This paper defines the beta function and other linear orbit parameters using
the exact equations of motion. The orbit functions are redefined using the
exact equations. Expressions are found for the transfer matrix and the
emittances. Differential equations are found for the beta function and the eta
function. New relationships between the linear orbit parameters are found.Comment: 14 pages, gzipped postscript paper (120k

Parzen, G.

English

arXiv

This paper defines the beta function and other linear orbit parameters using the exact equations of motion. The {beta}, {alpha} and {psi} functions are redefined using the exact equations. Expressions are found for the transfer matrix and the emittance. The differential equations for {eta} = x/{beta}{sup 1/2} is found. New relationships between {alpha}, {beta}, {psi} and {nu} are derived

UNT (University of North Texas) Digital Library

1995 Pxticlc Accelerator Conference and BNL-61170 c CQ n4- 939.5 la- d! International Coiifcrence on High-Energy Accelerators May 1-5, 1995, Dallas, Texas LINEAR ORBIT PARAMETERS FOR THE EXACT EQUATIONS OF MOTION* . .  . I . G. Parzen, Brookhaven National Laboratory, Upton, NY 11973 USA Abstract This paper defines the beta function and other linear orbit parameters using the exact equations of motion. The p, Q and .il, functions are redefined using the exact equations. Expressions are found for the transfer matrix and the emit- tance. The differential equations for = x /p' l2 is found. New relationships between a, /3, 7c, and Y are derived. I. INTRODUCTION This paper defines the beta function and the other linear orbit parameters using the exact equations of motion. The usual treatment [l] of the linear orbit parameters is based on the approximate equation of motion d2x ds2 - + K(S)X = 0 A5proximations are made in obtaining Eq. (1) which are usually valid for large accelerators. The exact linearized equations of motion can be written as x and p x  are the canonical coordinates in a curvilinear co- ordinate system based on a reference orbit and the Aij(s) are periodic in s with period L. The approximate Eq. (1) assumes that All = A22 = 0,  A12 = 1 and A21 = -K(s). The exact values of the Aij are given in [2]. A treatment of the linear orbit parameters based on the exact equations, Eqs. (2), rather than the approximate Eq. (1) may be desirable in the following situations: 1. Symplectic long term tracking using a procedure where the magnets are replaced by a sequence of point magnets and drift spaces. For the tracking to be sym- plectic, one has to use the solutions of the exact equa- tions of motion. The linearized equations of motion then have the form of Eq. (2). 2. Small accelerators where the approximations made in deriving Eq. (1) may not be valid. Many of the results found using the approximate equa- tions carry over for the exact equations. A few of the changed results are the following: *Work performed under the auspices of the U.S. Department of Energy. (3) u = Gl l C  ds- A 12 P where C is the circumference of the accelerator. I 11. EIGENFUNCTIONS O F  THE EXACT LINEAR EQUATIONS OF MOTION AND THE LINEAR ORBIT PARAMETERS The problem now is, given the exact linear equations of motion, Eq. (2), how does one define the linear orbit pa- rameters p, a, 7,.il,, u and the emittance E ,  and what are the relationships that hold between them. To do this, one has to repeat the well known treatment of the linear orbit parameters, and see where the definitions and relationships change for the exact equations. The treatment given below is believed to reduce the amount of algebraic manipulation required, and makes few assumptions about the Aij coeffi- cients in the linear equations. For the x motion, the linear equations are written as The transfer matrix M ( s ,  SO) obeys z =  M ( s ,  S 0 ) X O  = (p:) d -M = A M  ds One may note that the symbol x is used in 2 different ways. The meaning of x should be clear from the context. The matrix M is symplectic as the equations of motions are derived from a hamiltonian. [1,3] Thus M% = I N %  - M = S M S  s = ( 0  -1 0 I ) ,  I = ( ;  ;) u S is the transverse of S. Also /MI = 1; 1MI is the determi- nant of M. The one period transfer matrix is defined by n;r(s) = M ( s  + L, s )  (7) DISCLAIMER This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or use- fulness of any information, apparatus, product, or process disclosed, or represents that its usc would not infringe privately owned rights. Reference herein to any spe- cific commercial product, process, or service by trade name, trademark, manufac- turer, or otherwise does not necessarily constitute or imply its endorsement, recom- menddon, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof. DISCLAIMER Portions of this document may be illegible in electronic image products. Images are produced from the best available original document. c where L is the period of the A;j in Eq. (4). One can show that &(s) and &(SO)  are related by where X I ,  2 2  are two solutions of the equations of motion. Eq. (17) corresponds to the Wronskian in the treatment of the approximate equations of motion. For X I  and 2 2 ,  we use the two eigenfunctions x1 and x2 = 2; .  k(S) = M ( s ,  so)f i (so)M(so,  s) (8) The eigenfunctions and eigenvalues of M(s)  are defined by I .  