v FIRST ORDER STUQY G? SGME BEAM ANALYZWG SYSl’EMS 50R A MEMUM ENERGY CYCLOTRQN Thesis For the Mgr“ of M. S. MiCHIGAN STATE UNEVERSSTY Kai Kasai: i962 WWII!“lllllllllllfllllll’ll!)HIIIHHIINHIWHIIHI 301764 0396 LIBRARY Michigan State University ABSTRACT FIRST ORDER STUDY OF SOME BEAM ANALYZING SYSTEMS FOR A MEDIUM ENERGY CYCLOTRON by Kei Kosaka In order to have a useful output some distance away from an accelerator, it is necessary to use magnetic systems which focus, bend, and resolve the beam. When such systems are compounded, it is cumbersome to try to hand calculate the beam properties. A computer program is presented here which calculates the first order optical properties of any combi— nation of such systems and gives the combined effect of these systems on the beam properties. Results are presented for several typical systems of interest in handling cyclotron beams. FIRST ORDER STUDY OF SOME BEAM ANALYZING SYSTEMS FOR A MEDIUM ENERGY CYCLOTRON BY Kei Kosaka A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Physics and Astronomy 1962 35} ////L r/I “WM ACKNOWLEDGMENTS I would like to express my appreciation to Dr. H. G. Blosser and Dr. W. Johnson of this laboratory for their helpful suggestions. I am grateful to the United States Atomic Energy Commission and the National Science Foundation for making this work financially possible. Also, I would like to thank R. L. Dickenson for his work on the graphs. ii TABLE OF CONTENTS Page INTRODUCTION . . . . . . . . . . . . . . . . . . . l I. EQUATIONS OF MOTIONS . . . . . . . . . . . . 5 A. The Magnetic Quadrupole 6 B. Bending Magnets 13 C. Edge Effect for the Bending Magnet 21 D. Relative Orientation and Separation 28 E. Relation of Beam Properties to the Matrix Describing the Focusing Systems 30 II. THE COMPUTER PROGRAM FOR CALCULATING THE BEAM PROPERTIES . . . . . . . . . . . . . 34 III. RESULTS . . . . . . . . . . . . . . . . . . . 40 A. Single 900 Bending Magnets 40 B. Single Quadrupole Magnet 44 C. Bending Magnet Pairs 44 D. Quadrupole and Flat Bending Magnet Combinations 68 IV. APPENDIX . . . . . . . . . . . . . . . . . . . 77 A. Flow Chart for the Computer Code 78 B. Order Pairs for the Computer Code 79 REFERENCES . . . . . . . . . . . . . . . . . . . . . 107 iii Figure 1. LIST OF FIGURES Page Magnetic quadrupole geometry. d is the aperture. and L is the length of the quadrupole . . . . 6 Edge effect of the bending magnet. The central ray enters from the right and exits to the left, with angles Bl and 52, respectively, away from the normals to the entrance and exit edges of the magnet . . . . . . . . . . . . . . . . . . 22 Geometry of non-normal entry into a bending magnet . . . . . . . . . . . . . . . . . . . . 26 Diagram showing the relative positions of the object plane, focal planes, principal planes, image planes, and the focusing region . . . . 3l Radial resolution, radial image distance, and radial magnification of a single n I 0, p = 36 in., a = 36 in., a = 90° and 61 = 52 = O0 bending magnet plotted against the object distance . . . . . . . . . . . . . . . 41 Radial resolution, radial and axial image distances, and radial and axial magnification of a single n = 1/2, p = 36 in., a = 90° and $1 = 52 = 00 bending magnet plotted against the object distance . . . . . . . . . 43 Radial and axial focal lengths and distances focal length away from the focal plane of a single magnetic quadrupole plotted against K, where K was defined to be K2 E'gg and has units of in-l. . . . . . . . . 45 0 iv Figure Page 8. Radial resolution of a system consisting of two n = 0, p = 36 in., a = 90°, and B = B = 00 bending magnets oriented so that 9 = 0°. Plotted against the object distance in inches for the separation distances d = 10 in., 60 in., 110 in., and 160 in. . . . . . . . . . 47 9. Radial resolution for two n = 1/2, p = 36 in., a = 90°, and B = 62 = 00 bending magnets oriented so that G = 00. Plotted against the object distance for the separation distances d = 10 in., 60 in., 110 in., and 160 in. . . . . . . . . . . . . . . . . . 49 10. Radial and axial resolutions for two n = 0, p = 36 in., a = 90°, and Bl = 52 = 00 bending magnets oriented so that 6 = 450. The axial resolution is plotted with dashed lines. Plotted against the object distance for the separation distances d = 10 in., 60 in., 110 in., and 160 in. . . . . . . . . . . . . . 50 11. Radial and axial resolutions for two n = 1/2, p = 36 in., a = 90°, and 61 = 62 = 00 bending magnets oriented so that 9 = 450. Plotted against the object distance given in inches, for the separation distances d = 10 in., 60 in., 110 in., and 160 in. Axial resolutions are plotted with dashed lines . . . . . . . . 52 12. Radial and axial resolutions for two n = 0, p = 36 in., a = 900, and 31 = B2 = 00 bending magnets oriented so that G = 90°. The axial resolution is plotted with dashed lines. Plotted against the object distance for the separation distances d I 10 in., 60 in., 110 in., and 160 in. . . . . . . . . . 53 13. Radial and axial resolutions for two n = 1/2, p = 36 in., a = 90°, and 51 = 62 = O0 bending magnets oriented so that 9 = 900. Plotted against the object distance given in inches, for the separation distances d = 10 in., 60 in., 110 in., and 164.16 in. Axial resolutions are plotted with dashed lines . . 54 V Figure Page 14. Radial and axial resolutions for two n = 0, p = 36 in., a = 90°, and °1 = B2 = 0° bending magnets oriented so that 9 = 135°. The axial resolution is plotted with dashed lines. Plotted against the object distance for the separation distances d = 10 in., 60 in., 110 in., and 160 in. . . . . . . . . . . . . . . . 56 15. Radial and axial resolutions for two n = 1/2, p = 36 in., a = 90°, and B1 = B2 = 0° bending magnets oriented so that O = 135°. Plotted against the object distance given in inches, for the separation distances d I 10 in., 60 in., 110 in., and 160 in. Axial resolutions are plotted with dashed lines. . . . . . . . . 57 16. Radial resolution of a system consisting of two n = 0, p = 36 in., a = 90°, and B1 = B2 = 0° bending magnets oriented so that 6 = 180°. Plotted against the object distance in inches for the separation distances d - 10 in., 60 in., 110 in., and 160 in. . . . . . . . . . . . . . 58 1?. Radial resolution for two n = 1/2, p = 36 in., a = 900, and B = B2 = 0° bending magnets oriented so that e = 180°. Plotted against the object distance for the separation distances d = 10 in., 60 in., 110 in., and 160 in. . . . . . . . . . . . . .. . . . . . . 59 18. Radial magnification for the two n = 0, p = 36 in., a = 90°, and B1 = B2 = 0° bending magnets oriented so that 9 = 0° or 180°. Plotted against the object distance given in inches for the separation distances d = 10 in., 60 in., 110 in., and 160 in. . . . 60 19. Radial and axial magnifications for the two n = 0, p = 36 in., a = 900, and 61 = 62 = o0 bending magnets oriented so that 9 = 45° or 135°. Plotted against the object distance given in inches for the separation distances d = 10 in., 60 in., 110 in., and 160 in. Axial magnifications are plotted with dashed lines . . . . . . . . . . . . . . . . . 61 vi Figure Page 20. Radial and axial magnifications for the two n = 0, p = 36 in., a = 90°, and B = B = 00 bending magnets oriented so that 9 = 90°. Plotted against the object distance given in inches for the separation distances d = 10 in., 60 in., 110 in., and 160 in. Axial magnifications are plotted with dashed lines . . . . . . . . . . . . . . 62 21. Radial and axial magnifications of two n = 1/2, p = 36 in., a = 90°, and B1 = B2 = 0° bending magnets with any relative orientation plotted against the object distance given in inches for the separation distances d = 10 in., 60 in., 110 in., and 160 in. . . . . . 64 22. Radial image distance for the two n = 0, p - 36 in., a = 900, and B1 = B = 0° bending magnets oriented so that 6 = 0° or 180°. Plotted against the object distance given in inches for the separation distances d = 10 in., 60 in., 110 in., and 160 in. . . . . . . . . . . . . . 65 23. Radial and axial image distances for two n = 0, p = 36 in., a = 90°, and B1 = = 0° bending magnets oriented so that 9 = 45° or 135°. Plotted against the object distance given in inches for the separation distances d = 10 in., 60 in., 110 in., and 160 in. Axial image distances are plotted with dashed lines . . . 66 24. Radial and axial image distances for two n = 0, p = 36 in., a = 90°, and B1 = B2 = 0° bending magnets oriented so that 6 = 900. Plotted against the object distance given in inches for the separation distances d = 10 in., 60 in., 110 in., and 160 in. Axial image distances are plotted with dashed lines . . . 67 25. Radial and axial image distances for two n = 1/2, p = 36 in., a = 90°, and B1 = B2 = 0° bending magnets with any relative orientation, plotted against the object distance given in inches for the separation distances d = 10 in., 60 in., 110 in., and 160 in. . . . . . . . . . . . . . 69 vii Figure Page 26. Diagram showing the relative positions of the magnets in a system containing two n = 0 bending magnets and one axially focusing magnetic quadrupole . . . . . . . . . 70 27. Radial and axial magnifications for the system with two n - 0 bending magnets and an axially focusing quadrupole as shown in Figure 26. Plotted against the object distance given in inches for quadrupole fields specified by K = .04 in.'1, .045 in.'1, .05 in.'1, and .055 in."1 The axial magnifications are plotted with dashed lines . . . . . . . . . . 71 28. Radial and axial image distances for the system described in Figure 26, plotted against the field strength of the magnetic quadrupole. These are lines of unit magnification. The axial image distance is plotted with a dashed line . . . . . . . . . . . . . . . . . 72 29. Diagram of a system with two n = 0 bending magnets and three magnetic quadrupoles, one of which is radially focusing and the other two axially focusing . . . . . . . . . . . . . 74 30. Radial and axial magnifications for the system as described in Figure 29, plotted against the object distance given in inches for the several field strengths of the radially focusing quadrupole as specified by K. The axial magnifications are plotted with dashed lines . . . . . . . . . . . . . . . . . 75 31. Radial and axial image distances for the system as described in Figure 29, plotted against the field strength of the radially focusing quadrupole. These are lines of unit magnifi- cation. The axial image distances are plotted with dashed lines .. . . . . . . . . . . . . . 76 viii INTRODUCTI ON The output beam from an accelerator consists of particles with momenta distributed over some range. In nuclear experiments, it is usually desirable to have a beam with small momentum spread, high intensity and small cross sectional area. Usually, the type of experiment and the quality of the beam from the accelerator make it expedient to sacrifice one or more of these beam properties to improve the third. It is possible, by using bending magnets, to disperse particles of different momenta, and produce a spectrum at the image plane. If the image slit is to just include all of the particles with some selected momentum, it must have a width equal to the magnification times the source slit width.