Judd, D L

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MURA DLJ-1 Internal ANALYTICAL APPROXIEaATION IN MARK V SCALLOPED ORBITS AND TO RADIAL BETATRON OSCILLATIONS ABOUT THEM David L. Judd University of California Radiation Laboratory and Midwestern Universities Research Association* August 10, 1955 As a check on the work of Laslett, and because of apparent discrepancies between (1) Ridge Runner (2) Feckless Five, and (3) Laslett's analytical values obtained with the aid of his tables, in determining frequencies of radial betatron oscillations, it was decided to compare analytically these various approaches, using an independent method of procedure, for motion in the median plane, At the conclusion of this work, correction of errors in Laslettls anilysis produced satisfactory agreement, and all treatments of this prob- lem now agree except for some discrepancies in the co- efficients in xs below. The present work is believed to be more accurate at this point. Ia Com~arison of RR and FF. The Ridge Runner (rigorous) equations are for motion of a particle of mechanical momentum *Supported by the National Science Foundation MURA DLJ-1 Internal P = 4 8. (T;/~) JW Y; in the median plane of the field Here J The Feckless Five equations, which are approximate, have been supplied by Laslett, and, when specialized to motion in the median plane, assume the form = ( x  p, P; = - (A+/)x --t.(&+/)x% - " d f & + l ) ( & - Q X S /  L .I- Here J s  tan'(k + l)w, N P ~ N o + J - ~  Straightforward ex- pansion of the Ridge-Runner equations yields x l = ( l + X )  px + I *'?$) ' . = - ( ~ + 4 * .  - - & ( L + I ) X ~  , J ~ ( ~ + Q ( A - / )  x 2  - f w J e  (&+I) A - c - -.)-r&+,)&..U& -...&' Z f & - J N O + S ~ / ( ~ ~ ) -  9= rXa + "C&' I xj, where one term in x3 has been explicitly kept to compare B +- 1 with Feckless Five, and where ( 1 f x ) and e (k+ l)x have been used interchangably since Y Lc 1 and k 77 1. Thus a discrepancy to this order in x exists in the terms MURA DLJ-1 I n t e r n a l  i n  x b  , s ince  -C&P+~~)-J = - h e  +Q -&sW /N e + 4 1 with  the  second term being erroneously present  i n  FF and missing i n  the  r igorous RR. The e r r o r  i s  no doubt comparable with the  neglect of terms of order x3 fo r  the  small values of appropriate  t o  the  range of parameters of current  i n t e r e s t .  Va l id i ty  of the  neglect  of x3 terms i s  c e r t a i n l y  open t o  quest ion i n  the  study of non-linear e f f e c t s  i n  beta t ron o s c i l l a t i o n s  but does not appear t o  be s ign i f i can t  i n  l oca t i ng  the  closed o r b i t s  o r  i n  determining t h e  beta t ron frequency i n  l i n e a r  approximation, f o r  the  range of parameters of present  i n t e r e s t ,  XI. Analvt ical  Determination of Closed Orbits ,  The r igorous (RR) equations may be expanded without the  somewhat a r t i f i c i a l  in t roduct ion of t h e  quan t i ty  J- The r e s u l t ,  t o  the  order indica ted ,  i s  MURA DLJ-1 Internal r * Here Y : ;5 - 4 4 .  Neglected terms in the above equations are small with respect to the leading terms. In the first equation, v In the second, One may proceed to convert these highly accurate equations to a single second order differential equation: In order to proceed further analytically, it is necessary to expand in powers of 3 , even though considerable in- accuracy may result for some values of x/w. The terms neglected in the d3egree of approximation used below are of + I v - 2 -  order - Ly - 9 M ~ +  r !  fd relative to the largest'terms in the differential equation; this quantity 3 is less than 2 ~ 1 0 - ~ f o r  f& .), N11.50, l/w 10 but 3 may be as large as 3 for N 3 25, 1/w 2 3 x 10 ! ! MURA DLJ-1 I n t e r n a l  This  expansion, when ca r r ied  out ,  using the  abbreviat ion f E N f q  , yie lds  ~ " + ( & + j ) x  = f- a 9 -G 4 , - ( ~ t z j f  - j f i .  + p + + + 4  + f{(i+$(&,9 - 3 g ~ ~  - 2- L X ' - The l a r g e s t  neglect  i s  t h a t  of the  terms i n  r / w Y .  