Jones, L W

Pruett, C H

Terwilliger, K M

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~/~1IIIIII1111ltili MURA-212o llbD DD3!S2b 4 External MIDWESTERN UNIVERSITIES RESEARCH ASSOCIATION* EFFECTS OF RESONANCES IN THE RADIAL SECTOR FFAG MODEL Lo W. Jones+, Charles H. Pruett v and Kent M. Terwilliger+ December 21 9 1956 ABSTRACT:	 A survey of a portion of the V'x _~ plane ha s been made in the Michigan radial sector FFAG model to study the effects of resonances on the accelerated beam. No beam is observed on integral, half-integral, and linear sum resonances, as well as on the essentia 1 non-linear resonances 30;. = 21]; and 2 ~ + ~ ::: 2'7[;. Some other quadratic resonances attenuate the beam, but no effect is observed due to cubic or higher order resonances. <J-x ::: 2 ~ ,<:Ii' =~ and other difference resonances have no noticeable effect on the beam. + On leave from the University of Michigan * Supported by Contract AEC #AT(ll-l)-384 A.	 Method. A large section of the v:r~ UJstability region in the Michigan 1 8 sector FFAG electron betatron has been surveyed in detail to study the effects of varlOUS resonances on the accelerated beam. This survey was made by varying the working point of the machine by changing the effective value of k and the ratio of currents through the positive and negative field magnets. In this way a region of the?..;( -v":? plane was covered corresponding to UJ of from 1 to 3 and toL!~ of from 2.3 to 3 0 Data were taken by tuning with current ratios from 0.88 to 1.30? noting beam intensities as indicated by pulse heights on a scintillation detector at intervals of .003 to 0005 in the current ratio. Measurements of the betatron oscillation frequencies were made at intervals using the R.F. 2knockout technique so that the ~ for each data point could be inferred by interpolation. Five runs were made between extremes of current ratio~ each corresponding to different values of k. In addition, several shorter runs were made corresponding to a smaller range of current ratios v and other runs were made by varying k and holding the current ratio constant. 91 separate points were measured by the R.F. technique during the course of this survey. Typical data from one current ratio tuning run and one k-tuning run are illustrated in figures land 2? where the logarithm of the observed intensity is plotted versus the tuning parameter. Measured frequencies and the positions of resonances from interpo­1. MURA-LWJ-KMT-5 2 MURA-KMT-30 (1) lation between measured frequencies are indicated. The influence of resonances on the beam as summarized in table 1 and discussed below was inferred from such plots of the data. In figure 3 the data have been used to construct a contour map of beam in­tensity versus tuning with contours drawn every factor of ~ in intensity. It should be pointed out that the absence of accelerated beam can be due to a resonance or simply to a lack of beam capture at injection. With the pulsed expander operating~ however, initial capture should be independent of tuning. All of the data for this survey were taken with pulsed injection~ the expander and the same magnet configuration. Due to differences in peak energy, beam length~ and beam shape corresponding to different tuning positions~ the photomultiplier pulse height is not exactly pro­portional to the number of electrons. However these corrections should be small relative to the range of intensities studied. B. Particular Resonances. Resonances may be classified according to whether they arise due to particular terms in the differential equations, i.e. linear, quadratic~ etc. They may be further classified as to whether they would lead to an instability in a perfect machine (sector resonances) or whether they only give rise to large amplitudes of oscillation or instability due to irregularities in the eight sector accelera­tor at hand (machine resonances). We discuss below the observed effects of different classes of resonance. 1. Linear machine resonances. On integral and half integral values of vI~ and tI~ and along the lines vI~~vI~ = integer~ the beam is completely destroyed. (2) The widths of the stopbands (region where no beam is found) around integral resonances appear to be about ± .12 wide in units of LJ while they are only about _~ .01 wide around half-integral and sum resonances. It is observed generally that the resonances reinforce each other 9 for example the ~~ ~ 2 stopband is broader as 1lX approaches 3 9 so that the contours of figure 3 are in general rounded. No difference resonances 9 ~ .,;-v~ ~ integer were observed. 