* With such an image slit, and assuming the source slit is illuminated uniformly, the momentum distribution of the transmitted particles will be triangular with the peak *There is no advantage in making the slit narrower, since the transmitted intensity decreases while the width of the momentum distribution (measured at the usual half maximum intensity points) remains the same. If the slit is made wider, the width at half maximum, of course, increases. located at the selected momentum. The width of the triangular momentum distribution may be made smaller by either decreasing the magnification or increasing the momentum dispersion. Decreasing the magnifi— cation allows one to narrow the image slit and still pass through all of the particles with the selected momentum; increasing the momentum dispersion causes particles with momentum differing from the selected momentum, to be deflected through greater angles, so that some previously transmitted particles will now fail to pass through the image slit. Usually, for real systems, neither the magnification nor the momentum dispersion change independent of the other. The momentum spread across an image slit sized to just admit all particles with a selected momentum, is a well known figure of merit--designated resolution—-which com- bines the effect of both magnification and momentum dispersion. In the calculations that follow, the fractional half base width of the triangular momentum distribution at the image slit (equivalent to the fractional full width at half maximum of the distribution), is calculated. This quantity, calcu- lated for a unit source slit width, is defined as the resolution. For different systems, the resolution depends on the way in which the magnets are positioned, on the type of bending magnets used, and/or on the number of various magnets which compose the complete system. The computer program presented here enables one to study this property for various simple and combined systems, as well as to study the more familiar optical properties, and to make useful comparisons between them. Equations of motion for first order theory of the bending magnets were first developed in connection with magnetic spectrometers. Such a study of magnets giving general expressions for image distance, astigmatism, magnifi- cation, solid angle, dispersion, and resolution has been made by Judd.l* The basic equations are, however, essentially the same as the betatron equations developed by Kerst and Serber.2 The first order equations for the magnetic quadrupole developed here are much like those used by Enge.3 In such a first order treatment, the equations of trajectory are, of course, by definition linear in the initial displacement, angular spread, and momentum spread. This enables one to set up the problem in matrix form, which is convenient when several magnets are considered in series. The matrix method used here is based on the formalism developed by Penner. *References are listed at the end. The magnetic systems considered herein are the bending magnet and the magnetic quadrupole lens. Nonmagnetic systems such as the electrostatic quadrupole lens are not considered, since excessively high fields would be required in the energy range of interest (2 40 Mev protons). It would not be difficult to handle the electrostatic quadurpole in the computer program, however, since the equations are exactly the same as the magnetic quadrupole except that the force constant has to be redefined. In the examples calculated, comparisons are made between the performance of double focusing and flat field bending magnets. Combined systems focusing radially and axially have also been worked out using flat field bending magnets and magnetic quadrupoles. Such combined systems are attractive in that they combine double focusing with the usual advantages of flat field bending magnets, i.e., such magnets can easily be precisely stabilized by the use of a nuclear magnetic resonance probe. I. EQUATIONS OF MOTION In setting up a sequence of quadrupole lenses and bending magnets to form an analyzing system, one, of course, designs with respect to some particular momentum value: all of the lenses and bending magnets are positioned and their strengths adjusted such that a particular ray of the specified momentum follows a central path through the system. This path is designated the "optic axis" of the system. An arbitrary trajectory is specified at any point along the optic axis by giving perpendicular displacements (x, y) from the optic axis in two independent directions (customarily at right angles to each other), the corresponding conjugate, momentum (px, py), and the momentum difference (Ap = |§1 - IEBI) between the trajectory in question and the optic axis. Following the procedures of Penner4 and Livingood,5 equations of motion accurate to first order in these displacement coordinates are derived in the following subsections for the magnetic quadrupoles and bending magnets. In each case the matrix formulation is specifically exhibited. A. The Magnetic Quadrupole The geometry of the quadrupole is roughly shown in Figure l, the z axis in the figure being the optic axis. if Y ._ *— E//// Figure 1. Magnetic quadrupole geometry. d is the aperture and L is the length of the quadrupole. Hyperbolic equipotential lines, for field strengths less than saturation, are made possible by making the poles rectangular hyperbolic cylinders which are symmetrical about the x and y axes. If adjacent poles have opposite polarity, the gradients of the field components are constant and the force on the particle is proportional to the displacement from the z axis. The equipotential lines are hyperbolas expressed by the equation V = ny (l) where G is a constant to be evaluated. There is no field component along the z axis within the quadrupole (at the edges, it exists only over a small range compared to the length of normal quadrupoles), so that we need only to consider the other two components Q! Bx = dx = Gy B =IQ! = Gx y dy The gradients of these components are de —— = G dy = G 0.: Q: *Lw These are equal and positive, since BX is proportional to y and By is proportional to x. The x and y components of the Lorentz force relation F. = q V x‘B are FX =-q v B = - q vz Gx (2) q V Gy (3) F = q V B z y z x where the beam is moving parallel to the z axis. These equations show that with this choice of orientation of poles with respect to the x and y axes, the x component of the force drives the particle toward the z axis while the y component drives it away. It is not difficult to see that if the orientation were to be changed by a 900 rotation about the z axis, this situation would be reversed. The equation of motion in the x-z plane is from equation (2) since trajectories are considered only in magnetic fields, there is no energy change of the particles, and hence, no change in the velocity or mass of the individual particles. This makes it possible to take out the mass from the derivative in the above equation, i.e., 2 d x m'——§ = - q vz Gx dt This can be rewritten by noting that 2 6—5:.93—(Eaéz) d“:2 dtdtdz and n-g_z_g__=vg_ dt dt. dz 2 dZ as .93_ d_x__ mvz dZ (V2 dZ > _ q Vz Gx ’ (4) Since v 2 1/2 V = .V;2 "V 2 = v [:1 —(-E- + :] z x v .... equation (4) can be written to first order in x and vx as: Regrouping of the constants gives 2 §_§_ +-g§ x = 0 (5) 62 p where p is the magnitude of the particle momentum. In terms of the reference momentum, p0, equation (5) can be written and by defining K2 , 2 -l d g + K2 (:l.+é¥i) x:= o 0 dz Ill '0 o 1% r1. 3 m n which has a solution -1/2 -l/2 x = A cos K 1 + éR' Z + B sin K 1 + AE- Z po ' po Expanding the arguments of the trigonometric functions in a binomial series and keeping only terms of first order and lower, 1 x = A cos K (1 -'§ 0'“)? Z+BSinK(l-% .015 N Rewriting the trigonometric functions in terms of the sum of the angle relation x = A (cos K Z cos l'éE K Z + sin K Z sin l'éEK Z) 2 pO 2 p0 A A + B (sin K Z cos l'—EK Z - cos K Z sin"]""—E K Z) 2p 2p o o . . . . . QB . . Expanding the Sine and COSine functions which have in their 0 arguments, and only keeping terms of first order and lower A A x = A (cos K Z +-l'—E K Z sin K Z) + B (sin K Z - l'-2 K Z cos K Z) 2 pO 2 p0 whose derivative ' x" =°-3-=A (-K sinKz+—l-é9KsinKz+l-é91<2Zeous) dz 2 pO 2 po 2 , 1 . + B (K cos K Z -'3 K cos K Z +‘% K Z Sln K Z) .0»? .0 P3 10 along with the initial conditions, x = x0, x' = x; when 2 = 20 = 0, makes it possible to evaluate the constants. . x' o . These turn out to be A = x and B = - Expanding o 11.32 K(1—— ) 2 p0 B in a binomial series and only keeping first order or lower , I x A . terms, B = i2'(1 + %--E). Dropping all terms containing, po - AP. .92- . either of the products xO p or xO p , in the equation for o o x and x , and evaluating at Z = L, the terminating point of the quadrupole, gives, finally: ssh-4 x = x0 cos KL + x sin KL Q. (7) x' = -x0 K sin KL + x; cos KL Equation of motion in the y-z plane is by equation (3) g—'m° - v G dt Y g Y In similar fashion to the previous calculation, all terms containing Ap can be shown to be of second order. The linear equation of motion, hence, reduces to .Q_X - K2y = 0 (8) where K is defined in the same way as before. This has a solution y = C cosh KZ + D sinh KZ 11 whose derivative y' = “g? = CK sinh KZ + DK cosh KZ along with the initial conditions, y = yo and y' = y; at z = . Y2. zO - 0, yields C - Y0 and D = K , so that 1 = + I _. ° L y yO cosh KL yO K Slnh K (9) y' = y0 K sinh KL + y; cosh KL where, once again, L is the effective length of the quadrupole. It is possible to write both equations (7) and (9) in matrix form, since they are all linear equations. Equation (7), for example, becomes: ( \ K 1 ‘ -’ \ x cos KL '— sin KL x K o x' -K sin KL cos KL x' . \ ) \ / \ 0/ This arrangement has the attractive advantage that all information regarding the quadrupole is contained in the matrix entirely independent of any particular set of initial conditions. The effect of the quadrupole on any arbitrary trajectory is found simply by multiplying the phase space coordinate vector at entry by the matrix to obtain the phase space vector at exit. In similar fashion to (7), equation (9) can be written / \ f y cosh KL XIH y' K sinh KL \ I \ It is convenient for with the bending magnets, to make a single six by six matrix which includes 12 sinh KL y L I cosh K \Y°J calculations made in combination combine these two matrices to spread as the third and sixth coordinates. It has been shown, above that the coefficient in front of the is to first order zero. Hence, E term of the transformation matrix can be set equal to zero, the momentum the third and sixth columns except for the unit transformation coefficient which accom— panies each momentum coordinate. /\ K 1 . x cos KL E'Sln KL 0 0 0 x' -K sin KL cos KL 0 O 0 A —9= 0 0 1 0 0 pO 1 . y 0 0 0 coshKLESinhKL y' 0 0 0 Ksinh KL cosh KL $2 0 0 0 0 0 L9 \ \ O l J The combined matrix is / X P 1% \°/ \ O (10) Note that if the field direction is reversed or if the quadru- pole is rotated by 900, the same condition is achieved by 13 interchanging the two submatrices, i.e., F, N p l \ ./ \ x cosh KL E'sinh KL 0 0 0 0 x0 x' K sinh KL cosh KL 0 0 0 0 x; 93 0 0 1 0 0 0 92 PO po = 1 (11) y 0 0 0 cos KL E'Sln KL 0 yO y' 0 0 0 -K sin KL cos KL 0 y; $9— 0 0 0 0 0 1 $2 \ QJ \ ,J \ 9’ Equation (10) is converging in the x—z plane and diverging in the y-z plane. The reverse is true for equation (11). B. Bending Magnets Consider, first, the axial motion. Use cylindrical coordinates with origin at the center of curvature of the optic axis of the magnet. Within the bending magnet Be is zero, and edge phenomena will be treated separately later, so that the axial equation of motion is: mz = - q V Br (13) §_. dt 9 we are interested in the first order terms in the variables expressing variation from the particle following the optic axis. These variables are taken to be the l4 displacement coordinates x = r - rO and z, and their conjugate momenta pX and pZ and the total momentum displace- ment Ap = '5' - [55'. First, expand Br about 2 = o, and v in terms of v and v 9 x z 2 2 d l Vx + Vz dt mz = -q v 1 — V2 + - .. Br (r,z) + z=o [531i] 22 d2B Z —_ + -—— + I o 0 dz 2! [6122] 2:0 z=o Keeping only the linear terms in the variables mentioned above d [dBr amz=-qv Br(r,z) 4-ng- zzo z=o ll 0 Since the field is symmetrical about 2 = 0, [Br (r,z)] z=o Using the fact that the curl of B is zero for a source free region _C_3_ mZ = — q V ZEBZ (r’zfl d dt r z=o Expanding the field derivative about r = r0, and since the mass is constant in a magnetic field 2 ' dB (r,z) d2B (r,z) 2=_.g._v__z [Z ] +X Z 4-... p dr dr2 r=rO r=rO z-o z=o Expanding p in terms of Ap and v in terms of AV, which is 15 related to Ap, and keeping only linear terms q v2 dB (r,z) u o z z = — Z (14) p0 dr r=rO z=o The momentum of central particle, i.e., r = r0, 2 = o and Ap = o is given by p.=-qr.