positions of U1 and E0. A comparison of these 92 results with Fig. 12a Shows the qualitative similarity between the computer results and theoretical predictions; that is, the theory predicted that for ”pure one-corner opening” U1 and E0 would move toward each other along the d = 0 axis and at the same time, the two fixed points at d 2 + 120° move toward the a = + 90° axes. These same general features are observed in Figs. 16a, 16b and 16c. Since Figs. 16a, 16b and 16c represent a mixture of one and two-corner opening at U1 and U3, qualitatively the fixed point U3 experiences a force tending to move U3 out toward the d = - 90° axis and a weaker force tending to move U3 toward E0; therefore, it is expected and observed that U2 moves out toward the a = + 90° axis faster than U3 moves out toward the a = -90° axis. With the addition of the bump field, the stable region is determined by U1 and E0. As evidenced in Figs. 16a, 16b and 16c, as b increases, U1 and E0 move toward 1 each other and the stable region becomes smaller until it disappears for some critical value of b1. In Figs. 16a, 16b and 16c, the curves labeled @ are indicative of the nearly "pure one-corner opening” at U1; the curve labeled (3) shows the presence of the small ”two-corner Opening" at U1 and U3. 93 Consider next the effects of the field bump on the field B26.29A. Figs. 17a and 17b are phase plots for B26.29A with b = +.OO4 and +.O2 respectively. From the 1 figures, the stable region disappears for some bl be- tween these values. According to Eq. (3.5), the critical value of b for ”pure one-corner opening" is given by: 1 (b = eAOBO/2R = + .0117 = 159 gauss. 1)c For ”pure two-corner opening", the critical bump strength is three times this value (i.e. bl = +.O35l). A semi- quantitative linear interpolation from Figs. 15 and 17a gives bl = +.Ol3 as the critical value for which the stable region of B26.29A will disappear assuming "pure one-corner Opening”. It is seen that this is in agreement with the result obtained from Eq. (3.5). Since dl is again 180° and since d1 = l70.2° cor- responds here to ”pure one—corner opening” at U1 for field B26.29A, the fixed points of B26.29A undergo almost the same relative displacement as the fixed points of B26Al. As a result, the discussion given above for B26A1 serves equally well to explain the behavior Of the fixed points for B26.29A. Again the curves labeled @ indicate the one-corner opening effect at U1 and curves labeled @ indicate the small two-corner Opening effect at U1 and U3. 94 C. Detailed Comparison of Computer Results with the Theory: As was mentioned previously, a direct comparison of the computer results and the theoretical predictions will be made only for field B26Al. In section 2 of Chapter II, formulae were obtained governing the motion of the fixed points for a one-sector perturbation for the "pure one—corner opening” and ”pure two-corner open— ing” situations. In the present section, the results will be presented of computer studies aimed at checking in detail the accuracy of these formulae. From tables 3-4 and 3-5, if 01 = -l2.5°, d1 = 167.5° = do. According to the theory, when d1 = do, "pure one- corner opening” is obtained at U1 when y is positive and ”pure two—corner Opening" is obtained at U2 and U3 when 7 is negative where y 2 5/26. To check the theory of Chapter II, the flat one-sector bump field given by: b(r,0) = b cos (a+12.5°), 1 was used in a new set of orbit computations. With this choice, ”pure one~corner opening” should be Observed at U1 when bl is positive, and ”pure two-corner opening” should be Observed at U2 and U3 when bl is negative. For each value of b1 the Fixed Point Code was used to find all four of the fixed points for this field at 95 8:0. Each of the fixed points was then run through the General Orbit Code for one revolution to Obtain the fixed point orbit at all 489 values; a Fourier analysis was then made of each fixed point orbit. From the Fourier Analysis Code [23], the amplitude lel and the phase Oi of the first harmonic, as defined in Eq. (3.2a) and (3.2b) were obtained for each fixed point. Recall that at the beginning of this chapter it was pointed out the A = |x1| and q = dl — 8i where A and a are the parameters Specify- ing the fixed point orbit as defined in the theory of Chapter II. In table 3—6 the resultant values of the amplitude A and the phase a are given for all of the fixed points for each value of b1° In addition, the entry in parenthesis in table 3-6 was obtained from the theoretical formulae of Chapter II to give some idea of the difference between the theoretical predictions and the computer results at this value of bl where the greatest difference occurs. According to the theory, with bl positive and d1 = 167.5° "pure one-corner Opening” is Obtained at U1; moreover, both E0 and U1 should have a = O in this case. The theory also predicts that the amplitude of U2 and U3 will beuequaliand that U2 and U3 will have a equal in mag- nitude but opposite in Sign for a given bl; furthermore, as b1 > 0 increases, the value of a for U2 and U3 Should ll. Ill-I'll 'I {I‘ll-III 96 .6.mmH 6m6H6. o:.+ mmms6. 6666.- Aom.6sa-v Amosao.v Aom.osav Amosao. Aoov Aomoso.v Aoo.omav Aoomao.v Aosoo.-v om.6mH- mmmfio. om.6SH mmsfio. om.+ mmfiso. oe.mmfi 6HmH6. 6566.- om.6sfi- owmfio. 0S.mma mmmao. om.+ H6626. as.HwH Hmfiflo. 6666.- o6.6m7 mammo. om.mmH mmmmo. om.+ ssmmo. om.6wH smm66. 6:66.- o66$- msmmo. 06.6ma mammo. o6 msmmo. - 6 6 06.6HH- mmmmo. 0H.mHH mammo. o6 msmmo. 0H.- m6H66. 6666.+ os.wHH- :smmo. om.mHH m66m6. o6 6mmm6. 0H.- mmm66. 6H66.+ om.SHH- 6m6m6. om.SHH mH6m6. o6 oasmo. 0H.- mmsoo. 2H66.+ o6.SHH- H6Hm6. 0H.SHH Hm6m6. 0H.- smamo. 0H.- H6666. 6m66.+ om.6HH- msfimo. 03.6HH m6Hm6. 0H.- mmsfi6. 0H.- m6HH6. 6m66.