Brown, K L

Carey, David C

Servranckx, R V

CERN Document Server

Particle Accelerators1983 Vol. 13 pp. 199-2070031-2460/83/1303/0199$06.50/0© Gordon and Breach, Science Publishers, Inc.Printed in the United States of AmericaSECOND-ORDER PATH LENGTH TERMS FOR AGENERAL BENDING MAGNETDAVID C. CAREYFermi National Accelerator Laboratory, P.O. Box 500,Batavia, Illinois 60510 U.S.A.ROGER V. SERVRANCKXUniversity of Saskatchewan, Saskatoon, SK,Canada S7N OWOandKARL L. BROWNStanford Linear Accelerator Center, P.O. Box 4349, Stanford,California 94305 U.S.A.(Received December 28,1982)Second-order terms are derived for the path length through a non-uniform-field bending magnet. The field isspecified by its value on the reference trajectory, and its first and second derivatives with respect to the radialcoordinate. This extends the catalogue of second-order path length terms, as previously derived by Daviesand Helm. The results have been incorporated into the computer programs TRANSPORT, TURTLE, andDIMAT.1. INTRODUCTIONIn high-energy particle storage rings, attention must be given to preserving thelongitudinal bunch structure, so as to ensure adequate luminosity. Bunch structure isalso important in transferring the beam from an accelerator to a storage ring, orbetween storage devices, to maintain synchronism with the accelerating field. Innuclear and particle physics analyzing spectrometers, prycise timing information isuseful for determining particle identity. For these reasons, careful attention must bepaid to the path length the particles take through the various transport systems. Inparticular, it is useful to be able to calculate the second-order terms contributing topath length. Those for a quadrupole and for a uniform-field bending magnet have beencalculated by Davies,l while those for the fringe field of a general bending magnet havebeen calculated by Helm. 2 Here, results are presented for a non-uniform-field bendingmagnet. We assume that the magnet preserves midplane symmetry and that the fieldcan have both linear and quadratic dependences on the horizontal transversecoordinate. A similar second-order calculation has been performed independently byMatsuda, Matsuo, Ioanoviciu, Wollnik, and Rabbel. 3 However, we find our results to199200 D. C. CAREY, R. V. SERVRANCKX AND K. L. BROWNbe not entirely consistent with theirs, and deem it worthwhile to publish oursseparately.Our exposition begins with a brief review of the meaning and derivation of thetransverse second-order matrix elements. From there, we develop the formalismnecessary to derive the matrix elements for the path-length difference. Having done so,we give explicitly the form of the required transverse terms. We then list our results,which are the the second-order matrix elements for the path-length difference. Finally,we relate the concepts of path-length difference and longitudinal separation.II. TRANSVERSE COORDINATESThe coordinate system used to specify the position and direction ofan arbitrary particletrajectory is based on the reference trajectory. The reference trajectory is a particulartrajectory of a particle having the central design momentum of the beam line. It passesthrough quadrupoles on their magnetic axes and experiences uniform field in thecentral part of bending magnets.At any point on the central trajectory, we may construct a plane perpendicular to thecentral trajectory at that point. As the point moves along the central trajectory, theplane will sweep out the adjacent region of space. Any point not on the centraltrajectory may be identified with a point on the central trajectory if it lies in the planeperpendicular to the central trajectory at that point. Moveover, in any