[Bz] _ r-rO z-o which when substituted in equation (14) gives v2 dB (r,z) 2 = 0 Z z rO [B2] dr r=rO r=r z=o z=o v When an is defined as (0 E';9 o 2 r0 dB (r,z) z -u) 2 [Ba] dr r=rO r=rO z=o z=o Define a field index as 3B (r,z) z _ -—r z n (r) — [B (r,z)] Br (15) z z=o z=o Using this and (to conform with the notation of the previous section) writing y in place of 2, as the axial coordinate of the cylindrical system, the equation of motion simplifies to 16 H 2 y +w n (r0) y = o (16) This has a solution, 1/2 y = A cos n 1/2 wt+Bsinn not If a is taken as the angle through which the magnetic field is effective, 1/2 y = A cos n 1/2 a + B sin n a . Z Note that a can also be written';, so that .1/2 1/2 y' = fix = - An sin nl/2 a + Bn cos nl/2 0 dz p p With the initial conditions, y = y0 and y' = y; = 0, the _ = _Q_ constants are A — yO and B nl/z, and _. 1/2 . ____Q__ . 1/2 y — yO cos n a + yO nl/2 Sln n a (17) . nl/2 . 1/2 . 1/2 y = -y0 Sln n d + yO cos n d Equation (17) can be rewritten in a single three by three matrix equation. As in the case for the magnetic quadrupole, the momentum terms appear only in second or higher order, so that zeros may be placed for the coefficients of the momentum term. l7 / \ / \ /' \ y cos nl/2 d —£-— sin nl/2 a 0 y n1/2 o ' - — nl/Z sin nl/2 a cos nl/2 a 0 ' Y _ yo (18) A A BE 0 0 1 BB K QJ \ ./ K °/ Returning now to conventional custom,the axial coordinate of the cylindrical system is again written as Z, and the radial motion is considered. Since Be is zero, the radial component of the Lorentz force is Again the mass is independent of time and can be taken out of the time derivative of the momentum and the radial acceleration Ve can, hence, be written as f - r . The equation of motion is then 2 v q v B .0 e e r m Again, the equation is expanded in terms of the deviation from the optic axis and Ap. First, expanding v9, B , and r, z 2 V2 vx2 + vz2 x —(r +X)_——— l_ l_—-+ co. = dt2 0 r0 v2 r0 ‘3 1 vx2 + vz2 dB mvl-S V .. [32] Ha— 1.. 18 Keeping only the linear terms in the deviation from the optic axis, v2 x de Si-—<-——>=9-v [3] +x (20) r r m 2 dr r=r r=r O O Expand p, in terms of Ap and v in terms of the corresponding Av, ( )2 3{-_Vo+AV = r r O P 2 A de £‘~—(v +Av) 1- —E +--- B +x —— po 0 0 Z r If only the linear terms in the deviations from the central momentum and the optic axis is kept, this becomes 2 2 v v x 2 v Av 00 O O ___Q___ X — _—_ + 2 — r = r0 r o (21) o 2 2 de 2 A .g_ v [B:] + .g_ v x a——' - gt'v '—2 [B:] + pO 0 2 pO 0 r pO 0 pO 2 r=r r=r r=rO o o 2qv Av ° [le po r=r 0 V0 Using the relations p0 = -q rO [B°] and QJE E‘» r=ro 0 one can form the relations, l9 r=r v qv ran r=r o _ o . o o , and a): - o — I — - r B po 0 p0 [ 2] p0 r=r o r dB is + wzx - r0032 £2 + (sz—Q [—2-] = 0 (22) kaing use of the field index defined by equation (15) and writing p in place of rO yields §E+w2(1-n)x=pw2-A-E (23) po The general solution to this equation is x = i—g—E: %E~+A cos co (1 — n)]'/2 t + B sin u)(l - n)l/2t o = -—£——- A2' + A cos (1 - n)l/2 a + B sin (1 - n)l/2 1 - n pO a where a is the angle through which the magnetic field is effective. Examine the solution to the equation § +co2 (1 - n) s = 0 which is the same as equation (23) except that the right side has been set equal to zero. A solution to this equation is /2 1/2 . 1 s = A cos (1 — n) a + B Sln (1 - n) a 20 and the derivative is 1/2 s' = fig = —'Ll—:—°L A sin (1 - n)l/2 a + dz p 1/2 (1 - ) B cos (1 - n)l/2 a P The initial conditions, 3 = 80 and s' = s; at z = 20 = 0, s; p gives the constants, A = s and B = . Equations 0 1/2 (l-n) for s and s' are 98' 1 s = s cos (1 - n)l/2 a + 0 sin (1 - n) /2 a 0 1/2 (l-n) (1-- n)l/2 1/2 1/2 s‘ = - s sin (1 - n) d + 3' cos (1 - n) P 0 0 which in matrix notation can be written / / \/\ SW cos (1 - n)1/2 a -__—E_—-l/2 sin (1 - n)l/2 a s = (l-n) O g1 — n)1/2 . 1/2 1/2 Ls' - Sln (l - n) a cos (1 — n) a s' / \ P /\0/ Now change the variable, 8, to x. Since s = x - I—g—E AB po and s' = x', / n) / 12 1/2 x-i—‘g-a—B cos (l-n)/ a___2_1/2 sin (l-n) 0L po = (l - n) 1/2 1/2 x' _ (l n) sin (1 _ n)l/2 a cos (1 - n) \ / \ p X ____2_ AP. 0 l - n p o XI 0 3 21 Writing this out in terms of x and x', x = x0 cos (1 - n)l/2 a + x' ———£L—’ o (1 - n) 92' ——2—— (l - cos (1 - n)l/2 a) pO 1 - n 1/2 x' = - xo'il—E—El sin (1 - n)l/2 . 1/2 ép_ 1 1/2 Sln (l - n) a Po (1 - n) In matrix notation, this becomes \ (x F203 (1 - n)1/2 a ——_£———1/2 (1 - n) 1/2 x' = — Jl_%_fli sin (1 - n)l/2 AE' 0 \p0/ \ \ / '———Q—— [1 - cos (1 — n)l/2 % 1 - n l . 1/2 ——————— Sin (1 - n) a (1 - n)l/2 l J \ C. Edge Effect for the Bending Magnet AB P O 1/2 sin (1 " n)l sin (1 - n)1 a cos (1 - n)l / /2 a + x; cos (1 - n)1 O, /2 + O. (24) Consider now the effect of the edges of the magnet. In the general case, these may be non—normal to the central ray (Figure 2). + 22 fanny optic axis BhsinPl 19° \\ U” \\\\ .9 B B“ (:05 ‘ 2,. Figure 2. Edge effect of the bending magnet. The central ray enters from the right and exits to the left, with angles B1 and B2, respectively, away from the normals to the entrance and exit edges of the magnet. Consider, first the axial focusing effect of the edges. As long as 91 and 92, as shown in the figure, are less than 900, there is axial edge focusing. At any point displaced from the median plane, a component Bh of B exists due to the bulging field at the edge. Assume that this edge field exists over a very short range, and hence, assume that the force due to the field changes the direction of the particle but not its position. Using coordinates where the z - axis is along the optic axis, the axial equation of motion in the edge region is d._ _ . dt my — q (vX Eh cos Bl,2 vz Bh s1n 61,2) (25) Take points, 21 and 22, as in Figure 2, which are 23 outside the influence of the edge field Bh' Consider a rectangular path which lies in the y—z plane between 21 and 22. Let one side of the rectangle lie on the optic axis and the opposite side displaced by a distance, y, which is same as the axial coordinate of the particle as it passes through the edge region. Carrying a unit magnetic pole about this path, starting at z on the median plane, no work is l done in carrying the pole away from the plane through a distance y, since no field exists at 21. However, in going - z from 21 to 22 along the displaced path requires 1° 2 Bh z 1 cos B1 2 d2 of work. The work done in going back to the I median plane at z is -yB, but the work done in going from 2 22 to 21 along the median plane is zero, since the field is perpendicular to the median plane at all points. Since there are no sources in the loop, the total work done in going around the loop has to be zero, so that Z 2 z‘[’ Eh cos 61,2 dz =-yB (26) 1 Since the edge field has been assumed to extend only over a very short distance, change in the coordinates in crossing the edge can be neglected. Hence, the contour of (26) corresponds to the particle trajectory. 24 Going back to equation (25), the first term on the right is second order, since Bh is proportional to y by equation (26). The mass is, again, independent of time and may be taken out of the time derivative. §—-y may be written dt A. 911 as Vz (dz dt)’ so that L'_s . dz y ’ m Bh 51“ B1,2 Integrating this equation along the particle trajectory in the edge region, [22 22 21 d (y) m Sin 61,2 21 Bh dz m tan Bl,2 z I 2 z1 Bh cos B1,2 dz Using equation (26), this becomes Z2 y = .g yB tan B1 2 Z1 but y = QZ.QZ' = v y' and as before v = v to first order- dt d2. 2 I I z I» . so that Z2 I = 9— y mv y B tan Bl,2 21 Consider a deviation from the central momentum, and 25 expand in terms of Ap, 2 A = g__ 1 H —B' + ... y B tan B P 1,2 2 o 0 Keeping only first order terms in the small quantities, this becomes 1'2 = 'g— y B tan B po Using the relation, po =-q rO B _ _.X — 9 tan °1,2 . (27) 1 where rO has been replaced by p. Note that the equation holds for both entry and exit. Radially, the rotation of the pole edge tends to defocus the beam when 91 and 92 are less than 90°. Consider the entry into the magnet in Figure 3. DG is the central ray entering the magnet at E and AC is another ray entering at B. Assume that AB and DE are very nearly parallel, so that BHzx, and < JBA = < HBEzBl. Also assume that x is small enough so that EF==EH. When ray AC reaches B, ray DG from the same source has reached the point F. The angle ¢ 26 Figure 3. Geometry of non—normal entry into a bending magnet. 27 can be written BH sin B ¢=§§QE_BEsinH1 2!) and tance, is image radial radial magnification of a single n = 0, p = 36 Figure 5. t ing magne 90° and B1 = °2 = 0° bend plotted against the object distance. a: in., .42 aperture occurs because the beam dimension becomes larger than the aperture, due to the increase in object distance. The fractional half maximum width of the momentum distribution transmitted by a matched image slit is obtained by multiplying the resolution by the source slit width. For example, in Figure 5, the resolution has a value .0116 in ‘1 when the object distance is 50 inches. If the source slit width is .2 inches, the fractional half maximum width of the trans- mitted momentum distribution will be .232%. Figure 6 shows the properties of an n = 1/2 bending magnet with parameters otherwise the same as for n = o magnet. This magnet is completely double focusing, i.e., radial and axial optical properties are identical to each other. The focal planes are located 56.7 inches from the principal planes and 25.2 inches away from the magnet edges. When the object slit is placed a focal length in front of the fozal plane, we have unit magnification. The image distance is negative unless the object is placed, at least, beyond the focal plane. A parallel beam results, if the source slit is placed at the focal plane. Again the resolution improves with increasing object distance and is slightly better for a given object distance than that for an n = o magnet. Although the magnification is infinite when the object distance H i . ' . is 25.2 inches, the resolution has a finite value, .014 in ", .--,.-- .- --.. . . 1 I ..-. _ I _ ~ . n . a A . c s I I III I. .I r . . . ,. . I 01'! A- ,p 1 _ . . . _ . . 1 1 . u I I‘o II . ...... I40 ' image 1 magnificat a = 90° and B1 = ...... l20 radial and axial ‘7..- 20 Figure 6. A--"I—.—-.»~O‘ -I“. I . I v -D-“a-o- . . , _. .. L$I~ _ l Radial resolution, f ion 0 t the 81.118 distances, andlradial and axia a single n = '3, p = 36 in., B2 = 00 bending magnet plotted ag object distance. 44 since the image distance is also infinite at this point. The radial focusing of the n = 1/2 magnet is seen to be weak as compared to the n = o magnet. For example, unit magnification occurs at 81.9 inches for the n = 1/2 magnet and at 36 inches for the n = o magnet. B. Single Quadrupole Magnet Figure 7 shows the properties of a single, five inch quadruople as a function of field strength. Although not shown explicitly on the figure, when the object is placed a focal length before the focal plane, i.e., object distance equal to f + b, there is unit magnification as expected. C. Bending Magnet Pairs Consider, now, systems consisting of two bending magnets. The bending planes for two bending magnets in a series need not be the same. It is possible to rotate the bending plane of the second magnet about the optic axis joining the two magnets so that it has some angular relation to the bending plane of the first magnet. The angle between the two bending planes, specified by 6, is defined such that when 9 = 00 the two magnets bend the beam in the same plane and to the same side of the optic axis entering each magnet. For example, if there are two 900 bending magnets which are ,. 