+ o a o a o < o a an m6 «6 H6 6m .mcprOme QESQ mSOHLm> pom mD 6cm «ND «HD «om 90% 8 Gem < 90 mosam> "mum mqm03 U1, E0: n=o; y = A(l-A/AO) (3.10) 99 U2, U3: cos a = -AO/2A, (A>AO)§ v = -AO(l — A2/Ai ) (3.11) bl < 0, U1: d = 0; 7 = A(1 — A/AO) (3.12) E0: d = v; 7 = -A(l + A/AO) (3.13) U2, U3: cos a = — AO/2A, (A < A0); v = -Ao(i-22/A§ ) (3.14) Fig. 18 is a plot of y versus A; Fig. 19 is a plot of y versus a. The solid lines in each figure represent plots of the theoretical equations above. The fixed point which belongs with a certain section of the curve is indicated on the figure and the direction in which the fixed point is moving as a function of the bump strength is also shown by an arrow. The points on the figures represent the computer results obtained from table 3—6. Even though no phase plots were obtained for these cases, a comparison of the theory of the previous chapter with table 3—6 and Figs. 18 and 19 gives reasonable proof that "pure oneecorner opening" was obtained at Ul for positive b and nearly "pure two-corner opening” was obtained at 1 U2 and U3 for negative bl‘ lOO Figs. 18 and 19 indicate that some of the computer points do not fall exactly on the theoretical curves for bl < O; moreover, the more negative bl’ the more pro- nounced the difference between the theoretical curves and the computer results becomes. It is evident from Figs. 18 and 19 that for values of b near the stability 1 limit there is a shift from "pure two-corner Opening" at U2 and U3 to a mixture of two-corner opening at U2 and U3 and one-corner opening at U3. It should be emphasized however that the deviation from "pure two-corner opening" is very small. According to the theory, for "pure two- corner Opening” E0, U2 and U3 should meet on the a = 180° axis simultaneously when bl = -.OO81. Figs. 18 and 19 and the results of table 3-6 indicate that E0 and U3 merge first leaving only one fixed point U2 which will presumably move out toward the d = 180° axis. In Fig. 18, the dashed line Joins the computer results of table 3-6 for negative values of b from the figure, it can be assumed that E0 13 and U3 merge at a value of bl somewhat greater than -.OO8l; since ”pure two-corner Opening” is the extreme magnitude of b for destruction of the stable region, it is therefore 1 expected that U3 and E0 will meet for some value of bl less than this critical value. Since the theory of Chapter II is only semi-quanti- lOl tative, perfect agreement between the theory and computer results is not expected. Furthermore, it was previously pointed out that the flat one-sector field bump gives rise to a two-sector gradient perturbation which is about 1% in effect of the one—sector perturbation for field B26Al. Since 6' £ 0, it is expected that neglecting the two—sector gradient perturbation will have some effect on the theoretical curves. 3. Flat Two and Four-Sector Bump Field. A. General Considerations: The next two bump fields investigated in this report were the flat two-sector bump field and flat four-sector bump field given by: b(r,e) = bn cos n(9-On), (3.15) where n = 2 or 4, an is a constant and bn is constant over the radius of the machine. From the formulae of tables 2-1, 2-2 and the results of table 3-3, it is possible to calculate the parameters 6, 6', dl and d2 in terms of the two constants bn and 9n for the flat two and four-sector bump field. In the tables immediately below, the parameters 6, 6', dl and d2 have been evaluated in terms of these two constants for the n=2 and the n=4 case for both fields under investigation in this report. 102 TABLE 3-7: Evaluation of perturbation parameters in terms of the bump strength and phase angle for fields B26.29A and B26Al for the two- sector flat bump. n=2 B26.29A B26Al o (.0345)b2 (.0286)b2 dl 182.1°-282 84.3°-282 6' b2/1.OO79 b2/l.0187 d2 90° + 82 90° + 82 TABLE 3-8: Evaluation Of perturbation parameters in terms of the bump strength and phase angle for fields B26.29A and B26A1 for the four— sector flat bump. n=4 B26.29A B26Al o (.0109)b4 (.00692)b4 O 0 d1 494+17l.4 484+289.3 Note that 6 is independent of 9n for both fields. a) General Considerations For The Two—Sector Flat Bump Field: According to the theory of Chapter II, section 4, the quantity I6'/(46/AO)| gives an estimate of the relative effect on stability of the two-sector gradient perturbation in comparison to the one-sector perturbation. Immediately below, the quantity [6'/(46/AO)| 103 has been evaluated for fields B26.29A and B26Al using the flat two-sector bump field parameters given above. TABLE 3—9: Evaluation of the quantity [6'/(46/AO)| for fields B26.29A and B26Al for the two—sector flat bump. n=2 B26.29A B26A1 [6'/(4640” .594 .244 Unlike the n=1 case discussed in the previous section, the relative effect of the two—sector gradient perturba- tion on stability is no longer negligible in comparison to the one—sector perturbation; as a result, when dis- cussing the phase plots for n=2, the effect on stability Of the two-sector gradient perturbation cannot be neglected. The detailed theory given in Chapter II governing the motion of the fixed points as a function of the bump strength was limited to the cases where either 6 or 6' were identically equal to zero. As shown above, the two- sector flat bump introduces into the equations of motion both a one-sector perturbation and a two—sector gradient perturbation neither of which is negligible in compari- son to the other; as a result, the detailed theory on the motion of the fixed points given in sections 2 and 3 in the previous chapter is not quantitatively applicable 104 here. It was shown in Chapter II, section 4, that when both a one-sector perturbation and two—sector gradient pertur- bation are present the Simple criterion for stability is given by: 22 ; (6')2+(46/AO)2 + 26'(46/AO)cos(3dO-dl-2d2)<(4e)2. (3.16) It must be remembered however, that the above equation gives only an approximate value for the critical bump strength. If the results of table 3-7 are substituted into Eq. (3.16), it turns out that the function a as defined in section 4, Chapter II is independent of 82. Below is the evaluation of A in terms of b2 for fields B26.29A and B26A1. B26.29A: A = (.974) b2 (3.17) B26Al : A = (3.66) b2. (3.18) The critical value of b2 may then be found by setting A = 4e. If this is done, the critical value of b2 for fields B26.29A and B26Al turns out to be: B26.29A: (b = + .227 = 3100 gauss 2)c B26A1 : (b + .0419 = 571 gauss. 