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Monte Carlo simulation and aspects of the magnetostatic design of the TRIUMF second arm spectrometer Duncan, Fraser Andrew 1988

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Monte Carlo Simulation and Aspects of the Magnetostatic Design of the TRIUMF Second Arm Spectrometer by Eraser Andrew Duncan B .A .Sc . University of Brit ish Columbia, 1985 A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F M A S T E R O F A P P L I E D S C I E N C E in T H E F A C U L T Y O F G R A D U A T E S T U D I E S D E P A R T M E N T O F P H Y S I C S We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A August 1988 © E r a s e r Andrew Duncan, 1988 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. 1 further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of PHYSICS The University of British Columbia Vancouver, Canada Date £ OCT H 9 8 DE-6 (2/88) Abstract The optical design of the TRIUMF Second Arm Spectrometer (SASP) has been completed and the engineering design started. The effects of the dipole shape and field clamps on the aperture fringe fields were studied. It was determined that a field clamp would be necessary to achieve the field specifications over the desired range of dipole excitations. A specification of the dipole pole edges and field clamps for the SASP is made. A Monte Carlo simulator for the SASP was written. During the design this was used to study the profiles of rays passing through the SASP. These profiles were used in determining the positioning of the dipole vacuum boxes and the SASP detector arrays. The simulator is intended to assess experimental arrangements of the SASP. ii Contents A b s t r a c t ii L i s t o f T a b l e s v i i i L i s t o f F i g u r e s ix A c k n o w l e d g e m e n t s x i i 1 I n t r o d u c t i o n 1 1.1 W h y a Magnetic Spectrometer 2 1.2 T h e S A S P Spectrometer 5 1.3 Designing S A S P 7 1 Designing the SASP 12 2 S p e c t r o m e t e r s 13 2.1 Magneto-optical Systems 13 2.2 Magnetic Spectrometers 19 3 S A S P D i p o l e 24 3.1 R A Y T R A C E the Program 24 3.1.1 Describing Dipoles in R A Y T R A C E 26 3.1.2 Fringe Fields in R A Y T R A C E 30 3.2 Dipole Layout 31 3.3 Interior Fields 36 3.3.1 Magnetic Field of a Clamshell Dipole 36 3.3.2 The S A S P Dipole's Internal Field 38 3.3.3 Dipole Design 40 iii 3.3.4 Dipole Saturation 44 4 D i p o l e A p e r t u r e s 50 4.1 Fringe Fields • • 51 4.2 The SASP Dipole's Apertures 60 4.3 Pole Edge Boundary 64 4.4 Pole Edge Bevels 68 4.5 , Field Clamps 71 4.5.1 Modifying Fringe Fields 71 4.5.2 Modelling a Field Clamp 74 4.5.2.1 Clamp Position (Sp and Dp) 77 4.5.2.2 Coil Position (Sc and SCF) 79 4.5.2.3 Clamp Thickness (tF) and Return Yokes 80 4.5.3 Comparing Computer and Scale Models 82 4.6 Clamp Design 84 4.7 Fitting the Clamp and Pole Edge 87 4.7.1 Fitting Procedure 87 4.7.1.1 Measuring V F B P E and SF 88 4.7.1.2 Fitting P E B and F C B 91 4.7.2 P E B and F C B Tolerances 96 4.7.3 Saturation Effects 98 4.7.4 Quality of the Calculation 98 II Simulating the SASP 102 5 M o n t e C a r l o M e t h o d 103 5.1 E A S Y , a Monte Carlo Simulator 110 5.2 Modeling the SASP Optics 116 5.2.1 Global Matrices 117 5.2.2 Aperture Cuts 121 5.2.2.1 Quadrupoles 126 5.2.2.2 Dipole 127 5.2.2.3 Vacuum Boxes 129 5.2.3 EASY's sasp_optics Device 129 5.3 Two Body Kinematics 131 iv 6 M o n t e C a r l o S t u d i e s o f the S A S P 135 6.1 High Resolution Solid Angle 136 6.2 Limit ing Apertures 138 6.3 Target Plane Acceptance 141 6.4 Downstream Ray Profiles 144 6.5 Mapping the Focal Surface 144 7 S A S P D e t e c t o r s 152 7.1 Vertical Drift Chambers 154 7.1.1 V D C Construction 155 7.1.2 V D C Modelling 156 7.2 Front E n d Chamber 157 7.2.1 F E C Construction 157 7.2.2 F E C Modelling 158 7.3 Track Reconstruction 158 8 S p e c t r o m e t e r M o d e l s 159 8.1 S A S P Model 160 8.1.1 S A S P Device Definitions 161 8.1.2 S A S P E D A C Definitions 162 8.1.3 S A S P Track Reconstruction 165 8.2 M R S Model 165 III Summary and Appendices 168 9 S u m m a r y 169 A p p e n d i c e s 175 A R e s o l v i n g M i s s i n g M a s s 175 B D i s p e r s i o n M a t c h i n g 178 C M a g n e t o s t a t i c F i e l d M o d e l l i n g 181 C . l Solving Magnetostatic Field Equations 181 C.2 P O I S S O N the Program 184 C.3 Magnetic Materials 185 v D T h e L E P S 188 D . l Field Mapping 191 D.2 Modell ing the L E P S Fringe Fields 196 D.3 Modell ing the L E P S Interior Fields 199 D. 4 Summary of the L E P S Studies 203 E P O I S S O N I n p u t F i l e s 204 E . l S A S P Field Clamp 204 E.2 S A S P Dipole Interior Field 206 E.3 L E P S Dipole Fringe Field 207 E.4 L E P S Dipole Interior Field 209 E . 