1 . ,. v. v . - 500 .03 .04 .05 .06 .07 .02 Radial and axial focal lengths and distances, ‘ focal length away from the focal plane of a single Figure 7. .l. . n K. .1 e .r r. O m s W t hi I n K H t s s a n h .m .a g n a a w. tG t O 1 = p .52 e K 1 0 e p b u r 0 d t a m. d e n c .1 .i .r t e e .a m, s a a m W 46 oriented such that 0 = 00, the outgoing beam from the system will be antiparallel to the incoming beam, and if these 1800, the outgoing beam will magnets are oriented such that 6 be parallel to the incoming beam. Figures 8, 10, 12, 14, and 16 show the radial and axial resolutions of two n = o bending magnets, which have the same field strength and bending angle as the single magnet described in A. Several relative angular orientations of the two magnets are considered, and for each orientation, several separation distances are taken. Figures 9, 11, 13, 15, and 17 show corresponding results for two n = 1/2 magnets with the same field strengths and bending angles as the n = o magnets. In the several orientations taken for both the n = o and n = 1/2 magnet combinations, when the bending of the beam takes place in a single plane, there is first order resolution only in that plane. Figure 8 is a plot of the radial resolution for the n = 0 bending magnets with 9 = 00. This plot shows three peaks in each of the three out of four curves plotted. From the way in which the d = 10 inch curve corresponds to the other three, it can be seen that there exists a peak in this curve, also, for a greater object distance. These peaks correspond to maximum transmission and no resolution. For improved resolution the source slit has to be moved away from 133°4- —— d=|6 ll II=IND“ Figure 8. l n o l n .1—--_.._.- .-...g, .4 ——l in _'2 00 ._. - --r- o . ,\° . ,ib : Y 3' . . g . d .- -I I ° * I I ' . ($99 1 " I 93,, i I 03". F 09 "I I so. . , i I _' ‘ .L! I I I I g . { 1.- (I II i __.;;...»____I 40 60 _ 80 ICC .120 I40 Radial resolution of a system consisting of two n = 0, p = 36 in., a = 90°, and B = 02 = 0° bending magnets oriented so that 9 = 0°. Plotted against the object distance in inches for the separation distances d = 10 in;, 60 in., 110 in., and 160 in. 48 the positions where the peaks appear, and have to be placed either far away from the magnet or nearer the focal plane of the first magnet. The intensity is sacrificed with an increase in object distance, since the beam has a greater chance of having a larger dimension than the aperture of the magnet. Improved resolution without an increase in object distance is made possible by an increase in the separation distance, but when this is done, loss of the beam may occur at the entry to the second magnet. Figure 9 is a resolution plot for the n = 1/2 magnets which corresponds to the n = 0 case given on Figure 8. The peaks follow the same pattern as the n = O magnets but occur at greater object distances. The resolution, at a given distance away from a peak, however, is a little better than the n = O magnets. With the object slit placed at the focal plane of the first magnet, for both cases, the resolution is about a factor of two better for the n = 1/2 magnets. Figure 10 shows the resolutions of a system of two n = O bending magnets oriented so that each magnet bends the beam in different planes oriented 450 to each other. The first magnet resolves the beam in one plane, while the second magnet resolves the beam again in the other plane. In the axial direction of the first magnet, the resolution remains particularly good for reasonable separation between the two ——--IH 20 Figure 9. 60 80 IO IO IO _Radial resolution for two n =-%, p_= 36 in., a = 90°, and $1 = B = d’bending magnets oriented so that 9 = 0°. Plotted against the object distance for the separation distances d = 10 in., 60 in., 110 in., and 160 in. 3‘» .‘ gaso" d=l|0" .OB-j— \ \ -; \ .02 I . :wo“ I; I~ .OI-It‘ in’l Q - —.lfl 2 Figure 10. l 1-. - .-:.~_..__J 40 so so loo 'I‘2o" "i467- Radial and axial resolutions for two n = 0, p = 36 in., a = 90°, and B1 = 52-: d’bending magnets oriented so that 9 = 45°. The axial resolution is plotted with dashed lines. Plotted against the object distance for the separation distances d = 10 in., 60 in., 110 in., and 160 in. 51 magnets for practically all object distances. The peaks in the curves for the resolution in the radial direction occur for object distances which are slightly longer than the 9 = 00 case. The general character of the curves are, however, very much the same as those of Figure 8. The resolution, in the axial direction, for the corresponding orientation for the n = l/2 magnets as plotted on Figure 11, is not very good for short separation distances. The radial resolution curves are very much like those for 6 = 00 except for a slight shift in the object distances. It is not very easy to make a precise comparison between the n = O and n = 1/2 magnets for this orientation, since there is resolution in two planes for both cases. Figure 12 is the resolution curve for the case where 9 = 900 for the n = O magnets, and Figure 13 is the correspond— ing plot for the n = 1/2 magnets. The radial resolution, which is in the axial direction of the first magnet, is practically equal for both cases and is unchanged by a change in separation distance. The axial resolution for the n = 1/2 magnets is peaked in practically the same way as the 6 450 case, and reasonably good resolution is found when the object slit is placed at the focal plane, although it is about a factor of two inferior to the corresponding situation with 6 = 00. The resolutions for the n = 1/2 magnets, at .04-+— "-T‘"'—._.’ . .—-.. ~ -.. -_. . .02-r- .Ol in Figure 11. “46'“ so“ go "mo ”"126” I I I > ' ..; 5- --§ - - L- .... . . ...,1-... 44 .1. a-.. o~ -+r . . .. _— a — ' ' I 1 Radial and axial resolutions for-two n = '3 p = 36 in., a = 90° and B = 62 = d’bending magnets oriented so that 8 = 450. Plotted against the object distance given in inches, for the separation distances d = 10 in., 60 in., 110 in., and.160 in. Axial resolutions are plotted with dashed lines. .045- . . ! . “—3-- . «—_¢ —- ._..--¢-- u. -. I . l - ' I I I I ' I H... —.s—+-o . I .. I :I ! Figure 12. I ' . I , . . . I . . . . . . . . .- a- .. A___...-._J__.-._.__- ._..-... _..'._.-. . :. ‘ . . ' ' 'I t I I' I -— - ‘m: ..I . -k-Q-.~~—“ I ' l I ‘ ' I I z r I ’ ; I I I . I l I . . -.... ... l.-. . .. . .- ., _.-..'...........-...--I_ ”T ._ LI Q I I I C I " ' I .: a I. E . I I I ‘ I ' . ' '._E I I. . ' _, ' - 'u 'u ‘nw u. g I . . wflzl 50”on 1.1 -mi---._--I i-.._...-_-.I 80 _ 00 I20 I40 Radial and axial resolutions for two n = 0, p = 36 in., a = 90°, and 61 = g = Oobending magnets oriented so that 9 = 90 . The axial resolution is plotted with dashed lines. Plotted against the object distance for the separation distances d = 10 in., 60 in., 110 in., and 160 in. ng Plotted . ...lb.M..-I+. 41—2 90° and Bl = 62 = Oobendi magnets oriented so that 9 O. 36 in., Radial and axial resolutions for two n Figure 13. p 900. against the object distance given in inches, ‘ for the separation distances d - 10 in., 60 in., Axial resolutions are plotted with dashed lines. and 164.16 in. 110 in., 55 longer object distances and larger separations, are quite good, but there is a natural limit as discussed before. The n = 0 curves, in Figure 12, show no peaks for the resolution in either of the planes, and the resolution remains good for all separation distances considered and a very wide range of object distances. Figure 14 shows the resolution curves for the n = O magnets for 9 = 1350. The radial curves peak near the focal plane or the entry edge of the first magnet, and very little change is observed from one separation distance to another. The corresponding case for n = 1/2 magnets, shown in Figure 15, has considerably poorer resolution, particularly in the axial direction of the first magnet. Figures 16 and 17 are resolutions for n = O and n = 1/2 magnets, respectively, both with G = 1800. The magnets in both cases are oriented in the same plane, and hence, they resolve the beam only in one plane. For both the n = O and n = l/2 magnets, the resolution peaks when the object slit is placed at the focal planes of the first magnets, regardless of the separation distance. The way in which the curves tail off, as the object distance is increased. is approximately the same for the two cases. Figure 18 through 20 show the magnification for the n = O magnets for all the orientations discussed so far. In .04 - d =10" '1 9 . ' 1 1 t i! 1 ~ I “1 ffc‘ 9 W 9 . ; 1 - i ': .9 . 1 1 1 - 1 '“ 7 t 9 ' 1 : 3 ‘ 1 1. _ 11 ‘ 1 -i--.‘ '- ' . +1 -.—I ‘ . IT “1..., i -1-- ..._- v«- I ’j—H‘E ' 1 ' 9 " 1 3 1 i 1‘ 'i’.‘ 1 ¢ I II 9 i ‘ '.'..‘1 fill? ‘ 1 ' - l - ....;‘ _.' “I ......-.-- _- 1 ““1- *5 : n u a... 1 fl. ‘ .3; 9 1 1 Y 3 Q .‘ :0 I - I . @911 ' . M "I ~ ’ b‘i— u... REL h--**‘- - .. MW 0.: W" H“ ,1 ~~d-‘—- “_ _- “nu-u...- Ib-In-fi— -‘- In?“ I w _.7.._,.:--,-1.-.._...:___.....-1_. ' ”"1‘""1"" 1‘1“, W... b" -- o' gin 20 4O 60 I00 |20 I40 Figure 14. Radial and axial resolutions for two n = 0, p = 36 in., a = 90°, and Bl = =O°bending magnets oriented so that 6 = 135°. The axial resolution is plotted with dashed lines. Plotted against the object distance for the separation distances d - 10 in., 60 in., 110 in., and-160 in. .1 11 11.”'11 11% l‘o‘llfivt 1114 1 11. 11111.1 H1 u l..«11 n _ . .w i: .. a - .. _ 1 .. _ .i. 1 1; -.--1l M. i a. 1 ..... m . :.. .i w. .i u .. .. 0.. " ..... ..1*p..1 .1fl.. I _ ,n _ . -11111F1111 .11- 11 i 1 1111* t . 4 .. . . i i _ h . , . _ . . _ l . .. . . . . . . 1131- .1.11 .. 1L-1 11111-%111WY11 _ m . .o, . . . . . 1" w . .0: . . . _ _ w- -1 _ m 1 . .6 _ ., . fl . . . I; i _ . .5" . . ..1o. 11 120A 80 IOO 60 4O 20 Figure 15. 36 (gas 1:2n e e ith =dtc non neli Ob.Pn WAU .1 t 0 :0 n r 56 023V fBli g 8:: n e 019C .15. n t ta ud at lnhS oati S d 600 .rOSt 9 C 1 de a: 6.3 .1 .t.b xano a 8 0“...“ do nnot 8.1 St 1 t a e .1 n d g Pu a m p agains 10 in., 60 in., Axial resolutions are 11165 . for the separation distances d in. hed l and 160 in.) 110 c‘as plotted with l ._ .. 1 . l _ “u m _ . . 1 _ . .... ... 1-4:“ . H H. . . i . _ ._ . .. ... 0 71171111.- . w M . . _ h H t . H ... . O ... 11m - 1511111.“; .1W1111111v m F0.- 0 o _ m . H . II = H H . g ONU i . i n 28 I111 - 1 i 51 . _. . u = = . . nv.1 - -1 2 s 19 . H . II n B. . . J w.d.n .. 1 “1 -.- H n h . o w a t t I O -. .- 0 so 3 . _ i .. | YO . . ... S 9 d .M e .- . .. . . . - .111- a = t . _ . H M f a m -. - ..i... 0.0... _- . . a n ”a . . .11.. . O n _ n p . a i .1 .1 8 .- .. ...11...: H _. .. . ...---11-M11.....-: t t . - . w . . u H u "w m . .. .. i H . W .1 0m ._ m. .. 1 4.. _ ..1-.-11 . . . . W i i .. a. M . 6 e m M . m . _ m - .- .r 0.9 . I r 1-“..11 » . L1....!1.1-.11 l l n no a. 0 d = m. 4 k n b 2C) Figure 16. --—-un Plotted against the object distance in inches in., 60 in., for the separation distances d = 10 111. and 160 111., 110 _ . . . . . . 60 80 - IOO I 20 I40 40 in 20 Figure 17. n .1 s 6.t 3 e n =9 a o.m pg 1=2n ....m. . n ne o.b w... __ r 02 gray n__ o .1 1. tap u 1.0 o n 8a e r a o 1.0 any .1 .0: ha ted so that 6 = 180°. 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V I. ‘IQI I, |II . a ...H V volv. 7 u. . . . . I. I .II .. . .. . ., .. . . . ~ .. . . . — .w . k . . * . . _ .. . . . 1 . o . I II. . I I .Io.. II! I I1 I.r I . v. .. fi 1. . 1 . . * .... . . . .. . . . o H . _ . > F 1 o1 . .1 . H . I! ‘II I I o I I. tr ll-"1¢.l.’ I 1 .. II N I N fl » . .4"... . OHIYI “- LI .0 2 . 1 1 .... . o . . .. . . c - . .1 .. U . H - _ 1 .. . . . 1 .1. . . u .1 .N. w .. . ...w... m I .. 1 . . . I. FII # .I. .IIJWI. I..-II... - . .I.. I . * ........ m I IJJI I.‘ II ......“ .. .. LIN JIIII. I .I - - 1 1 .1 1 :1. .. 1 1 1 . . 1 1. 1 .I 1 .. 1 .. . - .. > ? III III‘I‘ P» It, ['1' III, III I ‘. III I. III- I ‘ I III. L . ..I' IIWIC'O'IIHI- III! I41“ In: . Io} IIIII 4.04.1 . 1 e . . 1 . .1. u . .1 . L 1 . 1 .1 - . 1 . . .. . h .. -1 1. . 1 -..-1. 1 1-.. 1. ..--.-.O . .1 1 1P . . . . 1 _ . .. . . 1 . LI. I -.I - . - - I-l. - I: O I- II III- II III- III? O! AII|I| ' IL M IJIIIIIUII o I JI- I . MI 1 w W .1 w . . .v. M. .. . 1 . . | . . . . . . . .1 p t .. .- ..1. .1 - . .-..- . ----1. LI .+1-. . 1. I .1 1 1 . ..1 ..1 . . 1 1. 1 ... _ 1 . L . e . 1. . .. o. 1. . ... .. 1.. « . . . .1 - 1 . +II-III -|I..I .I..II..I.II|. .- - IIIIIIIIIIL . I I I1. I 1 . “ M115... .. «I . I“...- . 1 . I. . . . 1 . . . .. . . 1 _ . 1 w . -0- 1. o . . Y. .1 .. . . . --¢_- .. 13- . .1 $39.. . .. ..1 u 1. _. . H . . .1 -. I. .. . m 1 . - .1. 1 u . . fl . . . .I o _ o I 4 V T V...“ ...‘Il’. . ..1. . 1 . . L.8 .. . a . . . .1 1 .. . . . . . ‘ 1 . Il“.-. "I _ _ o I I III -ctt. I'd-IOIIIII > _ d . . 1 n. 1 . .--I1III-I. 1 ... T‘Lf.‘ I -' .. ~17- *— '. ”-y- . ' . I . I ., i Y I. . -.1...“ y—.......'. l. I . . ‘ . 1.14? .