2)c 105 A comparison of the critical values for the flat two- sector field bump with those determined in the previous section for the flat one-sector field bump shows the critical values for the flat two-sector bump to be 15 or 20 times greater than the critical values for the flat one—sector bump. b) General Considerations for the Flat Four- Sector Field Bump: The flat four-sector bump field introduces only a one-sector perturbation into the equa- tions of motion; as a result, a quantitative comparison can be made between the theory of Chapter II, section 2 and the computer results. AS in the case of the flat one—sector field bump, Eq. (3.5) gives the critical value of b4 for ”pure one-corner opening" and Eq. (3.6) may be used to determine the critical value of b4 for "pure two-corner Opening". Using the results in table 3-8, the critical value of b4 obtained for "pure one-corner opening" for fields B26.29A and B26A1 is given by: B26.29A: (b4)C = .209 = 2850 gauss B26Al : (b4)c = .0788 = 1070 gauss. The critical value of b4 for "pure two-corner opening" is three times the critical value for "pure one—corner Opening”. For a mixture of one and two-corner opening, the critical value of b4 lies somewhere between these 106 two extreme values. A comparison of the critical val- ues for the flat four-sector bump with those of the flat one-sector bump indicates that the critical value for b4 is 18 times larger than (bl)C for field B26.29A and 27 times larger for field B26A1. B. Discussion of the Phase Plots: a) Flat Two-Sector Bump Field: All the phase plots to be presented in this section are for a flat two- sector bump field whichlunmabeen Specialized to the case where 02 = O. Substitution of this value of 02 into the formulae of table 3-7 gives the following results for B26.29A and B26A1. n=2 B26.29A B26Al d1 182.1° 84.3° o2 90° 90° The effect on the fixed points of each perturbation will be presented as if the other perturbation were not pre- sent and the net effect of the flat two-sector bump field will be determined from the phase plots. In the following phase plots, the new location of the four fixed points for the field being used will be designated by ”circles" (O) and the position of the four fixed points before any bump field was added will be designated by ”x'sfl 107 According to table 3-2, if d1 = 83.4° for field B26Al, a mixture of one and two-corner Opening should be observed with the two-corner opening U1 and U2 dominating over the one-corner opening at U2. This would be the situation if the two-sector perturbation were not present. According to this same table, with d2 = 90°, a mixture of two-corner Opening at U2 and U3 and one-corner opening at U3 would be observed for field B26Al were the one-sector perturbation not pre- sent. Figs. 20a and 20b are static phase plots for field B26A1 with b2 = +.OO4 and +.O2O reSpectively. From the two figures, the effect of the flat two-sector bump field appears to be nearly "pure one-corner Opening” at U2 which is not inconsistent with the considerations immediately above. The figures indicate that as b2 increases, E0 and U2 move toward each other and the stable region which is now determined by E0 and U2 is reduced in size; at the same time, U3 remains almost stationary and U1 moves away from E0. When b2 = +.02O the stable region is almost non-existent indicating that a further increase in b2 would result in the disap- pearance of the stable region. In Figs. 20a and 20b, the curves labeled @ indicate the one-corner Opening at U2; none of the curves shown indicate any two-corner opening. 108 According to Eq. (3.16), the critical value of b2 which will cause the stable region of B26A1 to disappear is: — +.O419 = 571 gauss; this value is almost twice as large as the value Obtained from the phase plots. If the two-sector gradient per- turbation is neglected, which is not too unrealistic for B26A1 (see table 3-9), the critical value of b2 according to Eqs. (3.5) and (3.6) lies between +.019O and +.O57l. It is evident from Figs. 20a and 20b that the stable region disappears when b2 is Slightly larger than +.02O which is a little larger than the critical value for ”pure one-corner opening" Obtained from Eq. (3.4). The critical value of b calculated from Eq. (3.16) is 2 approximately twice as large as it should be. Since the two perturbations produce nearly "pure one-corner opening" at U2, it is expected that the critical value of b2 will be less than the critical value determined from Eq. (3.16). Consider next the case of B26.29A; for this field, d1 = l82.l° and d2 = 90° as obtained from table 3-7. According to table 3—2, if d1 = l82.1° nearly "pure one- corner opening” would be Observed at U1 along with a small amount of two—corner opening at U1 and U3 if the two-sector gradient perturbation were not present; with 109 d2 = 90°, a mixture of two-corner Opening at U2 and U3 and one-corner opening at U3 should be observed if the one—sector perturbation were not present. Figs. 21a and 21b are static phase plots for B26.29A with b2 = +.O2O and +.lOO respectively. The two figures indicate that the flat two-sector bump field produces nearly ”pure one- corner opening” at U3 along with some two—corner Opening at U2 and U3; this result is not apparently consistent with the values given above in the notion of the vector addition of the two effects. In Figs. 21a and 21b, the curves labeled (:) indicate the one-corner opening at U3 and the curves labeled (:) indicate the small amount of two-corner opening at U2 and U3. AS b2 increases, E0 and U3 move toward each other and the stable region shrinks; at the same time, Ul moves away from E0 and U2 moves toward E0. From these figures, it is estimated that for some value of b2 Slightly greater than +.1OO the stable region disappears. According to Eq. (3.16), the critical value of b which will cause the stable 2 region to disappear is: = +.227 = 3100 gauss, which is again about twice the actual value observed. From the data of table 3—9, it is clear that both per— 110 turbations must play a significant role in determining stability and therefore it is not realistic to compute the critical bump strength assuming one or the other is negligible; as noted above, since the result of the two perturbations is nearly "pure one—corner Opening” it is expected that the critical bump strength will be less than the critical value determined from Eq. (3.16). b) Flat Four-Sector Bump Field: As in the n=1 and n=2 cases, the phase plots to be presented for the flat four-sector bump field have been specialized to the case where 84 = O. Substitution of this value of 84 into table 3-8 gives the following results for B26.29A and B26A1. n=4 B26.29A B26Al o (.0109)b4 (.00692)b4 31 171.4° 289.3° Consider first the case of B26Al; since dl=289.3° and from table 3-2, dO for U3 is 287.5°, it is expected that nearly "pure one-corner opening" Should be observed at U3. Fig. 22 is a static phase plot for B26A1 with b4 = +.040. The figure clearly Shows that nearly "pure one-corner opening” is indeed obtained at U3. In Fig. 22, the curves labeled (:) indicate the one-corner opening 111 at U3. AS b4 increases, U3 and E0 move toward each other very nearly along the axis which joins the initial position of U3 and E0. At the same time, U1 and U2 move away from the initial position of E0 with U2 moving somewhat faster than U1. As b4 increases, E0 and U3 move toward each other and the stable region shrinks. According to Eq. (3.5), for the case of ”pure one-corner Opening", the stable region will disappear when b4 = +.O788. It is possible to determine an approx- imate critical value of b4 from Fig. 22 and Eq. (2.15b). Using the displacement of U3 from its initial position in Eq. (2.15b) gives the extrapolated critical value of b4 as +.08 which agrees quite well with the critical. value of b4 determined from the theory. Consider next the field B26.29A; Figs. 23a and 23b are static phase plots for B26.29A with b4 = +.1O and +.3O reSpectively. It was pointed out in the table above that d1 = 171.4° for field B26.29A. According to table 3-2, if d1 = 170.2°, "pure one—corner opening" is Obtained at U1. It is therefore expected that nearly "pure one- corner opening" will be observed at U1 with a very small amount of two-corner Opening at U1 and U3. Figs. 23a and 23b Show that this is indeed the observed effect. In Figs. 23a and 23b, the curves labeled (:) indicate the 112 predominant one-corner opening at U1 and the curves labeled @ indicate the small two-corner Opening at U1 and U3. It is seen that as b4 increases, U1 and E0 move toward each other along the axis joining the initial position of U1 and E0. As b4 increases, U2 and U3 move away from E0 toward the d = + 90° as measured from the axis joining the initial position of U1 and E0; moreover, the relative displacement of U2 and U3 from their initial positions in the unbumped field is nearly equal as expected from the theory in Chapter II for ”pure one—corner opening". According to Eq. (3.5 ), the stable region should disappear when b4 = +.209. By extrapolation from the results shown in Fig. 23a, an approximate critical value of b4 may be Obtained. Using the displacement of E0 from its initial position in the unbumped field in Eq. (2.15b) gives an approximate critical value of b4 as +.20 which is in good agreement with the critical value of b4 determined from the theory. 4. Two-Sector Gradient Bump Field. A. General Considerations: In this section, the effect on stability of a two-sector gradient bump field will be investigated. In keeping with the definition 113 given at the beginning of this chapter, the two-sector gradient bump field is given by: b(r,8) = a(r-R) cos 2(8-82), (3.19) where'a"and 82 are constants and R is the mean radius of the equilibrium orbit. The values of a are in c.u. and a conversion factor is provided by the equation: a = 1 c.u. = 150.4 gauss/inch. The value of R is given in table 3-3 for fields B26.29A and B26Al. From the formulae in tables 2—1 and 2-2 and the results in table 3—3, the parameters 6, 6', dl and d2 may be determined directly in terms of the two con- stants a and 02. Assuming that 6 and 6' are positive, the following results are obtained for positive and neg- ative a for fields B26.29A and B26A1. TABLE 3-10: Evaluation Of the perturbation parameters for fields B26.29A and B26Al for the two- sector gradient perturbation. 114 n=2 B26.29A B26A1 6 (.000981)|a| (.000485)|a| dl 2OO.1°-292, a > O; 25.8° - 202, a > 0; 20.l° - 292, a < 0 205.80 — 292, a < 0 6 .1954|a| .1992|a| d2 90°+e2, a > 0; 92, a.< 0 90°+e2, a > 0; 92, a < 0 Note that 6 is independent of 8 According to the theory of Chapter II, section 4, the quantity [6'/(46/AO)| gives an estimate of the rela- tive effect on stability of the two-sector gradient per— turbation in comparison to the one-sector perturbation. In the table below, the quantity |6'/(46/AO)| has been evaluated using the results of table 3-10 for fields B26.29A and B26Al. TABLE 3-11: Evaluation of ISI/(4o/2O)| for fields B26.29A and B26A1 for the two-sector gradient bump field. n= B26.29A B2621 lot/(4o/20)| 4.11 2.92 115 Note that the quantity I6'/(46/AO)‘ is independent of a. It is evident from the above table that the two- sector gradient perturbation is the main factor deter- mining stability; however, the effect on stability of the one-sector perturbation is not negligible. According to Chapter II, section 4, when both a one-sector perturbation and two-sector gradient pertur— bation are present, the simple approximate criterion for stability is given by Eq. (3.16) which may be used to determine an approximate value for the critical bump strength. If the results of table 3-10 are substituted into this equation, it turns out that the quantity A is independent of 9 Below is the evaluation ofAinterms 2. of a for fields B26.29A and B26Al. B26.29A: A ll [.4 O‘) (I) V m B26Al : A II A [D 4:- O v 93 From Eq. (3.16), a critical value of a may be found by setting A0 = 46; then, the critical value of a for the two fields considered in this report is found to be: B26.29A: (a)C +1.31 (3.20) B26Al: (a)C = +.644. (3.21) It should be emphasized that the critical value of a 116 obtained from Eq. (3.16) is only an approximation. An alternative method may be used to determine an approxi- mate critical value of a; assuming that the one-sector perturbation is negligible, the critical value of a may be obtained directly from Eq. (2.18b). Using the results of table 3-10 and Eq. (2.18b), the critical value of a for the two fields is then given by: + 1.13 (3.22) + .769. (3.23) B26.29A: la|C B26A1: lalC The two critical values of a determined from Eqs. (3.16) and (2.18b) differ by less than 10% for B26.29A, and for B26A1 the two values differ by less than 20%. Note that these critical values of a when expressed in gauss/ inch represent