x1 = (2) For X I  one uses Eq. (16) and for pz l  one finds from the ('1 equations of motion 1 dxl  P x l  = A12 (x -A I P )  where mjj are the elements of M, and using ]MI = 1. Eqs. (9) shows that the two eigenvalues A I ,  A2 obey A l A 2  = 1, and for stable motion, 1x1 = 1 and A2 = A;, and we can write A1 = exp(ip) (11) Given the eigenfunction at so, x l ( s 0 )  one can find the eigenfunction or any other point s using 2i dll, A , 2 p d ,  w =  The beta function p is normalized by normalizing the eigenfunctions so that N* W =xl Sxl = 2i and x1(s) has the same eigenvalue AI. This follows from Eq. (9), using Eq. (8) to relate G(s)  and &(so). Also x1(s) obeys the linear equations of motion, which gives (22) d* A12 -- -d s -  /3 Eq. (22) replaces the familiar result @/ds = 1/p which is obtained when A12 = 1. From Eq. (22) one can find a result for the tune. Using 27rv = $(C) - $(O) where C is the circumference of the accelerator, one finds which follows from Eq. (12) and Eq. (5). One can show that a(.)/.\;'L = fl(S), (14) where fl(s + L )  = f l ( s ) .  This follows from From Eq. (19) we now find for p, l ,  1 pxl = -(i-a)exp(ill,) p' 12 (24) Thus, one can write 2 1  (SI = exp(ips/L)f1(.) fl(.+L) = fl(S) Eq. (15) can be rewritten as q(s) = ~ ( s ) l / ~  exp(i+t) 4(4 = P / L + S l ( S )  (16) Sl(S + L) = g1(s), P ( s  + L) = P ( s )  Eq. (16) defines the beta functions, /3(s), except for a normalization multiplyer, for the eigenfunction x1 (s). The normalization multiplyer will be defined below. It will be shown first that II, and /3 are related. To find this relation, one uses the Lagrange invariant 111 w =& S X l  (25) Eq. (25) provides the new definition for the a parameter, which replaces the familiar result a = -+dp/ds. At this point the definition of (Y may seem arbitrary. It will be seen to be the convenient definition of a when the emittance and transfer matrix are considered. The eigenfunctions can now be written as, using Eq. (16) and Eq. (25), 2 2  = 4 For the results for the emittance and transfer matrix, see P I .  111. DIFFERENTIAL EQUATIONS FOR THE LINEAR ORBIT PARAMETERS This section finds differential equations for p, and q A .  Second Order Digerential Equation for x From the first order differential equation for z , p , ,  Eq. (4), one can eliminate p ,  to find a second order equation for x .  See [2] for details It has been assumed that A11 = -A22. B. Diflerential Equation for ,O put the eigenfunction To find a differential equation for p, into Eq. (27) for I I = b exp(i$) b = p1I2 (28) We find then, see [2] for details, (29) Eq. (29) is a second order differential equation for b = B1l2. It can be compared to the result found when A12 = 1 and A11 = 0, d2b A12 ---- -0 ds2 b3 C. Diflerential Equation for q q and x are related by 2 = b q,  b = pl/ ’  (31) In the differential equation for q the independent variable is $ or 6 which are related to s by We find d z l d s  and d(ATidx /ds ) /ds  which are then sub- stituted into Eq. (27) to get the equation for q, using Eq. (29) to eliminate derivatives of b. This gives, see [2] for details, (33) d2 v -+ u2q = 0 dQ2 The differential equation for q is unchanged. IV. PERTURBATION THEORY USING THE DIFFERENTIAL EQUATION FOR q The equation for q,  Eq. (33) is often used as a starting point in finding the effects of a perturbing field. The par- ticle coordinates are measured relative to a reference orbit which is the particle motion in a known magnetic field with components Bi.  The exact equations of motion can then be written as dxi - ds = C A i j z j  + fi i = 1 , 4 , j  = 1 , 4  (34) where the f i  includes all the terms not included in E A i j x j .  These include terms due to fields not included in the refer- ence field Bi, which may be referred as ABj, and nonlinear terms due to the terms in the exact equations of motion that do not depend on Bi. One can see from the exact equations of motion, that the contributions to fj which depend explicitly on ABi,  when AB, = 0, are given by 1 f 2  = - (1+x /p )ABy  BP Repeating the above derivation of Eq. (33) for 77, including the fi terms, one finds the q equation for the 2-motion A similar equation can be found for the y motion, ( f3 ) (37) A33 f;( = f 4  + -f3 + - -A34 ds A34 For the case of a gradient perturbation one can use Eq. (36) to find the change in ux,Avx. One finds (39) This well known result for Av, is not changed by using the exact linear equations. References [l] E.D. Courant and H.S. Snyder, Theory of the Alternat- ing Gradient Synchrotron, Annals. of Physics, Vol. 3, p. l(1958).  123 G. Parzen, Linear Orbit Parameters for the Exact Equations of Motion, BNL Report AD/RHIC-127, BNL-60090 (1994). 131 F.T. Cole, Notes on Accelerator Theory, MURA report, TN-259 (1961). 

Linear orbit parameters for the exact equations of motion

George Parzen

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5 Linear Orbit Parameters for the Exact Equations of Motion*

This paper defines the beta function and the other linear orbit parameters using the exact equations of motion. Then may be desirable in the following situations: (1) symplectic long term tracking using a procedure where the magnets are replaced by a sequence of point magnets and drift spaces. For the tracking to be symplectic, one has to use the solutions of the exact equations of motion; (2) small accelerators where the approximations made in deriving the approximate equation of motion may not be valid

http://arxiv.org/pdf/acc-phys/9508001.pdf

Linear Orbit Parameters for the Exact Equations of Motion

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