- ... .. . . I t r _ . .. . I II IIIIII l I‘lfizII . . W. I 1 1 1 ' ltr‘ II ‘I‘-‘v| I. O .' z l ' 9 '- ...‘b-o «10*- «-to—M I». v D ' t O ‘ 1 I r . . 1 1 IIIII+1I.I. 1". o . a Ink.§"91 - . . . . . . . . Plotted against the 6’bending magnets oriented for the two n = O, = 00 or 180°. object distance given in inches for the separation 90°, and E1 = B2 distances d Radial magnification so that 9 a Figure 18. in., 60 in., 110 = 10 I I I —>in 20 Figure 19. I---I . I ' I ‘ I ,_ 4O 60 80 IOO I20 I40 Radial and axial magnifications for the two n=o,p=361n.,a=9o°,anda=B2=o° bending magnets oriented so that (9 = 450 or 135°. Plotted against the.object distance given in inches for the separation distances d ='lO in., 60 in., 110 in., and 160 in. Axial magnifications are plotted with dashed lines. _‘. .4 o ... . u A --l... ‘0. n I ii. I ' —-*in 20 Figure 20. -I I . o l . I ' I I I I r .- »v . . - , I I I I - ' I ' 2 I -L-..~.L.I.o .— - -.- . .‘. .. - c- k-“ ..-— -...I. . * d8lO',‘60',‘IIO','I60" .... .I. . . --. —. y .- ..-. . 4, .. 9. o.... -.-o. ‘ I I 40 6 so ”Ibo” I20 I40 Radial and axial magnifications for the two n=0,p=36in.,a=90°.andf31=B2=0° bending magnets oriented so thaté? = 909 Plotted against the object distance given in inches for the separation distances d = 10 in., 60 in., 110 in., and 160 in. Axial magnifications are plotted with dashed lines. I ' . _ . 1 I ‘. ‘ I ‘ ' I . 1 I I ' .l , | I “'3‘?" ‘ I ’* "I' ”I“ I” t ', - "'I' - f" ' I I I t I ‘ I I I ; I ' , 'I ‘. . . ' ‘ - 1 I f f I I ‘ I I ' z I I I ‘ I I I I r. ......o._.. —.- — E o—o---O '7‘.-. A. 1.. ,,,' » I... ..f...; ' I . I I I , I . I . ' I I ‘ I 1 I I :L- '1'". i ' f ' I I. I ' ‘ . I- r - :-- . ' ' ' . . I 2 I I , I I " ' I ' ' . -- ' _ ’. .--- --. - .. .v .. .1 ...-.- - 'I 4P””*" I'- I . t t I’ I I I . ' ; . I I ~ , I I . . .H L“ . _ ‘. I 1 .. - .,- I I A I i . I I-o I‘. . ' . ‘I v 1' I I ‘ . : ‘ I : i . , "if I ' . 1 ‘ I I I I ' "T“ I ” 1’ °_ t “ I "-’":‘_w 'I 'I' - “'Nfl‘ ;" ' ' 'i" I” ‘ ‘d r ”‘7“"' °’ ' I ~ 1 g ' 5 I : I ; . . = 4 . = - ; I . I . ‘ ' - ; ‘I _ E I , ; ' ' I l . I I . ‘ . p -- .._. .:?.. ..-. —. f .74 ”owes-J, .—.~ ‘— - . . .T J .g. . - y..— -' . ; f r... ---! . ‘4-._.-i.li ... I I I I ‘ i i I I . I ‘. . I '5 I 5 I .i ; I b..- it "I. .. of- .—.._. g ——o- ' I —o—.- l I - 9 »!‘,.. L i. ’ I ' I I I I I I z "" ' --v..-4 I ‘ ‘v I . h , I I ' : I i. .-.‘.-... . .... -Jiiw. . - .. Jill..-“ 63 general, the magnification blows up for small separations and small object distances. This makes it undesirable to consider placing the object slit at a distance which is left of the peaks in the resolution curves for most cases. There is no focusing in the axial direction when the magnets are oriented in the same plane. Figure 21 is the magnifications for the n = l/2 magnets, and the curves are same for all orientations. Also. since this system is double focusing, the radial and axial focusing properties are the same. Except for very small separation distances, the magnification for object distances not very far from the focal plane blows up. When the object and separation distances are fairly large, the magnification is well behaved. There is unit magnification for all cases when the object slit is placed at the focal plane of the first magnet. Figures 22 through 24 show the object distances for the n = O magnets. In general, for the n = O magnets. the image distances are in a more useful range for larger object and separation distances. In Figure 22, which describes the radial object distances for the n = 0 magnets for the orien- tation angles 9 = 00 and 1800, image distances less than 150 inches are available for separation distances greater than 110 inches. Figure 23 shows the object distances for the d=l60" '4.— -—--in 2 Figure 21. ' I . —'-.‘ v i t ,. l » ‘ t 4%,.- . _. .rq ‘ i ; . ,' \ l . . .. - ‘ ... . .-. .-. , . n I r - ' . o \ ..- r— ». -~t- h. _- .. d-suo” dr60' d=|0“ .. +3... - . ! a I _‘ . : f . ... ,-L.. .4 é ; ...—“.4...- —-—--— , r . o , , . . . - ..............+._._. .——L— ‘-4~..._‘_- ...—...- . . a Y . l ‘ . . . u x - .— a.-. L ; g 1. . g 4.0 60 80 IOO l2 I401. Radial and axial magnifications of two n ='—, .p = 36 in., a = 90°, and £31 = 62 = 0°bending magnets with any relative orientation plotted against the object distance given in inches for the separation distances d = 10 in., 60 in., 110 in., and 160 in. v , - , r‘ g .. I '00 ' o-— >— C In ——-in 20 Figure 22. .M—- ...—.0 - {—4 4 I ' ‘ l . I l , n 3 ‘ o . , -. $ 6 .— -'— _‘,, 4‘ < n >- ~~>-4~ >OC'---“ .._—. l , , l . l I ._ l . 9 __ . . l . l ' v t - ‘r o —— r—v-oool-«ofiro- --—-?o~.r¢-o-a g. c l ”.4. I lllll l L ' I n h~s~‘.‘—-‘. _. ...c...-s- - -_ . - ' A . I I n -| . 0 ....l ’ ‘ v—qhfi4<“i.._--f ...- i ' . v I | ......... . t ' ' -.-“..- _—r.—._.,_._..J._i.l_. -... _. ----.. ' ' I 0 « l ‘_ . ‘ . . ' . .jll V . I 5 o 1 . T " ‘ 5 I ’ I ':—-"- 1 ' r ',-. .. V 4 s.. .-.» ....4 . A» u -- . g. “...-.1 I . ‘ . . . , . . . \. ' . I : --. *— f. t I Y 1 I A D .T I J J I I ........... f f. l ‘ . 1 t , . . s » ‘ .--~M‘+"“- “ i l l . I . ~ I . l I I J .¢. 9 1 I. L G v o } l o _.—o.>.—Q. _.—.——- ' " . : . a. ' . . : A. z . I ...i._...._.-_..__ ......T" ‘ r ._ 9-4 . -. V. I 3 . . ‘ a. .- ‘ n . |‘ w' r l "‘ ' IV ;., . I .‘ . I o . : . . '. ., | I - I . . A . . _. ”4-. ,. .. --..._ . _. . .. .. .. -. - . g . . ' . . - a z . . . . I - ‘ u . . . V. . . - . . I ' ' ‘ . - I K . l . . I ' ' . ‘ . : . i . ' .- . .. A .- 4 4. § - ..4 _. ..1._...._. --~~~ ‘7' - --- 60 80 IOO IZO I40 Radial image distance for the two n = 0, p = 36 in., a = 90°, and E1 = 62 = d’bending magnets oriented so that 9 = 0° or 180°. Plotted against the object distance given in inches for the separation distances d = 10 in., 60 in., 110 in., and 160 in. YIlJfliili , _ 1. w .....- .. _. . . . i ”.5...“ ...- it .3“ v- . v. . . . .v .. +_a . i. .. m . n 4.. Q 11 J.-- w ..... i..- a a. 4 . . . ‘ ru-.l1l.+».liaxu-ib a??? _ _. M. h _ :7.-. .... 3.. . - .. .. _. .. I»! 41.1..-- M .. i l-.;-; . . ml m .7 . n . u o . . a -4 ...... r- . . ._ ~. . . . . rft oil! .-II. rr{$.I¢IAO, - 4 Il‘. l p i . . . 4. .. , _ . w, I. p ”I- 3. v .. :>\ Ivl . c . .. .i . .. ...... .v‘ I. I ...00 I 4 . .. . . ‘ 9- - - . . ..... _ .. , 13..-“. r..’ l.> . . .1329T1o0; ... . _ . . _ . . v c a m _ i ”1: .. . 4 , . #7 ...k . _ T ..... _ . — v . . l -. o o, .. H u L iii. -....lifoi- .2. {Littlltxi . . . . . i l i . w . . l 1 . 3 O 1....-. . ‘Oll't'tl 1| ' II?! .u‘j} I40 l20 tances for two n = 0, 80 IOO 60 Radial and axial p = 36 4O 20 Figure 23. ——.|n 111., ing d’bend 450 or 1350. Plotted against the object distance given in tances d.= 10 dis 90°, and 51 = 32 dis in. ICU image a: in., hes for the separat 111C magnets oriented so that 9 60 Axial image 110 in., and 160 ' in., tances are plotted With dashed lines. dis - I40 IZO D r..~—“r>~'.‘wh-'- T-— Hot-O IOO _ 200 .. trillrurtlull ..-! v- ... . ;_. "1"“‘f""':"L”" ]~__'_+ 80 It!» i . m 40 60 20 0' =9 n nu mm +.b rmu O. f: 820. 68.0 C 9 n: a = t1 88.9 fiat em». 9 +. a: mo Av .108 9 1.. d a: e .1 t ”ha n a e I.n d o nun O 3.1 S 16....— 338 i. n d: N. mnvmm 4 .4 e r. u g .1 F Plotted against the object distance given in 10 in., image ion distances d ‘and 160 in. distances are plotted with dashed lines. hes for the separat' inc 60 Axial 110 in., 111., 68 n = 0 magnets at 6 = 450 and 1350. The radial image distances are very much like the 9 = 00 case, but there is axial focusing as well as radial focusing for these cases. Unless the separation distance is made very large, double focusing is not possible for this system. Figure 24 shows the image distances for 6 = 90°. The positive image in the axial direction is located very close to the magnet edge for all object positions except when the object slit is placed very close to the magnet. The separation distance does not affect this axial image distance to any extent. Double focusing can be seen on the graph for the separation distances 10 inches and 60 inches at the intersection of the corresponding solid and dashed lines. Figure 25 shows the object distances for the n = l/2 magnets at all orientations. A short range of small positive image distances are available for object distances near the focal plane of the first magnet. Longer image distances are available for large separations when correspondingly large object distances are used. D. Quadrupole and Flat Field Bending Magnet Combinations Next, consider the combination of quadrupoles and n = O bending magnets. First case to be treated is a system .1I IIIIIM IIIIHIIMIIIIIWIIII- ,I.I I I I, I . .M..I IIIJTIIIWIIIIIHII u. . . _ . . .. . . m _ . . . i . “I. it... I h .I. u . . - . ”I ~I a I..% _ i. 1 .fi 4. i i w w. . .. :i .. i.“ ..H .. .« h . 1 . . . w. M I“! . . L --IIII I II.....IIII- I- I. L4! . IIIII 7 W 4 4 i . r _ .e . n a . I . . ~. W .. I. . :... .i a; i , .g. .; - :L . .. . 4 . . n i i v .... . c h . . . , . .II I . IIIII ...IILIIIIII.” ._ .. 2.3!... ---IJ. _ - .31 . . .4 U .m is w. W .M n * O . .. IN..I I ..m . b 0 ’ '— I‘ . ‘ - . I III-..-.. : H ,I - . i . w 4 h < a . I. m ...._04 m . . I. IIO'II i I 4 I 3 ' I r ‘f I I I 'H I I 0 .....-4444,..4I l I O- l I h. a -4» . . ; .4; .-i... I 1 , I I A I «I I 4W44; IOO o .— ' I I .— I I I =l 2 ing plotted inches for 60 in., tation, iven in in., distances for two n ive orien tance g 18 . -i. -.- 4..."... . . . i ‘ -. 4.4.4...-" I .....- , . 60 , and $1 = 32 = 0° bend image ject d in. a = 90° and 160 111., t the ob 111., 5 >04 v 0' > 0 an. 4o Radial and axial .“fl .... . . 4 , .4 .0 ;i.... a x . 1.3.-..:% w u u u H. . . I . . _ 4 a a... ... .4.. o... .. . . .. “I. . . .o. I . . i . . . 1;. . . .. . 4 .. _ a. .. w. II. .. . . ... .. . ....._. . . r .. M . . . . .. ... .II . II. II p I.. . I I ”II Irv. .III.I..ILIOIIII+.IIJ 91. .9qu I. II I . 41Iv.“ IoIIl I I #wIII . . . . . A I .. . I I.. . u .I . .. ... ¢o‘. 9.trv . . ... o I. I ..III II wIO..I .. . . .i 4. . _ .1 . . 1 _4I. .A —g._ ... ... . . ..~i ..‘I.I. ..Q ‘ . ... o. I. .. 1 . . . . .. . .4 . ._ .. . . . .... r ... ,I “I .. . . H.... ...4. w. _ v I .Ill I I!" IIIulI .. I — . t H . .HHLW. H .f. _. - .1.+ .YV. - . . p. ..I o .o. ..I- . IIIJIIIIIII IIIIIIII. I III .I I Io II I | I I II I o _ . .. I ...II .o. I... I . . . I. o. I u _ * . e . .r . .... . . ., .. ... I; .-I. .— . . n . aI.. _ .. o... . ..4 .. o .. O H . I .I. u . o I T . . ... I as . ._ . w I r I .V . II .0 II Isl I I. .I. II» I F I I‘VIAWIIIILII“?II $5 0. IIII'II I» I1 I q . i s . . fl I. . fl. ..9 t? . + I . . 7- . . o _ . - . . I. s a ...-.Q ... g ” s .’ O I. o. s . - . . v .. . o I—. ...... v . ... . I. ‘I . 4 < a . _ M . . a s. . 4. ~. IIU4 . Y I is. ... H I Ir m. . . I I I I. IVIII I .IIIIII IOIII+IQ|II III I L ii . II III III I I J . . III '1 IN V . r .7 _ a .I . ~N _ H H. .. W.I.r 9 Aw. A L 2 . I . . I . > . u A . I . . . _ n . . . . p * . . .. TM 1.,‘4. _ I. .. .w.._. ... .. . . . . 4 ,I ..... ,.. . . r . . o . i. v 0 9 A. .r. is v.. I. u. ,u... r .o .. r . v o H . . 'I .II IAI I.. 7. III 9.P0Ior II b” Qt ..VIIV I.l v I II 11IY'0ArI‘I OIIO».|1...A. OI... .Ibowv.‘Io 0'0. VAL. . . u . . . . _ .. i o. . H . . III I ... vo... I4 .... v.4 M «II. . .4... . . .... ., I . I .. o. ... . o ..4! . I. f . .... I .pi. . . 4 _ . 4 . . h . .. H......MA.. . 21.1... .1. . r . .— . "a. A.. ._ L. _ . . . . . . . . I“ . . _ ... ..i ... . I. A . a . .o o . . r . ... ,. ... I . . I? .. If. . I .hIIO .. p. L In p I u l P b ‘I I III".IIHIIIIIIQ Ill'.l ”I. +I.II¢IIIIIIIO II..IIIII+.IIIIIIII~IIIIIIIIII$“II OIIIWWIIWII _ . M . _ ..._. I u III“. .4. H I”. o. M. A I . . I. . .. o r. O . 1 . a .‘. . ._ . 4*. I .. . ... 4%.; .... ...... . I? r _.. 4 f “n . .1. . .. | v . . . I o. I A. .v 1.... v ..o .0 1.9 v a .I I I . . . « . — _ w . . . ~ ...v“. . . .... I I.’ h. o Io 4" OI...I . III. _nIII.- .I .. I I” .45 H. II . .4 . .. .I . _ . . I I. _ o I I . .ILTII IIvIv.IiIQI 7.4! III I 1» o I I I I r .5 . If “I AIIII. . .IIIoIIr j- 9.. IIMAHH I. vrPIIO . . .. ..Io. . ... I . ‘94. .II.. . L cc... ..v To. 0 I., . ... . I. I . . N . u w. 4‘ . .4 9.. .YI4M . v“.. ..#;<. A....I .03 HIV? OIA#4I. . I ... .. L”. 4. L H i. ..h4 . . . y I I . ..fltv 9.5.1..III#7...9 .... .... LII I I I v. o ... >0? Irerst.. st. or . .Y. ... h.