practicalsituation the distance of a particle from the central trajectory is small enough so thatthe identification will be unambiguous.Three coordinates, x, Y, and t, are used to specify the position of a particle on anytrajectory. The arc length along the central trajectory from the beginning of the beamline to the corresponding point is given by the coordinate t. The two transversecoordinates in the plane perpendicular to the central trajectory are denoted by x and y.The coordinate x is in the plane of bend of the central trajectory. If the bend plane ishorizontal, then, facing along the reference trajectory in the direction ofincreasing t, thex axis points to the left and the y axis points up. The three coordinates x, y, and t form aright-handed system.Two additional coordinates can be introduced to specify the direction of thetrajectory. They are the two direction parameters x' and y', which are the derivatives ofx and y with respect to t.The differential line element is given in terms of the coordinates as(1)where h is the curvature (or reciprocal of the radius of curvature) of the referencetrajectory. The path length of a portion of the trajectory is simply the integral of thisquantity over the part of the beam line under consideration. The path-length differenceI is obtained by subtracting the path length along the reference trajectory. The sixthcoordinate 8 is taken as the fractional deviation of the particle momentum P from thereference trajectory momentum Po. This is written asp = Po(l + 8). (2)Thus, a complete specification of a trajectory at any value of t is given by a six-SECOND-ORDER PATH LENGTH TERMScomponent vector X, wherexx'X=yy'1b201(3)In first order, the value of this vector at any point t l can be obtained from the initialvalue at to by multiplication by a six-by-six transfer matrix.Xl = RXo (4)In a static magnetic system with midplane symmetry, i.e., where all scalar magneticpotentials are odd in y, b is influenced by no other coordinate and I affects no othercoordinate. Further, the two transverse planes are independent, so that R is blockdiagonalized with two by two submatrices along the diagonal.The coordinates may alternatively be written asXl = (xlxo)xo + (xlxo')xo' + (xlb)bXl' = (x'lxo)xo + (x'lxo')xo' + (x'lb)bYl = (yl Yo)Yo + (y'l Yo')Yo'Yl' = (y'l Yo)Yo + (y'l Yo')Yo'1 = (llxo)xo + (llxo')xo' + (llb)b(5)The transverse elements are often called sinelike, cosinelike, or dispersion rays and aredenoted by the letters s, c, and d respectively, so that(xlxo) = cx(t)(x'lxo) = cx'(t)(yl Yo) = cy(t)(y'l Yo) = cy'(t)(xlxo') = sx(t)(x'lxo') = sx'(t)(yl Yo') = Sy(t)(y'l Yo') = sy'(t)(xlb) = dx(t)(x'lb) = dx'(t) (6)This formalism can be extended to include quadratic terms, so that for x we haveXl = xocx(t) + xo'sx(t) + bdx(t)+ (xlxo2)xo2 + (xlxoxo')xoxo' + (XIX O,2)XO,2+ (XI Y0 2)Y0 2 + (xlyoYo')yoyo' + (XIYo,2)yo,2+ (xlxob)xob + (xlxo'b)xo'b + (xlb 2)b 2(7)Only terms which survive midplane symmetry are shown. The expansion, using theabove notation, was first formulated by Streib4 and the actual second-order coefficientsfor the transverse terms were calculated by Brown. 5202 D. C. CAREY, R. V. SERVRANCKX AND K. L. BROWNIII. PATH LENGTH DIFFERENCESFrom Eq. (1), we see that the path-length difference is given by1= t {[X,2 + y,2 + (1 + hX)2]1/2 - 1} dt.Expanding by the binomial theorem to second order, we derive1 = t {hX(t) + ~[X'(t)2 + y'(tf] }dt.