2. Quadratic and cubic sector resonances. The resonances 3 cr-x ~ 27t: (VX' ~ 2.667) and OJ< + 2 a-~ ~ 2~ (1I~ + 2 ~, = 8) cause complete loss of beam with a stop­band width comparable to half-integral machine resonances. The resonance Q;;( = 2 ~ is not observed to effect the beam noticeably. The cubic resonance 2 o-~ = 2 ~ was near the limit of the tuning range and there it could only be studied close to integral stopbands; however in the 2 times it was crossed, no effect was observed. 3. Quadratic machine resonances. The resonances -zl;< + 2 ~ = integer almost always caused a reduction of beam intensity by a factor of 2 to 10. The resonances 3 v~ = integer were observed in some instances to cause some small reduction in beam intensity. The resonances 2.,}?(' + -v'~ = integer were observed to slightly effect the beam intensity, as would be expected from median plane symmetry of the ideal machine. The only resonance 3 -I'i = integer that could be studied except the sector resonanceox = 21? /3 was V?( ~ 2.333. It was (3) crossed only once, and then without an observable effect. Again no difference resonances we~e observed. 4.	 Cubic and higher order ffi2chine resonances. In the course of this survey none of the resonances fVx+tt~:6tJV wherelfl+-/~/ G 4 were ObSeTi,ed "[;'0 influence the beam. It appears that every observed fluctuation in intensity outside of experi­mental limits of error can be accounted for by the linear and quadratic resonances discussed above. We	 conclude that the beh~vior of this machine is consistant 3with what one would expect from general non-linear orbit theory. 3.	 FoA. sturrock, Static and Dynamic Electron Opticso (Cambridge University Press, 1955). (4 ) Resul1',5 of Reso!1ance Survey in the Michigan Model --'.'--:--.-..-_·...::::::::::~..::=:=='·::::.,~I··_·· .- ..::::::._:....:.......::" .._~. ,,:- " .",,~,,- '.. -,_.:-':- ·:r.:~· ..···· --.. ~=-:::. :=::::=.:j:l=-:::::':'======".==:.:;::::-= r-- i Percentage I' I R.esonance Nurr.ber of Number of 1 ; Deere) se ~n Stopb~nd I _ =--=-_..__-_.==~::========.=~_4+~-T-im-.,;;...e=s.=c,;:::;.-~r=0,;;..-.=...s_s-=:..e.=...._d.:.._-+~=~~=.m=e=s=s=e=e=n==:::t:! =I=n=t=e=n=S=i=t~y===I=W=i=d=t=.h====~F -;}i?:. - 10 0 -	 2 I 100 +0.08 --- Z/~ ~ 2. 0	 5' ··-···----+------;.5·-------4,--....;::;;;.;1o:::-:o:::-------t---:-±-;ii;:o...:..'1?>-2--;-­~~:_~_=-~__. - 1 100 -0.10 ~~x ~ 3.0 +-__--:-- 4 100 -0.12 ___~t..:...-_~,,;-__ .-----:i-----+---~-;i:----+_-___:_;:::...:..~---t-__JL~ - 105 I 4 4 1 1 00 ±'±OO.·OO~ 100?./~	 - 2. 5 ----4------,4~-_---+-____,.;4:_---+_-..:;::.,;,-~----+-----=~-:=:-;o:;---+-1/0' - 2.5 I 2 I 2 100 '±O.02 »);(-rV~ - 4 4; 4 I 100 ±.0.01 V)( + 2Vll = 8 0-";< +2 ~"t = 2 "/t. 2 2 100 .±0.01 2 1?<+ .~ 202 tJx + V1r- 6 2?4	 ~20 3 75 _' CI~ - '2/371: 3 0 '0 cr,;( _. 2/3?t 3 3 100 " ±O • 01 .. -----=:..... L})-'-:-'~...<.J,::..--+--- .....~vJ7X---"'7"C- 	 4----+------;;:.~-...,..----+---F-i<..,;,----i----.;;:~~=-----'_r_l?<-t .( 20 0 0 0 0 0 0 0 0 0	 0 Notes~ 1.	 Refers to the number of times a dip in intensity is observed in the neighborhood of a resonance. Questionable entries correspond to dips which may be due to nearby resonances or to intensity fluctuations un­related to resonances. 2.	 Relati~e to the general nearby intensity level. 30	 Width of region of complete absence of beam. 4.	 May be due to fluctuation unrelated to resonance. 5.	 M2V be (~ntirely due to 2 OX + crz. = 2 7]: • r-TABLE I It - TUNING RUN = ~ ~ 0­ ,J:~_ ~'t ~. """' '-t:l . :':;)" ~'6 ~ -~ ~ "'! ~ "-" ~ t I,,'tt.. ! ~f ~,,~ .....l ! ! 'i..oJ tt} I i \,} ""-"" ~·,1;, .?~., "'~ ~~,c..J1 If] ~ I I -4 ! .~~ -. (",,;;3 ~",:;'-I I, I I (1"'. ",!:~ f '", .~ 0 -~t ~ ......<s.." J.o~ I II /I I t--., II ",-........ I ~ ..­-1.1') Z ....,1 i'o­I.e I ... 1'<' r . N 4:: I ..... ~ '( w ~ .j I I I I I I, W :>-! I i ~ :5 ·1 ,.. 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Effects of resonances in the radial sector FFAG model

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