This equation may be solved by i t e r a t i o n  f o r  the  closed o r b i t ,  which i s  t h a t  so lu t ion  having period 2T/N  i n  0. x z  X , + X ,  + x z +  x j + - - - - I  f * d ' + ( ~ + / ) x ,  = f L - 4 ~  4 MURA DLJ - 1 I n t e r n a l  1 f [ / f 3 ~ - 5 ] c  . D,: ~ / C f d + t )  d = &a +< UI A I B * + ~ ,  W J ~ + * + S ,  ~ c ~ P N ~ + E ~  ~ 4 %  4+ f P3 a - 3 ~ 4 .  2-(4 , , l (d t2J  P , - [S - d m  0, '""I 9 Dl f "  i, : , )-,f2, (dtJ(#..y 44 P' p, l. r*+ -4' , ( A  4 a) + - I 3 (A+ IM+J) 3 L d r  2 4 D> ", T d 4  Here A ,  B,  ..... E a re  the  square brackets on the  r i g h t  s ide  of t h e  d i f f e r e n t i a l  equation being solved. To minimize the  labor  from here on, we note t h a t s  according t o  Symon's smooth approximation, 7 MURA DLJ - 1 s o  t h a t  f o r  reasonable values - ? / a ,  G@ -- T/Y 9 7( N 1  we have, approximately, - -1 , 4 t / 2- - since LJ Y .l I C 3 - f / I ,  , and D, - hl ' p 2 - - 9 N  a Thus the  ) f i r s t  terms i n  &,d, and d, a r e  the  l a r g e s t ,  l / u i  i s  an order of magnitude l a r g e r  than k, and the  l a r g e s t  terms i n  f i  and /I, are  those i n  f3/(w2 , ) a n  f %, ,. The l a r g e s t  terms on the  r i g h t  s ide  a r e  those coming from the  l a rge  p a r t  of A [- ' -5 times the  l a r g e s t  terms i n  x2 (- % N - , from t h e  l a r g e s t  term 9 Dl %n B ( - & 4 " ~ )  times xo times the  two _ l + a  - P3-- f 3  l a r g e s t  p a r t s  of xl ( - stm+,) o, SU, D , w  6 2 F  1, and from the  l a r g e s t  term i n  D ( yu r bf times 1/6 x z  We thus obta in  the  approximate equation ) has been kept ,  where only the  l a r g e s t  term i n  h, ( oC w a  and Dl& e 1/4. We thus ob ta in  8 MURA DLJ-1 F i n a l l y  we may c o l l e c t  a l l  our terms below and evaluate the  orders of magnitude of the  various terms, expressing a l l  quan t i t i e s  i n  terms of N through use of Symon's smooth approximation expressions with the  choices of & made above, 9 MURA DLJ-1  I n t e r n a l  N' To get  orders  of magnitude: take k e - k  - / 6 - a * ,  0 3  - # 0 t - -  I It i s  f e l t  unl ikely  t h a t  f o r  any reasonable design t h e  numerical re la t ionsh ips  used above w i l l  d i f f e r  by more than about a f a c t o r  of 2, or t h a t  the  orders  of mag- n i tude  of t h e  various terms ar r ived a t  w i l l  be changed by more than a f a c t o r  of 10, ?AURA DLJ-1 I n t e rna l  111. Linearize& Eauations f o r  Radial Betatron Osc i l l a t ionso  The per iodic  so lu t ion  jus t  obtained w i l l  heqceforth be denoted by xsd If we s e t  X - XS + , i n s e r t  t h i s  i n  the  r d i f f e r e n t i a l  equation, and r e t a i n  only l i n e a r  terms I n  , we ob ta in  C '+ i~+ j i e=At '  +Y,,c +cX;p1 + ( +  a d9?i; CI) + B (r: p '% Rearranging, Transforming t o  el iminate the  f i r s t  de r iva t ive  term, s e t  f"v &f a obtaining I + + - B - "': a We w i l l  now t r y  t o  evaluate the  square bracket i n  t h i s  equation. By use of the  same in t e r r e l a t i onsh ips  based on the  smooth approximation a s  were applied above t o  pick out l a rge  terms and r e j e c t  small ones, we may ve r i fy  t h a t  t h e  terms t o  be kept a re  f & I  +,- -9- ( A ~ S J  f*s 5 + ~ A + I J & + ~ J  eZ< A F f PTs* L-T 4- 3]fy - -f Lrr -9 .& , & After i n se r t i ng  xs a s  obtained e a r l i e r  and dropping small terms, we obta in  the  square bracket a s  11 MURA DLJ-1 I n t e r n a l  If- Making the  change of va r iab le  s/l/?--A, we may wr i te  t h e  l inear ized  equation f o r  v a s  wi th  2; + I)= bL.tan s3 was not evaluated s ince  D g  i s  l e s s  than ) 0.1% of B. From the  smooth approximation, orders  of magnitude are:  12 MURA DLJ-1 I n t e r n a l  The l a r g e s t  terms ca lcula ted  from the  o r i g i n a l  equation accurate through t h i r d  order terms i n  x, but not wr i t t en  above, a re  of order 1/1000 or  of order 1/n2 when compared i i t h  the  orders  of magnitude given immediately above. pg- l e c t  of. a l l  of the  terms of order 1 /N2  i s  not j u s t i f i e d  f o r  model parameters with small N;  t h i s  point  i s  inves- t i g a t e d  separa te ly  below.) We f i n a l l y  then obtain: One way s h i f t  t h e  o r