“.io~4vui . V b . w . . . . .u a» .. "17.” .w. ...3 T. .2 n . gr. I? I. 3n 3? TC H. I 3 g: E. t . i I | 200+ p ==36 the separation distances d - 10 magnets with any relat agains 110 Figure 25. 70 as diagramed in Figure 26. Figure 27 shows the magnifications for several quadrupole strengths as a function of object distance. For unit magnification, image distances are equal to object distance, and these can be plotted as a function of quadrupole strength, as can be seen on Figure 28. The intersection of these curves gives the condition for double focusing with unit magnification. At this point the radial resolution is found to be .01 in-l. The quadrupole strength for this condition is K = .053 in.1 which is quite moderate. n = O bending magnet. p = 36 in. a = 900 fi‘3pi=(f ’67.5 in. Axially focusing magnet quadrupole. L = 5 in. I 67.5 in. n = O bending magnet p = 36 in. a = 90° Pl = fiz-‘Oo Figure 26. Diagram showing the relative positions of the magnets in a system containing two n = O bending magnets and one axially focusing magnetic quadrupole. -l ........ 1n .055 fied by qnd ec1 1n71, lds sp .05 1.1.1: r M111- - - .5 wwmfi . _. M. Le}. . _ P 4.. . I‘D'.J'- I L. .w. I yvy‘. ll 19.! ‘ ..-; OIL .. . ..- L. . m. if... inflex . H. . 7. W.. .H 4:!“ 1.. ,LLL. ; . grwur ‘11. A .*. . . a .. .. w .... 3.“: .... ..- T. . uh 0'9er unll. . w . _ 4.. I l-,. ‘01.. o .. . H. c , .. .M 1 ca _. t n y O bending magnets and an axially drupole fie .045 inTl, 3a A -" O . .. s ‘ tonvla 0‘15 '1. Y! t .1 . . 1n hes for .04 ; - 7 I o 1 :an K The axial magnifications are plotted with dashed Radial and axial magnifications for the system lines. Plotted against the object distance given in focusing quadrupole as shown in Figure 26. with two n _ . ,- 4 u A y . u t- . . . . . . o . . . . .1”. . . . Y«'. l.l. .. v... .lvl .. I.) b V . . . . . _ . . . . . . 4 . . . a . . . . . . 1 . . ... s ... . . 4 . _ . w . . . ._ . .. ... L... .. . .. . . . . . . . . _ . . . . . _ . 4. .. . . . _ .. tl. . a . , I. up I ‘ . 1?.ltu _. . . nrlb I . t , .ub‘Io‘ Int. I;l'.4l’~ 1- y IV?! 10.. ‘0 < l a ‘ LI» L _ . . . .. . . 1. .. . . . . _ . . .u . . _ l . _ . . . . .. .. . .. .. 1. . .. . . . ._ h n . . . . ... . ... 1 . . , A . . ... . _ . . . . . . .. ..... . . . .. ... . . , . .. .. . ... .... . ._ s . . o . _ . fl Figure 27. . . . . . . . m c. .. 4 . .... .. . ..l .. .\ . ‘ ‘ . w m . 1 . VM .1..I¢Ar IP11 .-.uoyt . . .. . .0 .L. 1“ A U 79;» . . . . . ... .r.. o . . ., o . ... _ u i . a . . .. . . .. H . .. ...“.Q :15: 1.1.2.. 5.4- .57.". w . . _ . . . .. .. .. . . ..w .1 . . ... . . . . 1.... _. .o .. ‘tltl‘l II 1 0| 1 I. I .4 . 'v lit . ». IoI.‘ v... . ‘I I . 0|.0|..|Y|9|v|lll I« 1 >Ir b > . i \i . . .. . . i u l . .. Q: .. ”L .....3: 1... . . . _ _ .. . . .. 1 _ n . .. a. . .c . .. . . _ . x . _ . . . . .. ..... . ... u ...... . , ..pw‘ . . . ‘.A. . ... .‘ ._ .. . . y \. ~ _ w . . . . m . n . m . .cul.-llTvOt*‘.:v lo‘ ..Ok..l§;.u4o1¥1f.J.-}I¥. «O‘ov.o .H‘... . -o”.t.. .y...1.u¢| ~‘1‘¢o. 0.. . . . . ..fi...;_.. 1: .....t. 4...? ...........p ......z . . . m _ a . . . . , . e. . m . . ....‘L1..b_z.. -. ”....v. .. .M. . .... .u oo.¢Wvo”. ... c~f.~. .... on.o J... .. .. c .. — b — bl k—r . . . w . b. .n.. . .. w.y ..To». .. —_...,.. w..~...u .+H.»..A. .MIH.HA.oH. M...H... . m... ..H. ...;M..._ . .L . 1 LI» ' .050 image distances‘for the system gure 26, plotted against the f 1 1a Radial and ax Figure 28. 1eld i strength of the magnetic qudrupole. F lines of unit magnification. escribed in d e r e 39 a e m 3.1 e mu .1 X a e m 1ne o ' lotted,with a dashed 1' 18 p distance 73 This is a useful system, since the object, image distances are quite short for the double focusing condition, and the quadrupole strength is quite reasonable. This case can be compared with the two n = l/2 magnets where the object and image distances are 85 inches for unit magnification. The resolution is of the same order of magnitude for both cases, but the n = 1/2 magnets give a slightly superior resolution around .004 in.-1 as compared to .01 in.”1 for this case. Another combination was tried as shown in Figure 29. Figures 30 and 31 correspond to Figures 27 and 28 of the previous case, respectively. In this combination, the two outside quadrupoles are axially focusing and the one inside is radially focusing. The radially focusing magnetic quadrupole was made the variable magnet and the other two were set at a convenient field strength. Since we change the radially focusing quadrupole in this manner, the changes are mainly in the radial focusing properties, while the axial properties are practically unchanged. Unit magnification and double focusing occurs when K = .095 in.-1 for the radially focusing magnetic quadrupole. The radial resolution is .07 in.1 for when the axial magnification is -1, while it is .12 in—1 for when the axial magnification is 1. In both cases, however, the resolution is rapidly varying with object distance about 74 . . Axially n = 0 bending ‘30 lnifocusing magnet.' magnetic P ‘ 3601n. quadrupole a = 90 -1 p,‘ Pzgoa K = .07 in. L = 5 in. 67.5 in. ‘ Radially focusing magnetic quadrupole L = 5 in. I 67.5 in. V . Axially 30 . n = 0 bending focusing ln‘ magnet. “magnetic p = 36 in. quadrupole -l a = 90° 0 K = 007 in. Pl=pz=o L = 5 in. Figure 29. Diagram of a system with two n = 0 bending magnets and three magnetic quadrupoles. one of which is radially focusing and the other two axially focusing. these points. This can be compared to the two n = 1/2 magnets where the object and image distances are 25.2 inches. Other than the fact that the quadrupoles and n = 0 bending magnet system requires a little more physical dimension than the two n = 1/2 magnets, there is very little difference in the properties of these two systems for this situation. I f e . .. . --c—o-~ —— -—---~-a-+—_—o——o . . , - v . 60 Radial and axial magnifications for the system 1.1; . . . . .. . . .IH. . _ . -— - - . u v v o . _ . ... I! f I q . .. ... ,IO. .;I.ntll|uc 50 1': .... I u . » . M i _ _ . . Trlrit f. -- . ,..-I-.-I..lu1:!tLi-}t 4o 30 20 IO Figure 30. t the .1118 29, plotted aga Figure in as described object distance given in inches for the several field strengths of the radially focusing quadrupole The axial magnifications are plotted with dashed lines. as specified by K. i. “n . L—1—7-c- ——-—J-—-— - I— —--~-- —.- -.-—-—‘——‘—"‘_~ I I I” ' 7' I T . I P1 *‘r‘ ‘ I If 3 ; 4‘)**“ j - 1 In . ... I i - I . I - ‘ . l I . 304...... -. - . . . l O I 4< z I L..._..- .. .. ...-.- .-. _ -_ ,. III—.080 Figure 31. "0 I l I , : ' a I '. —I.—. A l ! II 1 - 1 I 3 1 ; ,. .| . ' l . _ - ,.. .. ‘ ¥ t I ‘ ' ; I 5 _ . i. _ I -. {..- .. “--...-.- + I } I 1 ' I I I I I I . .' I . . . . .‘ . . g I I . .‘ .. ......I " I I I .r‘ "HI-— T 1 - I s ' I 3 f 1 1 I I I . v o A l ‘ . I o . .- h— 1 $4 #4-“ ... ‘O— H-—v0 ' z . I . I I I ; ' I I I I ' ' I : . 1 s 1 . 1 .2 .--- -41 . ‘. .W.’. h, 1 H v I ' I. I I I . I I I ' I a , ' I: - 3‘ x '. 1 . 4 I . . I . . 1 I I I I g .- i - 1 - . 1 _ .-.:- - .I o “ ‘ ' ‘ I | ' I I a | . ' .. b . , ’ ; ; __" . I LMI.I .“ ”I ..L..-Ii M. “I iii ”HImLcml-il .085 .090 .095 .l00 .l05 Radial and axial image distances for the system as described in Figure 29, plotted against the field strength of the radially focusing quadrupole. These are lines of unit magnification. The axial image distances are plotted with dashed lines. IV . APPENDICES APPENDIX A. { choice at 1253. Black switch white switch 78 white switch FLOW CHART FOR.THE COMPUTER CODE DOI Black switch 26 I253 N Set the "pre- Read in the parameters and, if b = 1 lvious" matrix J~test for the parameter, b. to be unity. (SF) (SF) 1 I Calculate tan 5 if b = 2 ‘ 1'2 I and test for the Calculate the sub— parameter, n. (SS) matrices for the if n # 1 ’ if quadrupole, (1270— n = l 1323) Test for para— Test to see yes Calculate alculate meter, a, and form a~ . _ i . . . If n - 0. M for for 81* by.51x maIrlX (SS) ny= o and n = 1. (57) which 13 conSistent - - . store. With the parameter, a. lno (88) (1330) Print out . the parameters, d, 6, Test to see yes__Calculate K and L, and store if n < 0. M for .1 the matrix RQRD. 51 no ny< 0-(89) (1330, 885, and SF) I Test to see yes Calculate if (1 - n) Mx for < o. (55) (1 - n) < 0. (SK) no I r Calculate M Calculate and store. and (SS) store. (SS) Form the product Calculate M L with the "previous" 7 and store.~Y(SN)v £I matrix (SF) ‘ Print parameters, d, 9, and magnet . _ y parameters. Print Mk and My (885, & Test for the para- SJ) Store the matrix RMRD. (SF) meter, c, (SF) if c = l L rint out the total matrix if c = 0 product. (SF) ‘ Calculate and print out ther’ beam properties. (1070) ff , - m\\_. if N'< 0 Increment do and N and if N'> 0 test for N. (1070) B. 79 ORDER PAIRS FOR THE COMPUTER CODE S - Box Settings 00 3K 83 0 00 F 00 30F Parameters S4 1 00 F 00 60F Constants S5 2 00 F 00 100F tan B. Test for n. S6 3 00 F 00 1330F Quadrupole matrix S7 4 00 F 00 150E Rad. Mag. Matrix. n = 1. $8 5 00 F 00 182F Axial Mag. Matrix n = 0. $9 6 00 F 00 210E Axial Mag. Matrix n < 0 SK 7 00 F 00 600F Rad. Mag. Matrix n > 1 ss 8 00 F 00 670F Rad. Mag. Matrix 0 < n < 1 SN 9 00 F 00 730F Axial Mag. Matrix 0 < n SJ 10 00 F 00 780E Print Mag. Matrix SF 11 00 F 00 940E Combined matrix SL 12 00 F 00 3960F Subroutines S4 Constants Abs. Addr. Addr. Order Pairs Comments 00 60K 60 0 00 F 00 314159265000J w/lO 61 1 00 F 00 1000000000J 1/1000 62 2 00 F 00 F 63 3 00 F 00 F ‘64 4 00 F 00 F 65 5 00 F 00 100000001000J 66 6 00 F 00 99999999000J 67 7 00 F 00 1000J 68 8 00 F 00 F 69 9 00 F 00 100000000000J 1/10 70 10 00 F 00 10000000000J 1/100 71 11 00 F 00 9F 72 12 00 F 00 2F 73 13 00 F 00 3F 74 14 00 F 00 ~5F 75 15 00 F 00 143F 76 16 40 3F L5 300F 77 17 NO F 40 376F 78 18 L5 854 40 376F 79 19 L5 8S4 40 394E 80 20 41 F L5 309F 81 21 NO F 40 394E 82 22 00 F 00 11F 80 Abs. .Rel. Addr. .Addr. Order Pairs Comments 83 23 00 F 00 6F 84 24 00 F 00 340F 85 25 00 F 00 412F 86 26 00 F 00 484E 87 27 00 F 00 376F 88 28 00 F 00 520F 89 29 00 F 00 448F 90 30 26 SJ NO F 91 31 00 F 00 556F .92 32 00 F 00 71F 93 33 00 F 00 2000F 94 34 00 F 00 2036F An Appendage to SS 00 97K 97 0 L0 125F 40 100F 98 1 L5 llOF L0 125F Reset addresses 99 2 40 llOF 26 126E SS (Tan 6. Test for n. 00 100K 100 0 41 F L5 S3 101 l 10 4F 66 S4 102 2 SS F 00 4F 103 3 NO F 40 SF 104 4 50 F 50 4L 105 5 26 SL 40 6F Calc. tan 5. 106 6 LJ 5F 50 6L 107 7 26 SL 40 7F 108 8 50 6F 7J 154 109 9 66 7F SS F 110 10 NO F 40 254 111 11' F5 L 40 L 112 12 F5 10L 40 10L 113 13 L1 24L 36 16L 114 14 L5 24L L0 25L 115 15 40 24L 26 L 116 16 L1 24L 40 24L 117 17 L5 10L L0 25L 118 18 42 19L F5 19L 81 Abs. Rel. Addr. Addr. Order Pairs Comments 119 19 42 20L 50 F 120 20 NO F 7J F tan Bi tan Bi + l 121 21 66 154 55 F 122 22 NO F 40 454 123 23 L5 L 26 97F 124 24 00 F 00 1F 125 25 00 F 00 2F 126 26 L5 253 L0 554 127 27 32 29L L5 654 128 28 L0 253 32 29L if n = 1-————4.57 129 29 26 57 L5 754 130 30 L0 253 32 31L 131 31 26 33L L5 253 132 32 .L4 754 36 S8 if n = o-————,.58 133 33 L1 253 36 59 if n < o-———_..s9 134 34 L5 253 L0 684 if n > 1____4. SK 135 35 36 SK 26 55 57 Form Rad. Bend. Mag. Matrix for n = 1 00 150K 150 o 51 254 7J 453 1 151 1 66 954 55 F -—3-(1 + a tan 61) 152 2 L4 154 40 300F 10 153 3 50 353 7J 453 154 4 66 984 55 F 993 155 5 4o 301F so 301F 10 156 6 7J 453 10 1F a2 157 7 66 954 55 F 9 158 8 4o 302F 50 454 2 x 103 159 9 7J 154 66 353 160 10 7J 453 66 954 161 11 55 F 40 F 162 12 50 254 7J 154 163 13 66 353 55 F tan B1 tan 52 + tan $1 164 14 40 1F 50 354 103 a 103 165 15 7J 154 66 353 P p 166 16 55 F L4 F tan $2 167 17 L4 1F 40 303F + ——-3——- 168 18 50 354 7J 453 10 p 169 19 66 954 55 F 1' 170 20 4o -- F L4 154 103 (l + a tan 52) 171 21 4o 304F 50 154 172 22 7J 453 66 954 173 23 55 F 40 1F 82 Abs. Rel. Addr. Addr. Order Pairs Comments 174 24 50 F 7J 483 175 25 10 1F 66 954 _1_ (a + a tan 32) 176 26 S5 F L4 1F 103 2 177 27 4O 305F L5 BS4 178 28 40 306F 40 307F 179 29 L5 184 40 308F 180 30 26 SN NO F 38 Axial Bend. Mag. Matrix for n = O 00 182K 182 0 51 254 7J 453 1 183 1 66 954 55 F ———- (1 — a tan 61) 184 2 40 F L5 154 10 185 3 L0 F 40 309F 186 4 50 353 7J 453 .9g 187 5 66 954 55 F 103 188 6 40 310F L5 854 189 7 40 311F 50 454 190 8 7J 453 66 954 191 9 7J 154 66 353 192 10 55 F 40 F 193 11 50 254 7J 154 194 12 66 353 55 F 1 195 13 40 1F 50 354 P103 (a tan B1 tan B2 ' 196 14 7J 154 66 353 197 15 55 F 40 2F ianpr—tan 32) 198 16 L5 F L0 1F 199 17 L0 2F 40 '312F 200 18 50 354 7J 453 201 19 66 954 55 F 202 20 40 F L5 154 1 203 21 L0 F 40 313F 103 (1 ' a tan 62) 204 22 L5 854 40 314F 205 23 40 315F 40 316F 206 24 L5 154 40 317F 207 25 L5 3054 40 5855 208 26 26 55 N0 F 83 S 9 Axial Bend. Mag. Matrix for n < 0 Abs. Rel. Addr. Addr. Order Pairs Comments 00 210K 210 0 L7 253 10 4F 211 1 66 954 55 F 212 2 50 F 50 2L 213 3 26 30SL 40 4F 214 4 L5 453 10 4F 215 5 66 954 55 F 216 6 40 1F 50 1F 217 7 7J 4F 00 2F 218 ‘8 40 F L1 F 219 9 50 F 50 -9L 220 10 26 405L 40 F 221 11 50 F 7J F 222 12 40 F 50 F 223 13 7J F 40 F 224 14 50 F 7J F. 225 15 40 F 50 F 226 16 7J F 40 F 227 17 50 154 7J 954 228 18 66 F 55 F 229 19 40 1F 50 F 230 20 7J 954 40 F 231 21 50 F 7J 154 343_ 22 40 F L4 1F 233 23 10 F 40 2F 234 24 L5 1F L0 F 235 25 10 F 40 3F 236 26 L5 2F 66 954 237 27 55 F 40 2F 238 28 L5 3F 66 4F 239 29 55 F 10 2F 240 30 66 954 55 F 241 31 40 5F L5 254 242 32 66 954 7J 5F _1__(Cos h a lull/2 _ 243 33 66 1054 55 F 103 244 34 40 F L5 2F 1/2 245 35 L0 F 40 309F sin h 3 [nl- tan 51 246 36 50 5F 7J 353 -f 1/2 ) 247 37 66 154 55 F l/fln‘ 248 38 40 310F L5 854 sin h 0 [nl * 249 39 40 311F L7 253 103 |n|1/2 250 40 40 F 50 SF 251 41 7J F 40 F 84 Abs. Rel. Addr. Addr. Order Pairs Comments 252 42 50 F 7J 1054 253 43 66 353 55 F 254 44 40 6F 50 454 1/2 255 45 7J 5F 66 383 1 sin h a|n| tan 01 256 46 55 F L4 6F 3 1/2 257 47 40 6F 50 2F 10 p |nf 258 48 7J 254 66 353 259 49 55 F 40 7F tan B2 -1/2 . 