5 P O I S S O N Input File 210 F A p e r t u r e S p e c i f i c a t i o n 212 F . l Pole Edge Bevel 213 F.2 Vacuum Box Design 213 F.2.1 Entrance 213 F.2.2 Exit 216 F.3 Fie ld C lamp Design 219 F.4 P E B and F C B Specification 219 G R a n d o m N u m b e r s 229 H 2 B o d y K i n e m a t i c s 231 H . l Calculating C M Energies and Momentums 231 H. 2 A iming at a Detector 232 I E A S Y I n p u t F i l e s 234 I. 1 Solid Angle 234 1.2 Limit ing Aperture 235 1.3 Target Plane Acceptance 236 1.4 Downstream Ray Profiles 237 1.5 Focal Surface 239 J T r a c k R e c o n s t r u c t i o n U s i n g V D C s 240 K E A S Y ' s S A S P M o d e l 244 vi K . l Device Fi le 244 K.2 Data Acquisition Fi le 251 L E A S Y ' s M R S M o d e l 253 L . l Device Fi le 253 L.2 Data Acquisition File 259 B i b l i o g r a p h y 261 vii List of Tables 1.1 S A S P Specifications 6 3.1 S A S P Dipole R A Y T R A C E Parameters 32 3.2 Extreme S A S P Rays 40 4.1 V F B P E and SF Data 92 5.1 Standard Deviations of Errors in S A S P Matrices 120 5.2 The S A S P Aperture Cuts . 1 2 5 6.1 S A S P Solid Angles 138 6.2 Apertures Hit by Rays 139 6.3 S A S P Effective Apertures 140 8.1 S A S P Detector Addresses 163 8.2 S A S P Detector Booleans 164 8.3 Reconstructing the S A S P Canonical Ray 165 8.4 M R S Detector Addresses 167 8.5 M R S Detector Booleans 167 8.6 Reconstructing The M R S Canonical Ray 167 D . l L E P S Dipole R A Y T R A C E Parameters 190 D.2 L E P S Track Positions 191 F . l P E B and F C B Coefficients 223 F.2 Entrance Curves 224 F.3 Exit Curves 226 viii List of Figures 1.1 S A S P Elevation View 8 1.2 S A S P P lan View 9 1.3 S A S P Dipole Layout 11 2.1 A Magnetic System 14 3.1 R A Y T R A C E Dipole Layout 26 3.2 Clamshell Dipole 33 3.3 S A S P Dipole Pole Piece Geometry 34 3.4 Wedge Shaped Equipotential surfaces 37 3.5 R A Y T R A C E Trajectories 41 3.6 Interior Field of Dipole 43 3.7 Ratio of P O I S S O N Field to R A Y T R A C E field 44 3.8 New Dipole A i r Gap 45 3.9 S A S P Dipole Excitation Spectrum 47 4.1 Fringe Field Effects 53 4.2 Aperture Fil l ing 58 4.3 Saturation Shifts the V F B 60 4.4 Dipole Aperture Fringe Fields 62 4.5 Determining the Pole Edge Boundary 67 4.6 Rogowski Bevel 70 4.7 A Snake 72 4.8 A Field C lamp 73 4.9 Modell ing a Field Clamp in P O I S S O N 76 4.10 Fie ld C lamp Parameters 77 4.11 Fie ld C lamp Step Field 78 4.12 Clamped Field varying with D 79 4.13 C lamp Thicknesses 81 ix 4.14 Shorting the C lamp to the Dipole 83 4.15 Conceptual Field Clamp Design 86 4.16 P O I S S O N Model of S A S P Field Clamp 89 4.17 V F B P E Fi t 93 4.18 SF F i t 94 4.19 P O I S S O N V F B Subtracted From The Ideal V F B 97 4.20 Curve Residue V S Sp 99 5.1 Matr ix Errors 119 5.2 Ray Tracings 123 5.3 S A S P Apertures 124 5.4 S A S P Quadrupole Beam Pipe 126 5.5 Q2, Dipole, and Vacuum Boxes 130 5.6 Two Body Reaction 134 6.1 Target Plane Acceptance Contour Plots 142 6.2 Target Plane Acceptance Histogram For 0% 143 6.3 Downstream Ray Profiles 145 6.4 A n Ideal Focal Surface 146 6.5 Mapping the Focal Surface 149 6.6 S A S P Focal Surface 150 6.7 R A Y T R A C E Focal Surface 151 7.1 S A S P Detector Array 153 7.2 V D C Wire Planes 154 7.3 V D C Cross Section 156 C . l A rectangular grid 182 C.2 A Sample P O I S S O N Problem 186 C . 3 B - H Curves 187 D. l L E P S Dipole 189 D.2 L E P S Field M a p Tracks 192 D.3 M i d Excitation Profiles 194 D.4 Contour Plots of the L E P S Entrance Aperture Fields 195 D.5 P O I S S O N L E P S Fringe Fields 198 D.6 L E P S Cross Section 199 x D.7 LEPS Internal Field Excitation Spectrum 200 D.8 LEPS Excitation Curve 201 F . l SASP Dipole Bevel 214 F.2 Cutting The Bevel 215 F.3 Simplified Vacuum Box 215 F.4 Entrance Vacuum Box 217 F.5 Exit Vacuum Box 218 F.6 Entrance Field Clamp 220 F.7 Exit Field Clamp 221 F.8 Entrance Pole Edge 222 F.9 Exit Pole Edge 222 J . l V D C Track Reconstruction 241 xi Acknowledgements I thank the SASP group and especially my supervisors Ed Auld and Pat Walden for their guidance; A l Otter for showing me how to get POISSON to work; and Stan Yen for patiently supplying me with the results of countless computer runs. For moral, psychological, emotional, and air support, I thank the Friends of Mickey Mouse, a group of fearless free thinkers who will never let minor things like reality get in their way. And of course I thank my parents Donald and Clarice Duncan who are so tolerant of a son who ran like hell from horticulture. xii Chapter 1 Introduction The T R I U M F Second A r m Spectrometer (SASP) is a high resolution, large solid angle, large acceptance, Q Q C l a m (Quadrupole,Quadrupole, Clamshell Dipole) type magnetic spectrometer. W i t h construction about to start, the spectrometer should begin op-eration in mid 1990. Designed to complement the existing Med ium Resolution Spec-trometer (MRS) located in T R I U M F ' s Proton Hall at target 4 B T 2 , the S A S P will complete the Dual A r m Spectrometer System (DASS) allowing high resolution nuclear spectroscopy of three body reactions. This thesis describes aspects of the magnetostatic design of the S A S P ' s dipole and presents a Monte Carlo simulation of the D A S S / S A S P system. Part I of this thesis contains an examination of the S A S P dipole, a description of the magnetic field profiles specified by the optical design, and a study of how such fields can be created with the magnet steel geometry. The primary