fairly sizeable field gradients. B. Discussion of the Phase Plots: a) Phase Plots for B26Al with 9 = O: The phase 2 plots to be presented first in this section have been Specialized to the case where 62 = O and the constant a is either positive or negative. Consider first the case where a is positive; from table 3—10,when 62 = 0, d1 = 25.8° and d2 = 90°. According to table 3-2, if d1 = 25.8° for B26Al a mixture of two-corner Opening at U2 and U3 and one—corner Opening at U2 would be Observed 117 if the two-sector gradient perturbation were not pre- sent; moreover, the one—corner opening at U2 dominates over the two—corner opening at U2 and U3. If d2 = 90° for B26Al, a mixture of two-corner Opening at U2 and U3 and one-corner opening at U3 would be observed with the two-corner opening Slightly more dominant if the one- sector perturbation is neglected. It Should be kept in mind that the two-sector gradient perturbation has approx- imately three times the effect on stability as the one- sector perturbation; as a result, the phase plots Should tend to bring out the features due to the two-sector gradient perturbation more strongly than the features due to the one-sector perturbation. Figs. 24a, 24b, and 24c are static phase plots for B26Al with a = +.10, + .30, and +.6O respectively and 02 = O. In the phase plots, the new positions of the four fixed points in the bumped field are indicated by "circles" (O) and the initial positions of the four fixed points in the un- bumped field are indicated by ”x's". Figs. 24a, 24b and 24c show that the effect of the two-sector gradient field bump is nearly ”pure two—corner opening” at U2 and U3 along with a small amount of one-corner Opening at U2 which is not inconsistent with the expected results given above. In the figures, the curves labeled @ indicate 118 the predominant two-corner opening at U2 and U3 and the curve labeled @ in Fig. 24c indicates the one-corner opening at U2. As a increases, U2 and U3 move toward E0 and U1 moves away from E0. In the case being con- sidered here, the three fixed points E0, U2 and U3 determine the stable region; as a result, as a increases the stable region is reduced in size. From Figs. 24b and 24c, it is evident that for some value of a greater than +.3O but less than +.60, E0 and U2 coincide and the stable region disappears. From the Spacing of successive points on the curve labeled @ in Fig. 24c it is safe to say that the stable region disappeared for some value of a very close to +.60. According to Eq. (3.21), the critical value of a for which the stable region will disappear is +.644. It is possible to obtain an approximate critical value of a by extrapolation directly from Figs. 24a and 24b if the one-sector perturbation is neglected. It is assumed that the amplitude A of the first harmonic of U3 is equal to the relative displacement of U3 from E0 in the bumped field and that the distance from E0 to U3 in the unbumped field is A0. Thus it is possible to determine the ampli- tude A of U3 in terms of A0 at the two values of a in Figs. 24a and 24b. If 2 is plotted against 7' ; 6'/4e 119 for a = +.lO and a = +.30 and a parabolic curve is passed through these points, (assuming Eq. (2.16e) is valid), then the approximate critical value of a is found to be: [al = +.60. c This computer value of a is in fairly good agreement with the theoritical critical values given in Eqs. (3.21) and (3.23). Consider next the case where a is negative and 92 = 0. From table 3-10, d1 = 205.8° and d2 = 0°. According to table 3-2, if d1 = 205.8° for B26Al, a mixture of one- corner opening at U1 and two-corner Opening at U1 and U3 would be observed with the two-corner opening the dominant feature. This would be the Observed effect if the two- sector gradient perturbation were not present; since the two-sector gradient perturbation has approximately three times the effect on stability as the one—sector pertur- bation, it is expected that the features due to the two- sector gradient perturbation will be more pronounced. With d2 = 0°, a mixture of one-corner Opening at U1 and two-corner opening at U1 and U2 would be observed if the one-sector perturbation were absent. Figs. 24d and 24e are static phase plots for B26Al with a = -.30 and -.60 reSpectively and 62 = O. The figures indicate 120 that a mixture of one-corner Opening at U1 and two— corner opening at U1 and U2 is obtained which agrees with the results immediately above; moreover, the one— corner opening is the dominant effect. In Figs. 24d and 24e, the curves labeled Ca) indicate the one-corner opening at U1 and the curves labeled ® indicate the two-corner Opening at U1 and U2. It is seen from Figs. 24d and 24e that as [al increases, U1 and E0 move toward each other and U2 and U3 move away from E0 with U3 moving somewhat faster than U2. For some value of -60 < a < -.30, U1 and E0 coincide and the stable region disappears. In the case being discussed, it is possible to calculate an approximate critical value of a directly from Fig. 24d. Assume that the amplitude A of the first harmonic of U1 in the bumped field is equal to the relative displacement of U1 from E0. Assume further that a linear relation holds between A and y' E 6'/4€ as predicted by Eq. (2.16c) for ”pure one-corner opening”; this is not an unrealistic approach if the one-sector perturbation is small and nearly "pure one-corner Opening" is obtained which is nearly the case here. With these assumptions, the approximate critical value of a is then found to be: [ale 2 +.60, 121 which is in agreement with the theoretical critical values given in Eqs. (3.21) and (3.23) above. From Fig. 24e, the piling of the points in the region where E0 and U1 would be expected to lie indicates by itself that E0 and U1 will coincide for some value of Ial‘g +.60. According to the theory of Chapter II, section 3, if 6 = 0, E0 should remain fixed; it is evident from Figs. 24a, 24b, and 24d that this is not the case here. Consequently there must be some one-sector perturbation present; however, Since the displacement of E0 is small compared to that of the fixed points this one-sector perturbation must be small in comparison to the two-. sector gradient perturbation in agreement with the results immediately above. In the case of