92 260 50 50 2F 7J 354 + ln' 31"" a lnl 261 51 66 353 55 F B 1/2 262 52 40 F L5 6F ' C°s aln‘ 263 53 L0 7F LO F tan 61 - cos h alnll/Z 264 54 40 312F 50 5F 265 55 7J 384 66 184 tan 02 266 56 55 F 40 F 267 57 L5 2F L0 F 268 58 40 313F L5 854 269 59 40 314F 40 315F 270 60 40 316F L5 154 271 61 40 317F L5 3054 272 62 40 5888 26 55 SK Radial Bend. Mag. Matrix for n)>1 00 600K 600 0 L5 954 L0 253 601 1 40 F L7 F 602 2 10 4F 66 954 603 3 55 F N0 F 604 4 50 F 50 4L 605 5 26 305L 40 3F 606 6 L5 453 10 4F 607 7 66 954 55 F 608 8 40 F 50 F 609 9 7J 3F 00 2F 610 10 40 2F L1 2F 611 11 50 F 50 11L 612 12 26 408L 40 F 613 13 50 F 7J F 614 14 40 F 50 F 615 15 7J F‘ 40 F 616 16 50 F 7J F 617 17 40 F 50 F 85 Abs Rel. Addr. Addr Order Pahrs Comments 618 18 7J F 40 F 619 19 50 154 7J 934 623 20 66 F 55 F 621 21 40 1F 50 F 622 22 7J 934 40 F 623 23 50 F 73 154 624 24 43 F L4 1F 625 25 10 1F 40 2F 626 26 L5 1F L0 F 627 27 10 F 40 1F 628 28 L5 2F 66 954 629 29 55 F 40 2F 1 1/2 630 30 L5 1F 10 2F "3 (cos hI1—nl a + 631 31 66 3F 55 F 10“ 632 32 66 954 55 F 1/2 633 33 40 4F L5 254 sin hIl-nl a tfin B > 634 34 66 954 7J 4F '1_n‘1/2 “ 1 635 35 66 1054 55 F “ 636 36 L4 2F 40 330F 637 37 50 4F 7J 353 _ 638 38 40 301F L5 154 «9 sin h a L1-nLl/2 639 39 L0 2F 40 F 3 1/2 640 40 50 F 7J 353 10 ll"“' 641 41 40 F L5 954 642 42 L0 253 40 5F j/q 643 43 L5 F 66 5F _Q_.jiw:_pos h al1;n|* “) 644 44 35 F 66 1054 303 Il—nl 645 45 35 F 40 332F ‘ 646 46 50 4F 7J 154 647 47 66 353 55 F 648 48 40 F L7 SF 649 49 40 6F 50 F 650 50 7J 6F 66 954 651 51 55 F 40 6F 652 52 50 4F 7J 454 1 1/2 . 1/2 653 53 66 353 55 F 103('l'n' 31“ h a'l'n‘ 654 54 40 7F 50 2F “ 1/2 655 55 7J 254 66 353 sin h a \1-n| ‘ tan B] 656 53 55 F 40 8* +---~~~ {72 ‘ 657 57 50 2F 7J 35; |1—n|“ 658 58 66 353 55 F 659 59 L4 6F L4 7F tan 8 a 1 660 60 L4 8F 40 303F —~————— + cos h a‘l-n|"/A 661 61 50 4F 7J 354 1 2 662 62 66 154 55 F tan 61 + COS h ”'1’“. / tan 62) Abs. Rel. Addr. Addr. Order Pairs Comments 663 33 L4 2F 40 304F -—l§ (cos h <:L|1-n|1/2 + 664 64 50 302F 7J 354 10 665 65 66 353 55 F . 1 2 666 66 L4 4F 40 305F 31“ h all'nl / tan 52 . 667 67 L5 884 40 306F 1/2 ) 668 68 40 307F L5 154 Il‘n‘ 669 69 40 308F 26 SN 1 [sin h 611-n11/2 _—3' '1/2 + 10 |1—n| {l-cos h afl—nIl/z) I3 'l-n\ tan 2’ 88 Radial Bend. Mag. Matrix for 0 < n < 1 00 670K 670 0 L5 954 L0 253 671 1 10 8F 66 954 672 2 55 F N0 F 673 3 50 F 50 3L 674 4 26 308L 40 SF 675 5 50 5F 7J 453 676 6 66 54 55 F 677 7 00 4F 40 F 678 8 50 F 50 8L 679 9 26 SL 40 9F 680 10 LJ F 50 10L 681 11 26 5L 40 F 682 12 50 F 7J 154 683 13 00 1F 40 2F 1/2 684 14 50 9F 7J 254 -——§ cos (1-n) a + 685 15 10 3F 66 5F 10 686 16 55 F L4 2F 3 1/2 687 17 40 300E 50 9F 31“ (l'n) a tan BL) 688 18 7J 353 10 3F (1-n)l/2 689 19 66 5F 55 F 1/2 690 20 40 301F L5 954 p sin a g1—nL 691 21 L0 253 40 3F 3 1/2 692 22 50 353 7J 954 10 (I’m) 693 23 66 3F 55 F 694 24 40 4F L5 2F 695 25 66 3F 7J 353 696 26 66 1054 55 F 87 Abs. Rel. Addr. Addr. Order Pairs Comments 697 27 40 6F L5 4F 6 (1 _ 698 28 L0 6F 40 302F lo3 (l—n) 699 29 50 5F 7J 9F 700 30 40 7F 50 7F cosa41-n)l/2) 701 31 7J 154 00 SF 702 32 40 7F 50 7F 703 33 7J 154 66 353 704 34 55 F 40 7F __1_ [f (1-n)l/2 705 35 50 7F 7J 454 103 706 36 66 3F 55 F p 1/2 707 37 66 10S4 85 F sin (l—n) a + 708 38 40 8F 50 2F . 1/2 709 39 7J 254 66 353 $1“ ‘1‘“) a ta“ 31 710 40 55 F 40 10F 1/2 711 41 50 2F 7J 354 (l'n) 712 42 66 353 55 F tan B 1/2 713 43 L4 10F L4 8F + cos (l-n) a 714 44 L0 7F 40 303F 715 45 50 9F 7J 3S4 (tan Bl + tan 62) 716 46 66 5F 55 F 1/2 717 47 10 3F 40 8F 1 sin a (l—n) tan 52 718 48 L4 2F 40 304F 3 1/2 719 49 L5 154 L0 2F 10 (1-n) 720 50 66 3F 7J 354 + COS (1—n)l/2 a 721 51 66 1054 55 F 722 52 40 7F 50 301F 723 53 7J 154 66 353 724 54 55 F L4 7F 725 55 40 305F L5 884 726 56 40 306F 40 307F 727 57 L5 154 40 308F 728 58 N0 F 26 SN SN Axial Bend. Mag. Matrix for O < n . 00 730K 730 0 L5 253 10 8F 731 1 66 954 55 F ~732 2 50 F 50 2L 733 3 26 305L 40 SF 734 4 50 5F 7J 453 735 5 66 54 55 F 88 Abs. 1 Rel. Addr. Ader Order Pairs Comments 736 6 00 4F 40 F 737 7 50 F 50 7L 738 8 26 SL 40 9F 739 9 LJ F 50 9L 740 10 26 5L 40 F 741 11 50 F 73 154 742 12 00 1F 40 2F 743 13 50 9F 73 154 744 14 10 3F 66 5F 1 1/2 745 15 85 F 40 3F ———§[Cos n 01 - 746 16 50 3F 73 254 10 747 17 66 154 55 F . 1/2 748 18 40 4F L5 2F 31“ n a tan B1 749 19 L0 .4F 40 309F 1/2 750 20 50 3F 73 353 $/2 751 21 66 184 85 F Q sin n a 752 22 40 310F L5 854 103 n1/2 753 23 40 311F 50 9F 754 24 73 5F 40 4F 755 25 50 4F 73 154 756 26 00 5F 40 4F 757 27 50 4F 73 154 758 28 66 353 55 F 759 29 40 4F 50 9F 760 30 73 454 40 SF 761 31 50 5F 73 154 762 32 66 353 55 F 763 33 00 1F 40 5F 1 _ n1/2 sin a n1/2 + 764 34 50 2F 73 254 lo3p 765 35 66 353 55 F 766 36 40 6F 50 2F . 1/2 767 37 73 354 66 353 $1“ a n tan 61 tan B2 768 38 55 F 40 7F 1/2 769 39 L5 5F L0 4F . ’ °°S a n (tan B1 + 770 40 L0' 6F L0 7F tan 6 ] 771 41 40 312F 50 3F 2 772 42 73 354 66 154 .___ COS a n1/2 _ 773 43 55 F 40 4F 103 774 44 L5 2F LO 4F 1/2 775 45 40 313F L5 854 sin a n tan 62 776 46 40 314F 40 315F 1/2 ) 777 47 40 316F L5 154 n 778 48 40 317F 26 53 SJ Print Headings and Bend. Mag. Matrices Abs. Rel. lgggr. Addr. Order Pairs Comments 00 780K 780 0 92 131F 92 7F 781 1 92 259F 92 643F MAG 782 2 92 387F 92 579F 783 3 92 707F 92 139F 784 4 92 7F 92 963F 785 5 92 259F 92 387F 786 6 92 962F 92 2F 787 7 92 771F 92 387F ALPHA 788 8 92 707F 92 835F 789 9 92 963F L5 453 790 10 50 10F 50 10L Print a/lO 791 11 26 758L 92 131F 792 12 92 7F 92 971F 793 13 92 259F 92 258F 794 14 92 771F 92 578F RHO 795 15 92 707F 92 835F 796 16 92 963F L5 353 797 17 50 10F 50 17L Print p/103 798 18 26 758L 92 131F 799 19 92 7F 92 979F 800 20 92 259F 92 770F N 801 21 92 707F 92 835F 802 22 92 963F L5 253 803 23 50 10F 50 23L Print n/lO 804 24 26 755L 92 131F 805 25 92 7F 92 259F 806 26 92 195F 92 194F BETA, 1 807 27 92 322F 92 387F 808 28 92 707F 92 323F 809 29 92 66F 92 835F 810 30 92 963F L5 53 - 811 31 50 10F 50 31L Print Bl/lo 812 32 26 755L 92 131F 813 33 92 7F 92 259F 814 34 92 195F 92 194F 815 35 92 322F 92 387F BETA, 2 816 36 92 707F 92 323F 817 37 92 130F 92 835F 818 38 92 963F L5 153 819 39 50 10F 50 39L Print 52/10 90 223;. :2;;. Order Pairs Comments 820 40 26 758L 92 139F 821 41 92 7F 92 259F 822 42 92 643F 92 451F MX 823 43 92 707F 92 135F 824 44 92 7F L5 1684 825 45 42 48L L5 1384 826 46 40 4F L5 1484 827 47 40 5F 40 6F 828 48 N0 F L5 3OOF 829 49 50 10F 50 49L Print bending magnet 830 50 26 758L F5 48L matrices 831 51 42 48L L5 SF 832 52 L0 1284 40 SF 833 53 32 48L 92 131F 834 54 92 7F L5 1484 835 55 40 5F L5 6F 836 56 L0 1284 40 6F 837 57 32 48L L5 4F 838 58 L0 1284 40 4F 839 50 36 60L 22 63L 840 60 92 135F 92 259F MY 841 61 92 643F 92 386F 842 62 92 135F 92 707F 843 63 22 46L L5 1684 844 64 42 71L L4 1184 845 65 42 93L L5 2484 846 66 42 72L 42 78L 847 67 L4 1184 L4 1184 848 68 42 88L 42 94L 849 69 L5 1484 40 4F 850 70 L5 1484 40 2F 851 71 40 3F L5 300F 852 72 N0 F 40 34OF Restore matrices 853 73 F5 71L 42 71L 854 74 F5 72L 42 72L 855 75 F5 78L 42 78L 856 76 L5 2F L0 1284 857 77 40 2F 32 71L 858 78 L5 884 40 340F 859 79 F5 72L 42 72L 860 80 F5 78L 42 78L 861 81 L5 3F L0 1284 862 82 40 3F 36 78L 91 Abs. Rel. Addr. Addr. Order Pairs Comments 863 83 L5 4F L0 1284 864 84 40 4F 36 70L 865 85 L5 1484 40 4F 866 86 L5 1484 40 2F 867 87 40 3F NO F 868 88 L5 884 40 358F 869 89 F5 88L 42 88L 870 90 F5 94L 42 94L 871 91 L5 2F L0 1284 872 92 40 2F 36 88L 873 93 41 F L5 309F 874 94 N0 F 40 358F 875 95 F5 88L 4O 88L 876 96 F5 94L 42 94L 877 97 F5 93L 42 93L 878 98 L5 3F L0 1284 879 99 40 3F 36 93L 880 100 L5 4F L0 1284 881 101 40 4F 36 86L 882 102 NO F 26 238F Relative Orientation and Separation Matrices‘R,and D. 00 885K 885 0 L5 883 10 SF 886 1 66 S4 85 F 887 2 66 984 85 F 888 3 00 5F 40 SF 889 4 50 F 50 4L 890 5 26 SL 40 6F 891 6 LJ SF 50 6L 892 7 26 SL 40 7F 893 8 LS 2584 42 10L 894 9 L5 1584 40 2F 895 10 L5 884 40 412F 896 11 F5 10L 42 10L 897 12 L5 2F L0 1284 898 ‘ 13 40 2F 36 10L 899 14 L5 7F 40 412F 900 15 40 419E 40 ‘433F 901 16 40 440F 40 448F 902 17 40 455E 40 469F 92 Abs. Rel. Addr. Addr. Order Pairs Comments 903 18 40 476F L5 6F 904 19 40 415F 40 422E Form 3 and store 905 20 40 466F 40 473F 2 906 21 L1 6F 40 430E 907 22 40 437F 40 451F 908 23 40 458F 49 F 909 24 40 426F 40 447E 910 25 40 462E 40 483F 911 26 L5 2684 42 28L 912 27 L5 3284 40 F 913 28 L5 884 40 484E 914 29 F5 28L 40 28L 915 30 L5 F L0 1284 916 31 40 F 36 28L 917 32 L5 983 40 485E D 918 33 40 506F 50 184 Form‘——Z and store 919 34 7J 984 40 484E 10 920 35 40 491E 40 498F 921 36 40 505E 40 512E 922 37 40 519F L5 2284 923 38 40 25F 40 26F 924 39 92 139F 92 259F 925 40 92 67F 92 707F D 926 41 92 835F 92 963F 927 42 L5 983 N0 F 4 928 43 50 10F 50 43L Print d/10 929 44 26 758L 92 963F 930 45 92 259F 92 322F 931 46 92 771F 92 194F 932 47 92 322F 92 387F THETA 933 48 92 707F 92 835F 934 49 92 963F L5 883 2 935 50 50 10F 50 50L Print 9/10 936 51 26 758L 92 131F 937 52 92 7F 22 188F SF Combined Matrix . 00 940K 940 0 L5 2884 42 2L 941 1 L5 3284 40 F 942 2 L5 884 40 520F 93 Abs. Rel Addr. Addr. Order Pairs Comments 943 3 F5 2L 40 2L 944 4 L5 F L0 1284 Form a unit matrix 945 5 40 F 36 2L and store in place of 946 6 50 184 7J 984 "previous" matrix. 947 .7 40 520E 4O 527F 948 8 4O 534F 40 541F 949 9 40 548F 40 555F 950 10 50 F 50 10L Enter special input 951 11 N6 F 26 3860F‘ routine 952 12 L5 3384 L0 2884 953 13 L0 2884 42 20L. 954 14 L0 1284 42 18L. 955 15 L5 18L L4 1083 956 166 42 18L L5 20L 957 17 L4 1083 42 20L 958 18 26 885F 26 18L Test for parameter, b. 959 19 NO F 26 85 If b = 1 -——*-85 960 20 26 1270F 26 20L If h = 2 -—--1270 961 21 N0 F 26 37L 962 22 N0 F 26 58L 963 23 L5 2584 40 3901F 964 24 L5 2684 40 3902F Enter matrix multi. 965 25 L5 3184 40 3903F routine. 966 26 L5 2384 40 3904F A 967 27 50 F 50 27L Form 4 968 28 26 3900F L5 3284 2 x 10 969 29 40 F L5 3184 970 30 42 31L 42 32L 971 31 N0 F L5 556F 972 32 00 2F 40 556F 973 33 F5 31L 42 31L 974 34 42 32L L5 F 975 35 L0 1284 40 F 976 36 32 31L 22 20L 977 37 L5 2484 40 3901F 978 38 L5 3184 40 3902F Form 2MRD 979 39 L5 3484 40 3903F 104 980 40 L5 2384 40 3904F 981 41 50 F 50 41L 982 42 26 3900F L5 3284 983 43 40 ‘F L5 3484 984 44 42 45L 42 47L 985 45 NO F L5 2036F 986 46 66 184 85 F 94 Abs. Rel. Addr. Addr. Order Pairs Comments 987 47 N0 F 40 2036F 988 48 F5 45L 42 45L 989 49 42 47L L5 F 990 50 L0 1284 40 F 991 51 36 45L L5 2984 992 52 40 3901F L5 3454 Form 81mm 9993 53 40 3902F L5 3184 104 994 54 40 3903F L5 2384 995 55 40 3904F NO F 996 56 50 F 50 56L 997 57 26 3900F 22 79L 998 58 L5 2784 40 3901F 999 59 L5 3184 40 3902F 1000 60 L5 3484 40 3903F Form 2QRD 1001 61 L5 2384 40 3904F 10 1002 62 50 F 50 62L 1003 63 26 3900F- L5 3284 1004 64 40 F L5 3484 1005 65 42 66L 42 69L 1006 66 N0 F L5 2036F 1007 67 66 184 85 F 1008 68 66 984 85 F 1009 69 N0 F 40 2036F 1010 70 F5 66L 42 66L 1011 71 42 69L L5 F 1012 72 L0 1284 40 F 1013 73 36 66L L5 2954 Form fiQRD 1014 74 40 3901F L5 3484 104 1015 75 40 3902F L5 3184 1016 76 4O 3903F L5 2384 1017 77 40 3904F N0 F 1018 78 50 F 50 78L 1019 79 26 3900F L5 ”3184 1020 80 40 3901F L5 2884 1021 81 40 3902F L5 3484 Form the product .1022. 