tool for studying the dipole was a magnetostatic field simulation program called P O I S S O N . The aspects of the dipole design this work considers are the internal field profile and the aperture fields of the magnet. The field profiles predicted by simulations of the magnet are compared with the fields specified by the optical design. The correct internal field was produced by adjusting the orientation of the dipole's pole pieces. T h e desired aperture fields were produced by adjusting the shape of the magnet's steel at the apertures and by the use 1 CHAPTER 1. INTRODUCTION 2 of a device called a field clamp which limits the extent of the fringe fields. Preliminary engineering specifications for the dipole apertures and field clamps are appended to this thesis. Part II of this thesis presents a Monte Carlo simulation of the S A S P which is part of a general simulation program called E A S Y . E A S Y is designed to study the per-formance of the complete Dual A r m Spectrometer System and to access experimental arrangements. The simulation of the spectrometer uses transport matrices to propagate a particle from the target to the S A S P ' s focal surface, checking to see if the particle is blocked at various locations in the S A S P . T h e simulation is capable of modelling, in detail, the spectrometer's focal surface detectors including the wire chambers and scintillators. T h e detectors can be modelled to any level of sophistication up to and including the effects of multiple scattering in the wire chamber windows and gasses. Aspects of the S A S P investigated with the Monte Carlo simulation included the solid angle, the acceptance, the limiting apertures, and the downstream ray profiles. Studies of the limiting apertures were utilized in the design of the dipole, the vacuum system and the field clamps. Downstream ray profiles help position the spectrometer's detector array. 1.1 Why a Magnetic Spectrometer A n outline of the type of physics experiments proposed for the new D A S S / S A S P system is given in the proceedings of a two day workshop[l] which took place at T R I U M F in March 1986. A m o n g the areas of interest are: • P ion production, (p,7r), and associated pion production, (p,7rx). • Nucleon knockout (p,2p) and (p,p'n) • Charge exchange, (n,p) and (p,n) CHAPTER 1. INTRODUCTION Several experiments already approved for the D A S S / S A S P system are: " E E C No. Tit le Spokesman 416 Neutron knockout with S A S P C A . Miller T R I U M F 417 Survey of the (p, TT+) reaction in the A P.L. Walden resonance region T R I U M F 418 Nucleon effective polarization in C a , W . J . McDona ld Zr and P b U. of Alberta The S A S P with it's large solid angle and high resolution is a powerful instrument in itself, but it is the combination of the S A S P with the M R S to make the Dual A r m Spectrometer System that is the most interesting aspect of the project. While dual arm detector systems are not new (they are routinely made out of either a spectrometer and a detector telescope or two telescopes), a dual arm system consisting of two spectrometers operating together will achieve resolutions a factor of 10 to 100 times better than previously possible. Consider the following. The collision of a particle beam with a target can be classified by the number of particles produced in the reaction. A two body collision would be of the form mg + rriT —• rnji + mi and a three body collision mB + rriT —• mR + mx + m2 Where rng is the incident particle, m j the target nucleus, THR the recoil particle, and mi,rri2, etc. the scattered particles. Assuming the target nucleus does not break up in the collision, the scattered particles are usually simple entities (protons, electrons, pions, deuterons) of known masses , with the remains of the target nucleus making up the recoil particle. Often, the recoil nucleus is left in an excited state; the measurement CHAPTER 1. INTRODUCTION 4 of the energy, cross section, and reaction dependence of these excited states is called nuclear spectroscopy. Measuring the mass of the recoil particle directly is not possible since it usually does not leave the target material, hence the term "missing mass". The alternative is to measure the energy and momentum of the scattered particles and then, knowing the incident beam energy and momentum, PR = PB- PS ER = EB + ET — Es where the target particle is assumed to be at rest. Ps is the total momentum of all the scattered particles. Then the missing mass is given by, mR = y/E2R - PI If the masses of the scattered particles are known, it is only necessary to measure their trajectories and momentum. This can be done with detector telescopes or with a magnetic spectrometer. A detector telescope can typically resolve a momentum to one part in 1 0 - 2 ( 1 0 - 3 at best) while a magnetic spectrometer can do as well as one part in 1 0 - 4 . As discussed in appendix A, the missing mass resolution of a two arm system is completely dominated by the poorer detector. Thus a dual arm experiment using only one spectrometer