a ”pure” two—sector gradient pertur- bation with "pure one-corner opening" at U1, E0 and U1 will coincide at the critical bump strength leaving an unstable fixed point at the initial position of E0; more— over, as depicted in Fig. 13a, if the bump strength is increased beyond this value then a new stable fixed point should emerge from E0 and move toward U2 and U3. Since the bump being considered in this case does not produce a ”pure” two-sector gradient perturbation, no attempt was made to pursue this result further; however, another 122 reference will be made to this phenomenon later. b) Phase Plot for B26.29A with 02 = 0: Consider now the field B26.29A with a positive and 62 = 0. From table 3-2, if 0 = 0, d1 = 200.1° and d2 = 90°. According 2 to table 3-2, if d1 = 200.1° for B26.29A a mixture of one- corner opening at U1 and two-corner Opening at U1 and U3 would be observed in the absence of the two—sector gradient perturbation. According to this same table, if the one—sector perturbation were not present and d2 = 90°, a mixture of two-corner opening at U2 and U3 and one- corner opening at U3 would be observed with the two-corner opening dominant. It is expected that the features of the two—sector gradient perturbation will be more pro— nounced than the features of the one-sector perturbation because the two-sector gradient perturbation has roughly four times the effect of the one-sector perturbation on stability as noted in table 3-11. Fig. 25 is a static phase plot for B26.29A with a = +.30 and 9 = O. The figure shows the result of the 2 two-sector gradient bump to be nearly ”pure one-corner Opening" at U3 and what is believed to be two-corner opening at U2 and U3 which is consistent with the results immediately above. The curve labeled @ in Fig. 25 indicates the one-corner opening at U3. Although no 123 curves are shown in Fig. 25 which indicate two-corner opening at U2 and U3, it is highly probable that some two-corner opening exists. This conclusion is drawn from the position of the fixed points U1 and U2 relative to the curve labeled @ in Fig. 25. As a increases, U3 moves toward E0 and the stable region is reduced in size; at the same time, E0 remains almost stationary, Ul moves away from E0, and U2 moves away from its initial position but its relative diSplacement from E0 remains unchanged. From Eqs. (3 20) and (3.22), the calculated criti- cal value of a is +1.31 and +1.13 which are about four times greater than the strength of the field bump used in Fig. 25. Comparing the stable region of Figs. 24b, 24d and 25 which all have [a] = +.30, it is evident that it will take a much larger value of a to destroy the stable region of B26.29A than the stable region of B26Al. c) Phase Plots For B26A1 for Other 92 Values: Fig. 26 is a static phase plot for B26Al with a = -.30 and e = -l2.5°. Substitution of O = -12.5° into the de- 2 2 fining equation for dl and d2 in table 3-10 for negative a gives the values of dl and d2 as: $1 = 230.8° o2 347.5°. 124 This value of 8 was chosen in an attempt to illustrate 2 the "pure one-corner opening" at U1; according to the theory of Chapter II, section 3, ”pure one-corner opening" should be obtained at U1 when d2 = 347.5°, a value obtained for d2 when a < O and 92 = -l2.5° according to table 3—10. From table 3-2, if d1 = 230.8° and the two- sector gradient perturbation is not present, a mixture of two-corner Opening at U1 and U3 and one-corner Opening at U3 would be observed with the two-corner opening at U1 and U3 the dominant feature; however, the one-sector per- turbation is much smaller in effect than the two-sector gradient perturbation as noted previously. Fig. 26 shows the effect of the two—sector gradient field bump to be a dominance of one—corner Opening at U1 with some two-corner Opening at U1 and U3 Which is consistent with the forgoing results. In the figure, the one-corner opening at U1 is illustrated by the curve labeled @ and the two-corner opening at U1 and U3 is illustrated by the curve labeled @ . Comparing Fig. 24d with Fig. 26, indicates that for a change in 92 of 12.5° a shift from two-corner opening at U1 and U2 to two-corner Opening at U1 and U3 occurs; however, the dominant feature is still one-corner opening at U1 which is consistent with the expected results. 125 One more phase plot will be presented where a = -.30 and 82 = —l6.5°. Note that the only difference between this case and the previous case is a shift of 4° in 02. As it turns out, this small shift in 92 has a marked effect on the phase plot. Substituting 02 = -l6.5° into the formulae for dl and d2 in table 3—10 gives dl and d2 as: 238.8° $1 343.5°; 92 that is, dl is shifted by 8° and d2 is shifted by 4° from the corresponding values for the case immediately above. From table 3—2, if d2 = 343.5° nearly "pure one- corner opening" is obtained at Ul with a very small amount of two-corner opening at U1 and U3. From this same table, if d1 = 238.8° for B26Al a mixture of predominately two- corner opening at U1 and U3 and Slight one—corner opening at U3 would be observed in the absence of the two—sector gradient perturbation; recall again however, that the two-sector gradient perturbation is the dominant effect here. These conclusions are practically the same as those above where 82 = 12.5°. Fig. 27 is a phase plot for B26Al with a = -.30 and O = -l2.5°. From the fig- 2 ure it is evident that a mixture of one-corner opening 126 at U1 and two—corner opening at U1 and U3 is Obtained with the one-corner opening the dominant feature. This result is consistent with the discussion above. A comparison of Figs. 26 and 27 brings out the marked difference between the phase plots brought about by a shift of only 4° in 9 In Fig. 26, only four 2. fixed points exist with the stable region centered about E0; however, in Fig. 27, six fixed points exist; four unstable fixed points and two stable fixed points; more- over, two stable regions now exist centered around the two stable fixed points. The most probable explanation for the two extra fixed points is that E0 has Split into two stable fixed points and one unstable fixed point although no further investigation has been carried out to verify this hypothesis. This explanation seems reason- able in view of the fixed point