82 40 3903F L5 2384 1023 83 40 3904F N0 'F of YUM °Z Q) RD times 1024 84 50 F 50 84L 10 1025 85 26 3900F L5 3284 the "previous" matrix 1026 86 40 F L5 3484' and store as "previous" 1027 87 42 88L 42 91L matrix. 1028 88 N0 F L5 2036F 1029 89 66 184 85 F 95 Abs. Rel. 1 Addr. Addr. Order Pairs Comments 1030 90 66 984 85 F 1031 91 N0 F 40 2036F 1032 92 F5 88L 42 88L 1033 93 42 91L L5 F 1034 94 L0 1284 40 F 1035 95 36 88L L5 3284 1036 96 40 F L5 3484 1037 97 42 98L L5 2884 1038 98 42 99L L5 2036F 1039 99 N0 F 40 520F 1040 100 F5 98L 42 98L 1041 101 F5 99L 42 99L 1042 102 L5 F L0 1284 1043 103 40 F 32 98L 1044 104 L5 2684 L4 3184 1045 105 L4 2384 L4 1384 1046 106 42 108L L5 108L Test for parameter, C. 1047 107 L4 1183 42 108L If c = 0. go back to 10L 1048 108 N0 F 26 lO9L If c = 1, proceed with 1049 109 N0 F 26 10L print out 1050 110 L5 3484 42 115L 1051 111 92 135F 92 7F 1052 112 92 259F 92 643F M 1053 113 92 135F 92 707F 1054 114 L5 2284 40 SF 1055 115 40 6F L5 2036F 1056 116 50 9F 50 ll6L Print final matrix 1057 117 26 758L F5 115L 1058 118 42 115L L5 SF 1059 119 L0 1284 40 SF 1060 120 32 115L 92 131F 1061 121 92 7F L5 2284 1062 122 40 5F L5 6F 1063 123 L0 1284 40 6F 1064 124 32 115L 26 1070F Calculation of the Properties of the System and Their Print Out 00 1070K 1070 0 N0 F 92 131F 1071 1 92 7F 92 979F FX AX BX 1072 2 92 259F 92 898F 3 1073 92 451F 92 1003F 96 Abs. Rel. Addr. Addr. Order Pairs Comments 1074 4 92 387F 92 451F 1075 5 92 1003F 92 195F 1076 7 92 707F 92 7F 1078 8 N0 F L5 2042F 1079 9 66 954 55 F 1080 10 40 10F 50 154 Calculate fx 1081 11 75 154 40 F 1082 12 50 F 75 1054 1083 13 40 4F L7 4F 1084 14 L2 10F 36 153L 1085 15 L1 4F 66 10F 1086 16 55 F 40 10F 5 1087 17 50 10F 50 17L Print fx/lO 1088 18 26 758L L5 2043F 1089 19 66 954 55 F 1090 20 40 11F 50 11F Calculate ax 1091 21 75 10F 66 154 1092 22 55 F 40 11F 1093 23 50 10F 50 23L 5 1094 24 26 758L L5 2036F Print ax/lo 1095 25 66 954 55 F 1096 26 40 12F 50 12F 1097 27 75 10F 66 154 calculate bx 1098 28 55 F 40 12F 5 1099 29 50 10F 50 29L Print b /10 1100 30 26 758L 92 131F x 1101 31 92 7F 92 979F 1102 32 92 259F 92 898F 1103 33 92 386F 92 1003F FY AY BY 1104 34 92 387F 92 386F 1105 35 92 1003F 92 195F 1106 36 92 386F 92 131F 1107 37 92 707F 92 7F 1108 38 N0 F L5 2063F 1109 39 66 984 85 F Calculate f 1110 40 40 15F 50 154 Y 1111 41 75 154 40 F 1112 42 50 F 75 1054 1113 43 40 4F L7 4F 1114 44 L2 15F 32 155L 1115 45 L1 4F 66 15F 1116 46 55 F 40 15F 5 1117 47 50 10F 50 47L Print f /10 1118 48 26 755L L5 2064F Y 97 Abs. Rel. Addr. Addr. Order Pairs Comments 1119 49 66 954 55 F 1120 50 40 16F 50 16F Calculate a 1121 51 75 15F 66 154 Y 1122 52 55 F 40 16F 1123 53 50 10F 50 53L Print a /105 1124 54 26 758L L5 2057F Y 1125 55 66 954 55 F 1126 56 40 17F 50 17F 1127 57 75 15F 66 154 Calculate b 1128 58 55 F 40 17F Y 1129 59 50 10F 50 59L Print by/105 1130 60 26 758L 92 135F 1131 61 92 7F 92 967F 1132 62 92 259F 92 67F 1133 63 92 578F . 92 451F 1134 64 92 983F 92 643F 1135 65 92 451F ' 92 991F 1136 66 92 67F 92 514F- 1137 67 92 451F 92 975F 00x MX 01x 1138 68 92 258F 92 194F 1139 69 92 706F 92 707F RES. RAD. DOY 1140 70 92 643F 92 963F 1141 71 92 259F 92 258F MY DIY RES. Ax. 1142 72 92 387F 92 67F 1143 73 92 707F 92 643F 1144 74 92 259F 92 967F 1145 75 92 67F 92 578F 1146 76 92 386F 92 983F 1147 77 92 643F 92 386F 1148 78 92 99lF 92 67F 1149 79 92 514F 92 386F 1150 80 92 975F 92 258F 1151 81 92 194F 92 706F 1152 82 92 707F 92 643F 1153 83 92 963F 92 259F 1154 84 92 387F 92 451F 1155 85 92 707F 92 643F 1156 86 92 131F 92 7F 1157 87 L5 9OSL L0 165L 1158 88 40 905L L5 1353 5 1159 89 J0 7F 50 89L Print dOX/lO 1160 90 26 755L 26 158L 99 Abs. Rel. Addr. Addr. Order Pairs Comments 1161 91 N0 F L5 1353 1162 92 L0 11F 40 13F 1163 93 50 10F 75 154 1164 94 40 5F L7 5F Calc. M.X 1165 95 L2 13F 36 160L 1166 96 L5 5F 66 13F 1167 97 75 1054 40 13F 5 1168 98 50 8F 50 98L Print Mx/lo 1169 99 26 755L 50 10F 1170 100 75 13F 66 154 1171 101 55 F 66 1054 1172 102 55 F L4 12F Calc. d. 1173 103 40 14F N0 F 1X 5 1174 104 50 8F 50 104L Print dix/lO 1175 105 26 758L N0 F 1176 106 50 2044F 75 14F 1177 107 66 154 55 F a 1178 108 66 954 55 F 1179 109 40 F 50 2038F 1180 110 75 954 L4 F 5 1181 111 40 20F 50 13F Calc. radial resolution/10 1182 112 75 1054 40 6F 1183 113 L7 6F L2 20F 1184 114 N0 F 36 161L 1185 115 L5 6F 66 20F 1186 116 55 F 40 '6F 1187 117 L7 6F NO F 1188 118 JO 8F 50 118L Print radial resolution 1189 119 26 755L L5 1453 1190 120 30 7F 50 120L 1191 121 26 755L 26 162L Print d 5 1192 122 N0 F L5' 1453 °Y/10 1193 123 L0 16F 40 18F 1194 124 50 15F 75 154 1195 125 40 5F L7 SF 1196 126 L2 18F 36 152L Calc. M 1197 127 L5 5F 66 18F Y 1198 128 75 1054 40 18F 1199 129 50 8F 50 129L Print M /105 1200 130 26 758L 50 15F Y 1201 131 75 18F 66 154 1202 132 55 F 66 1054 1203 133 55 F L4 17F Calc. d. 1204 134 40 19F N0 F lY 1205 135 50 8F 50 135L 100 Abs. Rel. Addr. Addr. Order Pairs Comments 1206 136 26 758L 50 19F Print di /105 1207 137 75 2062F 66 954 Y 1208 138 85 F 66 184 1209 139 85 F 40 F 1210 140 50 2056F 75 984 1211 141 L4 F 40 21F 1212 142 50 18F 75 1084 1213 143 40 7F L7 7F 1214 144 L2 21F 36 152L Calc. axial resolution 1215 145 L5 7F 66 21F 1216 146 S5 F 40 7F 1217 147 L7 7F N0 F 5 1218 148 JO 8F 50 148L Print axial resolution/10 1219 149 26 758L N0 F 1220 150 L5 9OSL L4 165L 1221 151 40 9OSL 26 167L 1222 152 92 651F ‘26 150L 1223 153 F5 158L 42 158L 1224 154 F5 173L 42 173L 1225 155 22 30L F5 163L 1226 156 42 163L F5 178L' 1227 157 42 178L 22 60L 1228 158 N0 F 26 159L 1229 159 26 91L N0 F 1230 160 92 999F 92 999F 1231 161 92 995F 22 119L 1232 162 N0 F 26 163L 1233 163 26 122L N0 'F 1234 164 26 152L N0 F 1235 165 00 4F 00 F 1236 166 00 F 00 1F Constants 1237 167 L5 1383 L4 1583 Increment d and d 1238 168 40 1383 L5 1483 OX 0 1239 169 L4 1683 40 1483 1240 170 92 131F 92 7F 1241 171 L5 1283 L0 166L 1242 172 40 1283 36 87L 1243 173 N0 F 26 174L 1244 174 22 178L N0 F 1245 175 L5 158L L0 166L 1246 176 42 158L L5 173L 1247 177 L0 166L 42 173L 1248 178 N0 F 26 179L 1249 179 26 183L N0 F 101 Abs. Rel. Addr. Addr. Order Pairs Comments 1250 180 L5 163L L0 166L 1251 181 42 163L L5 178L 1252 182 L0 166L 42 178L 1253 183 24 SF 26 4071F Black sw., white sw. stop. Divergence Quadrupole Sub—matrix 00 1270K 1270 0 L5 3254 40 F 1271 1 L5 884 40 376F 1272 2 F5 1L 42 1L 1273 3 L5 F L0 1254 1274 4 40 F 36 1L 1275 5 50 553 73 683 1276 6 10 4F 66 154 1277 7 55 F 40 F 1278 8 L1 F N0 F 1279 9 50 F 50 9L 1280 10 26 4OSL 40 F 1281 11 50 F 73 F 1282 12 40 F 50 F 1283 13 73 F 40 F 1284 14 50 F 73 F 1285 15 40 F 50 F 1286 16 73 F 40 F 1287 17 50 154 73 954 1288 18 66 F 55 F 1289 19 40 1F 50 F 1290 20 73 954 40 F 1291 21 50 F 73 154 1292 22 40 F L4 1F cos h KL 1293 23 10 1F 40 6F 104 1294 24 L5 1F L0 F 1295 25 10 1F 40 3F sin h KL 1296 26 L5 3F 66 553 104 1297 27 73 954 40 7F 1298 28 50 3F 73 553 1_ sin h KL 1299 29 66 954 55 F K 104 1300 30 40 8F L5 2754 1301 31 42 1L 26 1330F 102 Convergence Quadrupole Sub—matrix Abs. Rel. Addr. Addr. Order Pairs Comments 00 1305K 1305 0 50 583 7J 683 1306 1 10 4F 66 S4 1307 2 85 F 66 1084 1308 3 85 F 00 4F 1309 4 40 4F NO F 1310 5 50 F 50 SL 1311 6 26 SL 40 SF 1312 7 50 5F 7J 184 1313 8 00 1F 40 5F 1314 9 LJ 4F 50 9L 1315 10 26 SL 40 6F 1316 11 50 6F 7J 184 cos KL 1317 12 00 1F 40 F ——-4_— 1318 13 50 F 7J 984 10 1319 14 40 6F 50 SF 1320 15 7J 1084 66 583 1_sin KL 1321 16 85 F 40 7F K 104 1322 17 50 5F 79 583 1323 18 40 8F 26 1339F -K sin KL 104 S6 Quadrupole Matrix '00 1330K 1330 0 L5 2884 L4 2684 1331 1 L4 2484 L0 1384 1332 2 42 BL 42 9L 1333 3 L4 1284 42 10L 1334 4 42 11L L5 BL 1335 5 L4 783 42 8L 1336 6 42 9L L5 10L 1337 7 L4 783 42 10L 1338 8 42 11L 26 11L 1339 9 N0 F 22 11L Test for parameter. a 1340 10 N0 F 26 13L and transfer to appropriate 1341 11 N0 F 22 13L locations. 1342 12 26 16L 26 22L 1343 13 26 22L 26 16L 1344 14 26 1305F 26 '26L 1345 15 26 26L 26 1305F 1346 16 50 184 7J 9S4 103 Abs. Rel. Addr. . Addr. Order Pairs Comments 1347 17 40 390F 4O 411F Form radial part of 1348 18 L5 6F 40 376F quadrupole matrix. 1349 19 40 383F L5 7F 1350 20 40 377E L5 8F 1351 21 40 382E 26 10L 1352 22 L5 6F 40 397F 1353 23 40 404F L5 7F Form axial part of 1354 24 40 398E L5 8F quadrupole matrix 1355 25 40 403F 26 11L 1356 26 92 135F 92 7F 1357 27 92 259F 92 642F K 1358 28 92 707F 92 835F 1359 29 92 967F L5 583 1360 30 50 10F 50 30L Print K/10 1361 31 26 758L 92 975F 1362 32 92 259F 92 962F L 1363 33 92 707F 92 835F 1364 34 92 967F L5 683 2 1365 35 50 10F 50 35L Print L/10 1366 36 26 758L 92 971F 1367 37 92 259F 92 387F 1368 38 92 707F 92 579F Print a = 1 or a = 2 1369 39 L5 2884 L4 2884 1370 40 L4 2484 L0 1484 1371 41 L0 1284 42 43L 1372 42 L5 43L L4 783 1373 43 42 43L 26 43L 1374 44 92 66F 22 45L 1375 45 92 130F 92 131F 1376 46 92 7F 26 238F Special Input Sub—routine 00 3860K 3860 0 K5 F L4 32L 3861 1 42 29L 46 24L 3862 2 10 2F 46 29L 3863 3 10 2F 46 25L 3864 4 41 36L 41 33L 3865 5 50 33L F5 33L 3866 6 40 34L 81 4F 3867 7 L0 30L 36 27L 104 Abs. Rel. Addr. Addr. Order Pairs Comments 3868 8 L4 30L 10 3F 3869 9 L4 33L 00 2F 3870 10 L4 33L 00 1F 3871 11 40 33L L5 34L 3872 12 00 2F L4 34L 3873 13 00 1F 40 34L 3874 14 91 4F 32 8L 3875 15 L4 31L 40 35L 3876 16 F3 35L 36 22L 3877 17 L3 35L 32 19L 3878 18 L5 33L 66 34L 3879 19 26 20L 50 33L 3880 20 L3 36L 32 23L 3881 21 81 F 26 24L 3882 22 L5 33L 42 24L 3883 23 26 4L 85 F 3884 24 N2 F 40 F 3885 25 N2 F F5 24L 3886 26 42 24L 26 4L 3887 27 40 36L L5 34L 3888 28 L0 36L 32 6L 3889 29 N6 F 26 F 3890 30 00 F 00 10F 3891 31 7L 4095L LL 4086F 3892 32 00 F 00 1F Matrix Multiplication Sub-routine. 00 3900K 3900 0 K5 F 26 15L 3901 1 00 F 00 F‘raal where 311' bll and 3902 2 00 F 00 F.K . 3903 3 00 F 00 F‘ bll are the locations K of the 1st element 3904 4 00 F 00 6F c . 3905 5 00 F 00 F 11 of matrices A, B. 3906 6 00 F 00 1F n and C,where the pro— 3907 7 00 F 00 2F duc? AB = C 13 3908 8 00 F 00 11F aegired' 3909 9 00 F 00 11F A' B? and C 3r? 3910 10 00 F 00 11F all n x n matrices. 3911 11 00 F 00 11F Constants 3912 12 00 F 00 9F 3913 13 00 F 00 F 3914 14 00 F 00 F 105 Abs. Rel. Addr. Addr. Order Pairs Comments 3915 15 42 55L L5 4L 3916 16 L4 4L L0 6L 3917 17 40 BL 40 9L 3918 18 40 10L 40 11L 3919 19 L5 8L L0 7L 3920 20 40 12L L5 1L 3921 21 40 13L 42 24L 3922 22 L5 2L 40 14L 3923 23 42 25L L5 3L 3924 24 42 40L 50 F 3925 25 N0 F 7J F 3926 26 N0 F 40 F 3927 27 F5 24L 42 24L 3928 28 L5 25L L4 4L 3929 29 40 25L F5 26L 3930 30 40 26L L5 9L 3931 31 L0 7L 40 9L 3932 32 32 24L L5 BL 3933 33 40 9L L5 1F 3934 34 L4 F 40 F 3935 35 F5 33L 42 33L 3936 36 L5 12L L0 7L 3937 37 40 12L 32 33L 3938 38 L5 8L L0 7L 3939 39 40 12L L5 F 3940 40 N0 F 40 F 3941 41 L5 13L 42 24L 3942 42 F5 14L 42 14L 3943 43 42 25L L5 SL 3944 44 42 26L L5 6L 3945 45 42 33L F5 40L 3946 46 42 40L L5 10L 3947 47 L0 7L 40 10L 3948 48 32 24L L5 BL 3949 49 40 10L L5 13L 3950 50 L4 4L 40 13L 3951 51 42 24L L5 2L 3952 52 40 14L 42 25L 3953 53 L5 11L L0 7L 3954 54 40 11L 32 24L 3955 55 N0 F 22 F 106 Abs. Addr. Sub-routines 3960 T5 Sine, Cosine Sub-routine. 3990 R1 Square root sub—routine. 4000 64 - 82 Exponent Sub-routine. 4035 P2 Print out Sub-routine. REFERENCES Judd, David L., "Focusing Properties of a Generalized Magnetic Spectrometer," Review of Scientific Instruments, 21, 213 (1950). Kerst, D. W., and Serber, R., "Electronic Orbits in the Induction Accelerator," Physical Review, 60, 53 (1941). Enge, H. A., "Ion Focusing Properties of a Quadrupole Lens Pair," Review of Scientific Instruments, 30, 248 (1959). Penner, 8., "Calculations of Properties of Magnetic Deflection Systems," Review of Scientific Instruments, 32, 150 (1961). Livingood, J. J. Principles of Cyclic Particle Accelerators, Princeton, D. Van Nostrand, 1961. 107 HICHIGRN STQTE UNIV. LIBRQRIES 31293017640396