(8)(9)In first order, only the first term inside the braces survives. The first-order matrixelements for the longitudinal coordinate 1 are then the integrals of the first-ordermatrix elements for the transverse coordinate x. Considering only the first term in thebrackets, we similarly see that one contribution to the second-order matrix elements for1is the integral of the second-order matrix elements for x. The remaining contributionscome from the terms in the brackets. Since the expression for 1is already quadratic inthe two derivatives x' and y', we need consider only first-order contributions to theseterms. We can expand x('t) and the two derivatives x' and y' in terms of the initialcoordinates and the appropriate matrix elements to derive explicit expressions for eachof the longitudinal second-order matrix elements.Because of midplane symmetry, there are only nine such matrix elements. They arethe nine which correspond to the nine second-order matrix elements for the transversecoordinate x. The expressions for the second-order matrix elements for 1are(llxo2)= t[h(XIX02)+~C/Jdt (10)(llxoxo') = t [h(xlxoxo') + cx'sx'] dt2 It [ 2 1 2J(llxo' ) = 0 h(xlxo') + "2 sx' dt(lI Yo 2) = f~[h(XIY02) + ~C/Jdt(llyoYo') = t [h(xIYoYo') + cy'sy'] dt2 It [ 2 1 2J(llyo' ) = 0 h(xlyo') + "2 sy' dt(llxoo) = t [h(xlxoo) + cx'dx'] dt(llxo'o) = f~ [h(xlxo'o) + sx'dx'] dt(110 2 ) = t [h(X102 ) + ~ dX ,2JdtSECOND-ORDER PATH LENGTH TERMSIV. SECOND-ORDER TRANSVERSE COORDINATES203The magnetic field on the midplane of a general bending magnet has no x component.The y component may be expanded to second order asThe characteristic rays or first-order transfer matrix elements are given by(11)sAt) = : sin kxtxSy(t) = : sin kyt,y(12)wherekx 2 = (1 - n)h2ky 2 = nh2 •If the value of k2 in either plane becomes negative, the functions cos and sin are replacedcosh and sinh respectively.If the second-order matrix elements are expressed in terms of the characteristicfunctions, then their form does not depend on the sign of k2 in either plane. Taking thework of Brown, and simplifying a bit, the second-order terms for x areh2(xlx0 2 ) = "6 dx(t) [2(n - ~) + (5n - 2~ - 3)(1 + cx(t))J (13)h(xlxoxo') = 3(1 _ n) sAt) [2(n - ~) - (5n - 2~ - 3)cx(t)]2 1(xlxo' ) = 6(1 _ n) dAt) [2(n - ~) - (5n - 2~ - 3)cAt)](xjy/) = h2 [-21(5n 2 - 6n~ + 2~ - n)dAt) - ~nhs/(t)]1 - 5n(xlyoYo') = 1 ~~5n (cy(t)sy(t) - sx(t))(XIYo,2) = 1 ~ 5n [h~S/(t) + ~ (5n - 4~ - 1)dAt)]h(xjxoo) = 6(1 _ n) [ -4(n - ~)dAt) - 2h(5n - 2~ - 3)s/(t)+ 3(n(n + 1) - 2~)htsx(t)J204 D. C. CAREY, R. V. SERVRANCKX AND K. L. BROWN(xlxo'o) = 6(1 ~ n)2 [2(5n - 2~ - 3)sx(t)(cx(t) - 1)+ 3(n(n + 1) - 2B)(sx(t) - tcx(t))J1(xlo 2 ) = 6(1 ~ n)2 [8(» - ~)dAt) + (5)> - 2~ - 3)hsx2(t)- 3(n(n + 1) - 2B)htsx(t)].v. SECOND-ORDER PATH LENGTH DIFFERENCESHaving obtained both first- and second-order expressions for the transverse coor-dinates, we may now substitute the expressions from Eqs. (13) into Eqs. (10). The resultsare the second-order expressions for the path-length difference.h2(llxo2) = - 12(1 _ n) [(n(3n - 1) - 2~)(t - sx(t)cx(t)) + 4(n - ~)(t - sx(t))h(llxoxo') = - 6(1 _ n) [4(n - ~)dx(t) - (n(1 - 3n) + 2~)hs/(t)J(llxO,2) = - 12(1 ~ n)2 [2(7n - 4~ - 3)(t - sAt))1- (5n - 2~ - 3)(t - sx(t)cx(t))J - 4: (t + sx(t)cx(t))2 1 [ 1 - 3n ] 2( )(lIYo ) = - 2(1 _ n) 2~ 1 _ 5n - n h t - sx(t)+ ~C~5n - n) h2(t - sy(t)cy(t))(llyoYo') == C! 5n - ~ n) h2s/(t) + 2h 1 ! 5n dAt)(lIYo,2) = - 2n(1 ~ 5n) (t - sy(t)cy(t))+ 2(1 ~ n) C~~5n + 1) (t - sAt)) - ~ (t + Sy(t)cy(t))(llxoo) = (1 ~ n)2 [ - ~ (n(n + 1) - 2~)h(sAt) - tcx(t))+ ~ (n - ~)h(t - sAt)) + ~ (n(3n - 1) - 2~)h(t - sx(t)cx(t)) ]SECOND-ORDER PATH LENGTH TERMS 205(/102 ) = - (1 ~ n)3 [~(n - ~)(t - sx(t))1 1 ]- "2 (n(n + 1) - 2~)(sAt) - teAt)) + 12 (3n 2 - n - 2~)(t - cAt)sAt))The expressions have been checked in a variety of ways. First, they reduce to theexpressions given by Davies for quadrupoles and uniform-field bending magnets.Second, they give identical results independent of whether a