is not able to use that spectrometer's resolution efficiently. With the Dual Arm Spectrometer System however, TRIUMF will have a detector system capable of taking full advantage of the inherent resolution of both spectrometers. CHAPTER 1. INTRODUCTION 5 1.2 The SASP Spectrometer The proposed T R I U M F Second A r m Spectrometer (SASP) is a high resolution mag-netic spectrometer designed to complement the existing Medium Resolution Spectrom-eter (MRS) at target 4 B T 2 in T R I U M F ' s Proton Hall . The two spectrometer system is called D A S S (Dual A r m Spectrometer System). The spectrometers can be operated independently in single arm modes or together in a dual arm mode which makes possi-ble high resolution studies of three body reactions. The specifications of the S A S P are given in table 1.1. The S A S P has a Q Q C l a m configuration, which consists of two quadrupoles fol-lowed by a 90° bend clamshell dipole. Unlike a regular dipole where the pole faces are set parallel to each other, a clamshell dipole has the pole faces set in a wedge. This adds additional focussing to the system. A particle scattered from the target in the 4 B T 2 target chamber enters the S A S P horizontally where it is first focussed by the quadrupoles and then deflected 90° upwards by the dipole towards the detec-tor array located at the spectrometer's focal surface 1.5 metres above the dipole exit. The detector array consists of four sets of instruments: three Vertical Drift Chambers ( V D C s ) , a set of trigger paddle scintillators, another scintillator further downstream and either a third scintillator or a Cerenkov counter. There is a front end chamber ( F E C ) which can be lowered into the beam between the target chamber and the first quadrupole. Although the S A S P is capable of achieving high momentum resolution without it, this chamber is useful for calibrating the spectrometer and for gathering additional trajectory information. T h e S A S P can be made into a neutron detector using a C H target to convert the neutrons into protons which the spectrometer detects. A t small angles, it is necessary to sweep the primary beam away from the target and in order to fit the sweeping magnet and conversion target before the S A S P , Q l must be removed and Q2 shifted CHAPTER 1. INTRODUCTION 6 SASP Specifications Central Momentum (P0) Design 660 M e V / c Maximum 759 M e V / c Momentum Bite - 10% to +15% A P / P o Solid Angle at -10% A P / P 0 13.1 msr at - 5 % A P / P 0 14.5 msr at 0 % A P / P o 15.4 msr at +5% A P / P o 14.7 msr at +10% A P / P o 13.9 msr at +15% A P / P o 13.2 msr Resolution @660 M e v / c (with 2mr multiple scattering 0.02% A P / P at focal plane) ( F W H M ) D / M 4.70 c m / % Flight path @660 M e V / c 7.02 m Angular Acceptance bend plane ± 1 0 2 mr non-bend plane ± 4 2 mr Focal Plane Ti l t 44.3° Total Bend Angle 90° Angular Range 14°-156° Angular Resolution (1 m m beam spot with 2 mr no front end chamber) Max imum target spot size vertical 10 cm (full acceptance) horizontal 4 cm Min imum opening angle with 40° the M R S Min imum opening angle with 14° the beam line Table 1.1: S A S P Specifications CHAPTER 1. INTRODUCTION 7 30cm back towards the dipole [2]. The vacuum box between Q2 and the dipole must be able to accommodate this. T h e relationship between the S A S P and the M R S is shown in figures 1.1 and 1.2. Both spectrometers rotate about the 4 B T 2 target chamber. T h e M R S pivots on a spherical bearing attached to the target chamber support pillar, while the S A S P travels on two tracks using a raised guide rail between them to keep the spectrometer oriented on the target chamber. The only contact between the S A S P and the target chamber is the vacuum pipe leading to the first quadrupole. Th is design, while technically more difficult than the M R S ' s pivot arrangement, allows the S A S P to be installed without modifying the M R S support structure minimizing disruption of the M R S ' s operations. The vacuum pipe to the S A S P will be attached to the target chamber with sliding seals that will allow it to rotate without breaking the vacuum. This is an improvement over the scattering chamber currently used for the M R S which uses a series of ports and bellows, making angel changes quite complicated. It is important that the smallest angle the two spectrometers can make with each other and with the beam line is as small as possible. Many experiments need to make measurements near 0° scattering (from the beam line) and others need to make measurements with a small angle between the scattered particles (i.e. a small angle between the spectrometers). T h e M R S is capable of approaching the beamline to less than 12° in its small angle configuration. While the S A S P is a more compact device, it is also closer to the target chamber and so will have a closest angle of approach to the beamline of 14°. The closest angle of approach between the two spectrometers will be 40°. 