evolution depicted in Fig. 13a; furthermore, the figure "8" nature of the flow lines near the center is of the type associated with a 2/2 resonance with a substantial frequency shifting term in the equation of motion. For convenience, the two stable fixed points Shall be designated as E01 and E02 where E01 is closest to the original E0 and the unstable fixed point midway between the two stable fixed points shall be designated UE. The three remaining unstable fixed points shall still be designated as U1, U2 and U3 as 127 indicated in Fig. 27. It is expected as la' increases, first U1 and E01 will coincide and vanish leaving UE, E02, U3 and U2 and then later U3 and E02 will coincide and vanish leaving the two unstable fixed points U2 and UE as the only remaining fixed points. This result is consistent with the one-corner opening case for a two- sector gradient perturbation depicted in Fig. 13a. To obtain a "pure” two—sector gradient perturbation, a mixed field bump must be constructed to nullify the effect of the one-sector perturbation. One such mixed field bump is given by: b(r,0) = a[r-re(0)] cos 2(8-02), (3.24) where the quantities in this equation have been defined previously. Since this field bump is identically equal to zero along the E.0., no displacement of this orbit can result. Preliminary investigations with this type of field bump indicate that E0 remains fixed. It is hoped that in the near future the investigation will be completed and a direct comparison made between the computer results and the detailed theory of Chapter II, section 3 as was done for the one—sector perturbation in section 2 of this chapter. 128 APPENDIX Extension of Theory to a Four-Sector Geometry. For a four-sector cyclotron it is the Vr = 4/4 non-linear resonance which imposes stability limits on the radial motion in the absence of a bump field, and the discussion of Chapter 11 must be modified accordingly. In this note a quite brief description is given of the results Obtained when the theory of Chapter II is applied to a four-sector geometry; all "sections" referred to here are those of Chapter II. The transformation from x(9) to y(d) in section (1) is quite general and applies equally well to a four- sector geometry. The near equivalence of the n=1 com- ponent of x(0) with the n=1 component of y(d), as dis- cussed for Eq. (2.8), is again valid since the n=3 and n=5 components of these functions are relatively small. The differential equation for the first harmonic (or "quasi-first harmonic”) component of y(d) in the absence of a field bump now has the form: n 2 3 y + vry = Ky cos 4(d-do), (A-l) where K and d0 are constants, and the non—linear driving 129 force is again semi-empirical in form. The form of the perturbation forces arising from the bump field, 6F(y,d), is exactly the same here as in Eq. (2.10). Extending the discussion of the fixed point orbits given in section (1A), the form of y(d) here is again given by Eq. (2.11). The general equation for A and d is, instead Of Eq. (2.12), now given by: Aeia = (AB/A§)e-i3a + 7 exp i(¢o-¢1) +y'Ae‘i° exp 21(20-22). (2.2) where y = O/2E, y' = 6'/4e as before, but here: 2 20 = (l6e)/K. (2.3) In the absence of a field bump (y = y' = 0), there are now four unstable fixed points all having A=AO, but with a = 0, i 90°, 180°. Following the discussion of section (1B), the approximate phase invariant H(d, A2) is now given by; H/e = 22 - 1/2(2“/2§) cos 4a -2yAcos(d+dl-do) -y'A2 cos2(d+d2—do) (A.4) Following the discussion of section (2) for the 130 one-sector perturbation case (y' = 0), there are two symmetric situations, namely: d1 = do plus an inte- gral multiple of 90°, for which "one-corner Opening" occurs; and d1 = do plus an odd-integral multiple of 45°, for which ”two-corner Opening” occurs. In the first—mentioned situation (see Fig. 28), one of the fixed points and E.O. move directly toward each other and merge at A = Ao/V33 the critical value of y is then: rYCl = (2B/9)AO° (A.5) Accompanying the one-corner Opening in this Situation, there is also two-corner Opening at the two fixed points initially t 90° from the one which disappears. In the other symmetric situation, (see Fig. 28), there is two-corner Opening at two adjacent fixed points; these two fixed points move inward and toward each other, while the E.O. moves outward to meet them. These three fixed points merge at A = Ao/f3'for a value of y given by: v, = (413/9) 2,, (2.6) leaving one unstable fixed point as a result of the merger, and no stable region. For any value of dl inter- mediate between the eight special values given above, the 131 critical value of y-will lie between the two values Of (A.5) and (A.6). Consider next the two—sector gradient perturbation acting by itself (y = 0) corresponding to the discussion of section (3); here again the critical value of y' is effectively yo = l, at which value the E.0. becomes unstable. For the four—sector main field case, however, the behavior of the fixed points and phase invariants is quite different. The simplest case to analyse is when d2 = do plus an integral multiple of 90°; this case correSponds to a "double” one-corner Opening (see Fig. 29). As y' increases two of the opposite fixed points (say, the a = 0 and d = 180°) move toward the E.O. and define the shrinking stability region between them; when y' = 1, they merge with E.O. leaving an unstable fixed point there. The other two fixed points (a = t 90°) are disconnected and move outward without changing their a values. The other situation which is amenable to analysis is when d2 = do plus an odd integral multiple of 45°, (see Fig. 29). Consider first that the d vaers are shifted by 45° so that the fixed points are initially at a = t 45°, t 135°. As y' increases the two fixed 132 points 0:45° and a = —45° move toward a = 0, while those at d = +135° and d = — 135° move toward 180°, all maintaining A = Ao. For y' > 1 the E.O. splits into an unstable E.0. plus two stable fixed points which move outward along the a = O and 180° directions, respectively. When y' ==f2 , two triangular closed ”stability" regions are formed on each side of E.O.; for y' >‘(2 each triangle undergoes two-corner opening; when y' = 2, each stable fixed point merges with its companion pair of unstable