bending magnet iscalculated as a single entity or has been segmented into several smaller magnets. Thissegmentation test has been done on bending magnets which are zero-gradient, weakfocusing, and strongly focusing or defocusing, and with both zero and non-zero valuesof~. Finally, all but the (1182 ) term vanish for the Brown achromat,6 which was analysedby Carey and Servranckx.7Second-order longitudinal matrix elements have now been incorporated into severalcomputer programs. These include the matrix calculation and fitting programTRANSPORT,s the Monte-Carlo beam simulation program TURTLE,9 and thecircular machine analysis program DIMAT. 1aVI. LONGITUDINAL SEPARATIONThe difference in path length corresponds to the longitudinal separation of the particlesonly for a highly relativistic particle beam, where the velocities of the particles areessentially that of light. For a non-relativistic beam, we must consider the fact thatparticles of different momentum will have different velocities. The longitudinalseparation will receive contributions both from the path-length difference and from thevelocity difference. To calculate the combined effect, we multiply the central velocity bythe time difference.1 = vo(~ _La)v va(15)Ifwe expand this expression in terms of the differences L - La and ~v = v - Va, andretain only terms which are at most second order in those differences, then we derive~V ~v (~V)21 = L - La - La - - (L - La) - - La - .Va va va(16)It remains to determine an expression for the fractional velocity deviation ~v/va,which is valid to second order in the fractional momentum deviation 8 = ~P/Pa. Theparticle velocity is given by ~ times c, the velocity of light, where ~ can be calculated interms of the particle energy E and momentum p. The energy, in turn, can be expressed interms of the momentum so that we can derive the following sequence of expressions.206 D. C. CAREY, R. V. SERVRANCKX AND K. L. BROWNpe 2v=~e=­EJm2e4 + p2e2poe 2(1 + 8)(17)Jm2e4 + Po2e2(1 + 8)2If we now take the quotient !iv/vo and eliminate Po and m using the ~ and y of thereference trajectory, we derive!iv 1 + 8 . _ 1Vo J1 + 2P82 + p282 .Expanding the denominator using the binomial theorem finally yields(18)(19)The complete expression for longitudinal separation of two particles now becomes( 8 ) 8 [ 1 3 (P)2] 2I = (L - L o) 1 - y2 - L o y2 + Lo y4 + 2 Y 8. (20)For the non-relativistic case, only those second-order terms involving the fractionalmomentum deviation 8 need be modified. We now label with a subscript R, forrelativistic, the second-order matrix elements for path-length difference. In terms of therelativistic matrix -elements, the matrix elements for the longitudinal separation aregiven by1(1Ixo8) = (1Ixo8)R - (llxo) 2"y(llxo'8) = (llxo'8)R - (llxo')~y(l18 2 ) = (l18 2 )R - (118) y\ + Lo [:4 + ~ (~YJThere is also a single additional longitudinal second-order matrix elementREFERENCES1. W. G. Davies, Nucl. lnstrum. Methods 169, 337 (1980).2. R. H. Helm, SLAC Report No. 24, 1963.(21)(22)SECOND-ORDER PATH LENGTH TERMS 2073. H. Matsuda, T. Matsuo, D. Ioanoviciu, H. Wollnik, and V. Rabbel, private communication'.4. 1. F. Streib, HEPL Report No. 104, Stanford University, Stanford, California, 1960.5. Karl L. Brown, SLAC Report No. 75, 1982.6. Karl L. Brown, SLAC-PUB-2257, 1979.7. David C. Carey, Nucl. Instrum. Methods 189, 365 (1981). R. V. Servranckx, PEP Technical Memo,PTM-249, 1982.8. K. L. Brown, D. C. Carey, Ch. Iselin, and F. Rothacker, SLAC Report No. 91, NAL Report No. 91,CERN 80-40.9. D. C. Carey, NAL Report No. 64, 1978.10. R. V. Servranckx, DIMAT.

Second Order Path Length Terms For A General Bending Magnet

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