1.3 Designing SASP T h e process of designing the S A S P started with an optical design that best met the physics requirements, which were the need for large solid angle, high resolution, and CHAPTER 1. INTRODUCTION 8 Figure 1.1: Elevation view of the S A S P and M R S . The target chamber and the S A S P ' s detector arrays are only conceptual in this drawing. This figure reproduced with the permission of TRIUMF Figure 1.2: P lan View of the S A S P and M R S beamline and 40° of the M R S . T h e S A S P can come within 14° of the CHAPTER 1. INTRODUCTION 10 large momentum range. Optically the S A S P consists of three elements, two quadrupoles and a dipole. T h e quadrupoles act as focussing elements while the dipole has both fo-cussing and dispersive properties. The attributes of the quadrupoles that could be varied during the design process were their length, and the strengths of each harmonic field component (in fact the "quadrupoles" should really be called multipoles). The basic dimensions of the dipole were fixed early in the design but fine tuning and correct-ing aberrations in the optics could be accomplished by adjusting the dipole's internal field profile and by altering the curved entrance and exit apertures. Working from a clamshell magnet design first suggested by H. A . Enge in conjunction with S. Yen, the optical design of the S A S P was finalized by S. Yen [3]. T h e engineering design began after the optical design of the spectrometer was completed. T h e dipole design has three aspects: specification of the pole pieces such that they produce the desired field shape and are large enough to accept the particles passing through the spectrometer; design of the dipole side plates, return yokes, and coils used to produce the specified field strengths; and the concept of the dipole vacuum box used to allow particles to pass completely through the spectrometer in vacuum. T h e design of the S A S P dipole is shown in figure 1.3. It is an H magnet with openings in the return yoke for the entrance and exit apertures. The dipole vacuum vessel consists of an aluminum ring (following the pole piece) shaped to form a vacuum seal between the pole pieces. Horns protrude from the dipole vacuum box at the entrance and exit apertures to connect it to the rest of the system. A coil encircles each pole piece outside the vacuum vessel. The basic structure of the dipole including the return yokes, coils, and vacuum box was designed by A . Otter[4], while the orientation of the pole pieces and the curvatures of the dipole apertures are specified in this work. Also specified here are field clamps — devices used to shape and limit the extent of the aperture fringe fields. CHAPTER 1. INTRODUCTION 11 Figure 1.3: The layout of the S A S P dipole. This drawing reproduced with the permission of TRIUMF Not shown are the entrance and exit vacuum horns. Part I Designing the SASP 12 Chapter 2 Spectrometers 2.1 Magneto-optical Systems Consider a charged particle passing through a system of magnetic fields distributed symmetrically about a median plane (figure 2.1). In the case of the SASP spectrometer, this median plane bisects the dipole and quadrupole magnets. The z — x planes of the entrance and exit coordinate systems lie on the median plane with j / i parallel to y2. It is conventional with bending magnets for x to point in the direction of increasing radius of the magnet. Keeping this convention for the SASP, x points vertically downward before the spectrometer with y horizontal in the laboratory reference frame. After the spectrometer, both x and y are horizontal with z pointing vertically up. The particle-field system is analogous to a light ray traversing an optical system. The trajectory of the particle is equivalent to the light ray [5]. The ray entering the system is described by Oi r\ = Vi *1 8 13 Figure 2.1: A system of magnetic fields (in this case the S A S P ) . A ray (the thick arrow) is transported from the xuyi plane (trajectory f = (x ,0 ,y , 4>)) to the x2,y2 plane. CHAPTER 2. SPECTROMETERS 15 and after traversing the system by r2 = #2 V2 <i>2 8 where 8 is the percentage deviation of the charged particle momentum P from the central momentum of the system Po-A s shown in figure 2.1, the components x and y give the position of the particle in the x — y plane. 8 is the angle between the projection of the ray onto the z — x plane and the z axis while (j) is the angle between the projection of the ray onto the z — y plane and the z axis. Note that there is no z component specified for the ray. This is because the trajectory of the particle is only examined at known planes along the "central trajectory" (the path followed by a ray with 8 = 0 entering the system along the zi axis). Distances (x,y) are expressed in cm and angles (6,(f>) in mrad. T h e central momentum and coordinate systems are defined such that if the ray enters the system along Zi it will leave along z2. This notation for describing rays was developed for T R A N S P O R T [6] — a popular magneto-optics program. T h e final trajectory, r*2, is a function of the initial trajectory r\ and can