fixed points leaving just one unstable fixed point in each case at A = Ao on the a = O and 180° axes; for y' > 2, these latter points move out away from E.0. The derivation of the simple stability criterion given in section (4) can be readily extended to the four-sector case. The criterion is again, |A| g 46, where now: 2 A = (6')2 + (362/eA2)2 O +26'(362/6A§) cos (420-221-222). (4.7) to lowest (non-vanishing) order in 6 and 6'. Note that in contrast to the three sector case, the effect of 6' adds vectorially with the "Squared" effect of 6. The results given in section (5) for calculating 133 the perturbation parameters in terms of the physical characteristics of a given bump field can be extended without difficulty to the four-sector case. It is clear that a one-sector perturbation can arise here not only from the bump field harmonic n=1, but also n=3 and n=5, to a lesser extent. Likewise, a two—sector gradient perturbation arises mainly from the harmonic n=2, but also much more weakly from n=6. For this perturbation the n=2 harmonic contributes both directly and also indirectly through coupling with the four- sector field. 134 +.28‘ + 26' *U2 +.247 +.22[ U3 +.20v +.14' +.12" +.lol l J -0‘08 -066 “003+ -.O'2 6) +002 +064 +36 +068 P r Figure 14: Phase plot for the unbumped field B26.29A; the "x's" repre- sent the three unstable fixed points (Ul,U2,U3) and the stable equilibrium orbit (E0). 135 +.26u +.244 +.22‘ a: (”‘7' +.2O‘L ‘ +.18" U1 +.l6:r +.14i' +0121, +.10(- -.08 —.06 -.04 —.02 P0 +.02 +.04 +.06 +.08 1’" Figure 15: Phase Plot for the unbumped field B26Al; the "x's" repre— sent the three unstable fixed points (Ul,U2,U3) and the stable equilibrium orbit (E0). +.26 +.l4 136 Figure 16a: Phase plot for B26Al with: bl = +.001, 6 ‘r 44 «(L 1' ° ' 6‘ 4 .K 9 v 9 ‘T -.08 -.06 —.04 -.02 0 +.02 +.04 +.06 +.08 P r +.18‘ +.14 +.12 137 r J. .L J. -.08 -.06 —.04 .02 0 +.02 +.04 +.06 +.08 Pr Figure 16b: Phase plot for B26Al with: bl = +.OO2, 61 = O. +.28 +.26' +.24 +.22‘ +.20‘ +.18l +.l6) +.l4“ +.lO‘ Figure 16c: 138 ‘f 4 I l n A n v ' 1 -.O4 -.O2 0 +.O2 +.O4 +.O6 -+. 08 Phase plot for B26Al with: b = +.OO4, 6 = O. 1 +.22” +.20“ -.88 -.66 42 4 l j v V U —.04 -.02 0 +.02 Figure 17a: Phase plot for B26.29A with: b 1 +. L v 04 +.06 +.OO4, 9 i V 1 139 O. l_ +.08 +.28- +.26‘ +.244 +.22‘ +.20‘ +.16‘ +.l% +.1d +.08 —.08 ..06 Figure 17b: -.04 Phase plot for 4 I +.O2 w<54 l" B26.29A with: b 1 ..L +.O4 = +.O2, 6 140 141 +.010*- 7 E0 U1 U2, U3 +.OO51' 0 . . 1 Amplitude (A) .01 .02 .03 .04 -.005-- \ - . 010 " E0 U3 U2 U1 \ / -.020*~ \/ —.O25” Figure 18; Amplitude (A) as a function of bump strength; solid lines represent theoretical curves; dots represent computer results. +.Olm"y +.OOS“ U1, E0 142 U3 dl(Degrees) -.005“ -.010‘ -.015‘ -.O20" —.O25- +40.o +80° U1 U2 +160° EO -80° .40° U3 Figure 19: Plot of d as a‘function of bump strength; the solid lines represent the theoretical curves; the dots represent the computer results. +.28( +.26‘ +.24‘ +.18. +.14 - +.12J 143 I. 69 l l I r fl I -.08 -.O6 -.O4 -.O2 0 +.O2 +.O4 +.06 +.08 P I" Figure 20a: Phase plot for B26Al with: b2 = +.OO4, 92 = 0. 144 I" +.28:- +.26 1’ +.24 ‘- +o22 " '° 0 9 +.2O ‘r X +018 J. 3‘ o +.16 " +.14 '- +.12 '* +010 .. -.08 -.06 -.04 -.02 O +.02 +.04 +.06 +.08 P I” Figure 20b: Phase Plot for B26A1 with: b2 = +.020, 02 = 0. +.2 l r V I :08 -.06 -.04 -.02 Figure 21a: Phase Plot for B26.29A with: b '13on 145 2 =+oO20, 9 _ O. +.24) +.18‘ +.16) +.14- +.lO- +.08‘ 146 1 -.O4 -.O2 4)- I I -.06 Figure 21b: Phase plot I ; +.O2 +.O4 P 8 c3-. for B26.29A with: b = +.100, 6 - O. 2 +.28‘ +.18 u +.16 +.14 —.08 Figure 22: Phase plot for B26A1 with: b4 : +.040, 64 = 0. -.06 O +.O2 Pr 147 +.28.. +.26:~ +.20‘" +.16“ +.12“ +.1OT 148 Figure 23a: -.04 -.02 0 +.02 Phase plot for B26.29A with: ill-{I‘ll}! : I 149 .28- .24“L .22" .20" .181 .16' .14” . 12“ .10" A -.08 -.06 -.04 —.02 0 +.02 +.04 +.06 +.08 Figure 23b: Phase plot for B26.29A with: b4 = +.300, 94 = O. +.28) +.26“ +.24t +.22” +.18” +016 '4' +.14“ 150 '068 -0016 -OO'LI’ .002 O' "1"002I +061+ +066 +008 P r Figure 24a: Phase plot for B26Al with: a = +.lO, 8 — O. +.24“ +.10s- 151 n L n L l L T V V f ' v T -.08 -.06 -.04 —.02 0 +.02 +.04 +.06 +.08 Pr Figure 24b: Phase plot for B26Al with: a = +.30, 8 +.26 ' l I n n I I -.08 -.O6 -.04 Figure 240: Phase 152 J j I A A. r I I "002 0 +002 ‘|"0O)"h +006 +008 Pr plot for B26Al with: a = +.60, 6 = 0. +.22 153 r O *h 4 o 1. ‘f' t : fl: : ; ‘r 1 *- 1 ‘008 ’006 ’004 "002 0 +002 +004 +006 +008 Pr Figure 24d: Phase plot for B26Al with: a = -.30, 9 = O. 2 154 r +.284 +.26 “ +.24 J- o +.22 “ $4 +.2O 1* +.18 r +.16 3 9 +.14 ‘ 6 +.l2 " +.10 “ '008 '006 -004 ‘002 0 +002 +004 +006 +008 Pr Figure 24e: Phase plot for B26Al with: a = -.60, 9 = O. +.28 +.2m +.22 +.18 +.16 +.l¢ r *- 155 A I A I I l r ' T v V V -.08 -.O6 -.O4 -.O2 0 +.O2 Figure 25: Phase plot of B26.29A with: a +.284 +.26 : +.24‘ +.184 +.12 ‘ 156 I’ p L + X -:08 -.06 -.04 -.02 o +.02 +.04 +.06 +.08 Pr Figure 26: Phase plot for B26Al with: a = -.30, 9 = -12.5°. 157 +.24-- +.2O v +.18 : +.16 4. +.14 » +.l2 ' +.lO “ -.08 -.O6 -.O4 -.02 O +.02 +.O4 +.06 +.08 Pr Figure 27: Phase plot for B26A1 with; a = -.30, 6 — 2 — -16.5°. 158 ing ctor phase plots with increas -sector perturbation: (a)-(o), ¢l=¢o; (d)-(f), 1 = on + 45°. Figure 28: Evolution of four-se ¢ [l] [2] [3] [4] [8] [9] 160 REFERENCES P.A. Sturrock, Annals of Physics 3 (1958), pp. 113-189. R. Hagedorn, M.G.N. Hine, and A. Schoch, CERN Symposium, Vol. I (Cern, Geneva, 1956), pp. 237-253. M.M. Gordon and T.A. Welton, Nuclear Instruments and Methods 6 (No. 3, 1960), pp. 221-233. P. Stahelin, Technical Report No. 1, Physics Department, University of Illinois, Urbana, Illinois. L. Smith and A.A. Garren,"0rbit Dynamics in the Spiral- Ridged Cyclotronf'(Lawrence Radiation Laboratory, Berkeley, Jan. 12, 1961). N.F. Verster and H.L. Hagedoorn, Nat. Lab., Verslag Nr.3623 (Philips Res. Lab., Eindhoven-Netherlands 1960). L. Smith, Sector—Focused Cyclotrons NAS-NRC 656 (Washington: National Academy of Sciences-National Research Council, 1959), pp. 45-47. H.G. Blosser, Sector-Focused Cyclotrons NAS-NRC 656 (Washington: National Academy of Sciences-National Research Council, 1959), pp. 59-65. L.J. Laslett and K.R. 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