ig in  of' t an amount 4 so a s  t o  I make the  new $, = 0; then ' can be obtained from a 13 MUttn DLJ-1 I n t e r n a l  the  equation i- S, - Ld;: I f  LL &'a L.k 2, There appears t o  be l i t t l e  point  i n  evaluat ing t h i s  expression f u r t h e r  a t  present  s ince t ab l e s  do not e x i s t  I which cover values of $ other  than 0 or For t h e  cases s tudied t h i s  f a r  numerically, 4 i s  l e s s  than about 0.15 radian,  and i s  l e s s  than t r i c e  $ , I ;L so tha t ,  does not exceed 0.1 radian. % For model aarameters use the  following formulas: A = same a s  f o r  l a rge  machines where aca k t  3) f a  = , q " w D ,  t 3) k 2 D ,  D, (The underlined t e rn s  may be omitted i n  l a rge  scale  machines) 14 MURA DLJ-1 Internal IV. Numerical Comuarisons, The closed orbit results may be put in the form - -g - f i  $*- d' b hJ+*-6& 1~ $'+terms<l$ of 6 . -A* The results of Laslett nay be written as: P I N " -  ( 4 7 1 )  + -  ; (A,) a The results of the present investigation have been quoted earlier. A numerical comparison has been made for the representative parameters f = .25, N=5O, 3 k.=100, w-I = 3 x 10 . 4 13 K- S .-14 Laslett ( 1.25 x 1.01 x lo=-4 -10 0 The discrepancies indicate that the values calculated here may more accurately predict fixed points, Results for A, B, and C in the linearlized radial betatron oscillation equation have been computed for six cases, and one tabulated below, together with valueg MURA 1 5  DLJ-1 I n t e r n a l  of q/T obtained from t h e  L a s l e t t  t a b l e s  and from the  I l l i a c .  E a r l i e r  discrepancies,  l a rge ly  due t o  e r r o r s  i n  L a s l e t t ' s  e a r l i e r  work, have disappeared upon in t roduct ion of h i s  corrected values,  which a re  e s s e n t i a l l y  t h e  same a s  those obtained here. Resul ts  f o r  using h i s  corrected values a re  a l so  tabula ted  f o r  complete- ness; good agreement i s  again obtained. * This value obtained by l i n e a r  i n t e rpo l a t i on  i n  C from p l o t s  of l e v i e s  of constant  B on A vs  cos (Tgraphs. Linear i n t e rpo l a t i on  i s  not too  well  j u s t i f i e d  here ,  and i s  espec ia l ly  bad from p l o t s  of l i n e s  of constant  d- on A vs B graphs i n  t h i s  case ly ing  near a  s t a b i l i t y  boundary; the  valve obtained i n  t h i s  way i s  -92, MURA DLJ-1 In t e rna l  I n  the tab le  above it should be noted t h a t  the reading of values of from graphs made from the Las le t t  t ab les  i s  uncertain t o  a t  l e a s t  one and probably two d i g i t s  i n  the second decimal place i n  many places. V. Conclusions. It i s  concluded (1) t h a t  the Ridge Runner and Feckless Five equations agree i n  the median plane through terms of second order i n  x and x except fo r  a small discrepancy which w i l l  be unimportant f o r  small kev; (2 ) t h a t  a re-  l i a b l e  expression, accurate t o  be t t e r  than of the l a r -  gest  term, f o r  the closed o r b i t s  has been obtained; (3)  t h a t  the coef f ic ien ts  A, B, and C f o r  the  r a d i a l  l inear-  ized betatron o s c i l l a t i o n  as  obtained here and independently by Las l e t t  are now i n  sa t i s f ac to ry  agreement; and (4) t h a t  the  frequencies obtained from inser t ing  these i n t o  the Las le t t  t ab le s  are i n  sa t i s f ac to ry  agreement with the  f r e -  quencies denied from I l l i a c  computations, The pr incipal  recommendation a r i s ing  from t h i s  work i s  t h a t  the  sect ion of the t ab le s  from C = -0.5 t o  C a f 0.5 be extended a t  In te rva ls  of Oal If it Is an t ic ip la ted  t h a t  extensive use of these tab les  i s  expected i n  the future .  It may a l s o  be repeated t h a t  the coeff ic ients  of xs obtained here a re  believed t o  be more accurate than those O f  Las l e t t  and may have some u t i l i t y  i n  more precise locat ion of f ixed points  i n  the machine calculat ions.  

Analytical approximation in Mark V scalloped orbits and to radial betatron oscillations about them

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