be written as, 8 = P-Po Po x 100% x2 = x2(xi, &i,y\,<f>i,8) (2.1) 62 = 02(zi,0i,yi.0i»£) V2 = V2{x\,Q\,y\,<l>\,S) CHAPTER 2. SPECTROMETERS 16 <f>2 = <(>2(xi,0i,yi,<f>i,S) where 8 remains constant if the system contains no electric fields (ignoring radiative effects). These relationships can be written as Taylor expansions, for example, dx2 dx2 dx2 dx2 dx2 d2x2 2 d2x2  X 2 = ^ 1 X l + W1dl + Wlyi + Wi<f>1 + ~d6-S+2Wx-^ + 2 l c ^ X i e i + ---where the partial derivatives are evaluated at f{ = 0. Because the order of differen-tiation does not matter, many of the second and higher order partial derivatives are equal. For example, the derivatives with respect to x and 6: d2x2 d2x2 2\dx1d91 2\d01dx1 The partial derivatives are written in a symbolic form that recognizes this property: { X / X ) S trT (x/x6)=. 2 9 2 X 2 2!dx1d01 where (x/x6) combines both of the x,8 derivatives making it unnecessary to have a (x/9x) term — note that this notation does not use subscripts. The equation for x2 is rewritten as, x2 = (x/x)x1 + (x/9)81 + {x/y)yi + (x/fifa + (x/6)8 (2.2) +(x/x2)x2 + (x/x9)xi0i + (x/xy)xiy1 + (x / X(j>)xi<f)i + (x/x8)x\8 CHAPTER 2. SPECTROMETERS 17 +(x/82)8\ + (x/ey)0m + (x/oWifa + These equations can be written in matrix notation, where, R i _ (x/x) (x/8) (x/y) (x/<f>) (x/l) (x/6) (8/x) (6/6) (6/y) (8/4>) (8/1) (8/6) (y/x) (y/8) (y/y) (y/<f>) (y/l) (y/6) Wx) (<t>/8) (4>/y) (cf>/<t>) (<f>/l) ($/*) (l/x) (1/8) (l/y) (1/4) (l/l) (1/6) [ (6/x) (6/8) (6/y) (6/<f>) (6/1) (8/8) The definition of f has been changed to conform to the convention used by TRANS-PORT. x 6 y 4> I 6 r = where / is the difference between the distance travelled through the system by f and the central ray (f{ = 0). It has no effect on the other components and is not used in CHAPTER 2. SPECTROMETERS 18 this work. R2 is a 6 x 36 element array and f\ is a 36 element column vector, Because the notation combines partial derivatives, half the terms in R2 drop out making it an upper diagonal matrix. Many of the terms in the matrices are in fact zero. For instance, because the momentum of the particle does not change, the only nonzero 6 term is (S/S) = 1. More terms drop out because the magnetic fields are distributed symmetrically about the median plane. To see this, consider the R1 matrix for a simple system consisting of a homogeneous magnetic field whose vector is parallel to y. A ray injected into the system will follow a helical path with an axis parallel to the y axis. Assuming the field fills the entire region between the initial and final planes, a projection of the ray's trajectory onto the x — z plane is a segment of a circular arc starting at xt and ending at x2. The projection of the incident ray is tangent to the arc at xx making an angle of with respect to z\. Similarly, the final ray is tangent to the circle at x2 making an angle of 82 with respect to z2 (which is not necessarily parallel to Z\). T h e final position and angle of the projection, x2,62 are dependent only on X\ and 0i and not on xt8i xiyi Xi(f>i Xth XiSi 6xxi e2 CHAPTER 2. SPECTROMETERS 19 2/i and <f>i. Similarly y2 and <f>2 are only dependent on y\ and <f>\. R1 then reduces to, '(x/x) (x/8) 0 0 0 (x/6) (8/x) (6/6) 0 0 0 (6/6) p i _ 0 0 (y/y) (y/cf>) 0 0 0 0 (<f>/y) (4>/<f>) 0 0 (l/x) (1/8) 0 0 1 (1/6) . 0 0 0 0 0 1 2.2 Magnetic Spectrometers A typical nuclear physics experiment has a particle beam colliding with a stationary target producing various scattered particles. The purpose of a magnetic spectrome-ter is to measure the momentum vector, P$ (magnitude and direction), of a charged scattered particle. T h e magnitude of the momentum is determined by passing the par-ticle through a dispersive magnetic system which produces a correlation between one or more of the components of the particle's final trajectory and the momentum mag-nitude. T h e momentum direction is determined either using position sensors before the spectrometer or by again producing correlations between the trajectory after the spectrometer and the initial trajectory before. T h e performance of the spectrometer is a function of the characteristics of the particle beam hitting the target, the kinematics of the reaction and the geometry of the apparatus. Consider a spectrometer with optics described by the first order equation, r 2 = R*r\ where f i is the ray at the target, f2 the ray at some plane after the spectrometer, and R} is given by equation 2.3. Determining the scattered particle momentum vector, Ps is therefore the same as measuring the components of the initial ray, r 1 : Wi th appropriately placed detectors after the spectrometer, the trajectory (x2,82,y2,4>2) of (2.3) CHAPTER 2. SPECTROMETERS 20 the ray as it passes through the z2 = 0 plane can be reconstructed. From equation 2.3, x2 and 82 are functions of 8 (the momentum magnitude), x2 = (x/x)x1 + (x/8)81 + (x/8)8 (2.4) 92 = (8/x)x, + (0/9)9, + (8/8)8 If X\ and #i were known, 8 could be determined directly with a position sensor mea-suring x2 in the z2 — Q plane. This plane is called the spectrometer's focal •plane. In general, it is not possible to determine f\ completely using only the final trajec-tory. Equations 2.1 give the basic transfer functions through the spectrometer. There are four equations (x2(),92(),y2(), 4>2()) and five unknowns (xi, #i, y i , (f>i, 8), making the system indeterminate. T o get a solution, it is necessary to place constraints on the initial ray, forego some of the information, or use front end detectors to measure the initial trajectory of the ray. For many experiments the last option is undesirable for two reasons. First , these detectors alter the momentum and trajectory of the particle, decreasing the momentum magnitude resolution of the spectrometer. Second, being near the target and beam line, they receive a large flux of particles and are likely to saturate, l imiting the rate at which the spectrometer can process data. Constraining the ray entering the spectrometer can be done by various techniques. The "spot size" (xj, yi) can be made arbitrarily small by decreasing the size of the par-ticle beam hitting the target. Alternately, the angular range of the scattered particles can be constrained with a collimator (such as the entrance aperture of the spectrome-ter). Constraining r\ does not mean the values of its components are known exactly, only that they lie within a known range. The remaining uncertainty will produce an error in the reconstructed momentum vector of the scattered particle, Ps. Whether this error is tolerable depends on the objectives of the experiment. CHAPTER 2. SPECTROMETERS 21 The last approach is to lose some of the information about r\. As with constrain-ing r*x, this will produce an error when reconstructing Ps- The particle beam hitting the target is not perfectly monochromatic, meaning that while the average momentum of the beam will be some value PB, a given particle will have a momentum deviating from the central momentum by a value, A P . Th is deviation is represented as a percentage of the central momentum. A P 8B = — x 100% P B For the T R I U M F cyclotron, 8B is normally distributed with a standard deviation of 0.1%. Ideally, the spectrometer should measure the momentum of the scattered particle, Ps, as a function of the beam particle momentum, P B , but the smearing of the beam particle momentum will cause a smearing of the scattered particle momentum reducing the resolution of the spectrometer. B y adjusting the beam and spectrometer optics using a technique called Dispersion Matching (see appendix B) , the effects of 8B can be eliminated, improving the spectrometer resolution. The trade off in dispersion matching is the loss of the ability to determine'x x from the ray trajectory after the spectrometer (it can still be measured with detectors placed before the spectrometer). Equation 2.4 shows x2 as a function of x l 5 9,, and 8. Call ing (x/x) the Magnifi-cation, M, of the spectrometer, and (x/8) the Dispersion, D, equation 2.4 is rewritten as, x2 = Mxx + {x/9)91 + D8 (2.5) where 8 is actually composed of two components (appendix B) , 8 = 8S + A8B Here, 8s is momentum of the scattered particle for the monochromatic beam expressed CHAPTER 2. SPECTROMETERS 2 2 as a percentage deviation from the central momentum of the spectrometer, Po, 6S = P s ~ P ° x 100% •ro and A is a constant that depends on the reaction kinematics and the experiment's geometry. The beam is said to be dispersion matched when Xi is made dependent on 6B, XL = ~MASB Note that at the target, the position of the beam particle, XB is the same as the position of the scattered particle, X i . Equation 2.5 reduces to, x2 = (x/8)8x + D8S To make x 2 a function of 6s only, the (x/8) term is made as small as possible. This is called point to point focussing (since a position in the focal surface depends only on x\ and 8, not Now consider the second and higher order terms in the Taylor expansion for x2 (Equation.2.2). Since they are not wanted, any nonzero terms are called aberrations. The most important aberration is (x/xd). This is the simplest in a series of aberrations which we can write in the form, (x/x86i) with i — 0,1,2, In the S A S P optics design [3], aberrations of the form (x/x68n), n = 0 to 4 are set to zero as well as other aberrations such as (x/<^2), (x/83), (x/8<f>2) and (x/x82). T h e aberrations are zeroed by adjusting the shapes of the S A S P ' s quadrupoles (thus introducing multipole components) and by changing the curvatures of the dipole's entrance and exit apertures. The non-chromatic aberrations (those without a 8 term) CHAPTER 2. SPECTROMETERS 23 axe corrected using the two quadrupoles and the dipole entrance. T h e aberrations (x/x88n) are corrected using the dipole exit curvature. There are an infinite number of aberrations in the spectrometer optics and only a few parameters that can be adjusted to eliminate them. While it is impossible to eliminate all the aberrations, in general they become less important as the order increases. After as many aberrations as possible are accounted for with the spectrometer hardware, software corrections can be applied. T h e particle momentum can be written as, <*S = + / (S 2 , 02, y2i^2) + 0 (3l , 01,1/1, M where / is a function of the ray trajectory after the spectrometer, and g is a function of the trajectory before. For the ideal spectrometer / and g are both zero. Assuming there are no front end detectors, g is indeterminate and will always produce an error in 8s; f on the other hand can (at least in theory) be determined. / is written as a polynomial, / = ax2 + b82 + cy2 + d<j>2 + ex] + fx282 + gx2y2 + hx2<f>2 + %8\ + ... Combining this polynomial with the focal plane position to determine 8s is called a software correction. To find the various coefficients in / , rays of known momentum are sent through the spectrometer and a least squares or other parameter fitting routine used to find the best fit. For the S A S P , aberrations are mostly corrected for central momentum rays using the hardware while software corrections become important for correcting the aberrations at the extreme ends of the momentum bite. Chapter 3 SASP Dipole 3.1 R A Y T R A C E the Program R A Y T R A C E is a general "ion-optics" computer code, written in F O R T R A N , devel-oped at M I T over the last 20 years[7]. The program user describes the various optical elements in the system to be simulated with standard devices supplied by R A Y T R A C E . These include multipole and dipole magnets, solenoids, and velocity selectors. In addi-tion to tracing the trajectory of the charged particle (called a ray) through the system, R A Y T R A C E produces first and second order transport matrices for the system as well as some higher order terms, up to fifth order. —• A particle with charge q and velocity v passing through electric (E) and magnetic (B) fields is acted upon by the Lorentz force, F = q(E + v x B) The fields the particle passes through are completely specified by R A Y T R A C E thus, if the velocity is known, the force acting on the particle anywhere in the system can be calculated. R A Y T R A C E solves the equations of motion using a fourth order Runge-K u t t a stepwise integration technique [8]. A stepwise integration can be thought of as follows: suppose the particle is moving predominately parallel to the z axis with 24 CHAPTER 3. SASP DIPOLE 25 a known initial position and velocity. The force acting on the particle is determined and used to calculate the deflection of the particle from its original trajectory at a point further along the z axis. The Runge-Kutta method expands on this approach by incorporating the force acting on the particle from several previous points. As the trajectory is extrapolated further from the starting point, the estimates become less accurate. Th is accumulated error can be made arbitrarily small by decreasing the increments of the steps — at the cost of increased calculation time. There are two other sources of error in this method: the inaccurate description of the magnetic fields; and the roundoff error carried through the calculation. T h e roundoff error is made negligible by using double precision arithmetic in the routines. T h e models of the magnetic fields used in R A Y T R A C E are idealized since the program does not account for the minor field variations of an actual system or the major effects of the saturation of a dipole magnet. The various elements described in R A Y T R A C E are modelled with idealized an-alytic fields. For example, a quadrupole field extends an infinite distance from the optical axis and if a particle was to try to pass through the quadrupole element 10m from the axis it would still behave as if in an ideal quadrupole magnet. R A Y T R A C E does not consider the physical devices required to create such a field, making it a poor program to examine the effects of apertures. Other programs such as E A S Y (section 5.1) can be used for this. Because R A Y T R A C E is not designed to consider the feasi-bility of the devices it models, the user must be careful not to stake optical designs on devices that are impossible to build. R A Y T R A C E does take into account the fringing fields at the apertures of mag-nets. The shape of the fringe field is specified with parameters input to R A Y T R A C E . To obtain the fringe field shape for a given problem, the user can either measure the field shape of an existing magnet or use a program like P O I S S O N (appendix C) to Figure 3.1: T h e R A Y T R A C E dipole layout. The two coordinate systems used in this work are B and C. model the magnet. These R A Y T R A C E fringe fields do not accurately model the vari-ous effects found on real magnets. For example, the fringe field of a dipole will change at the edges of the aperture. The R A Y T R A C E model does not consider this effect. Because of the large amount of computer time required for a calculation, R A Y -T R A C E is not usually used to do the complete design of a magnetic system, but rather in conjunction with a faster (but less accurate) program such as T R A N S P O R T which would rough out the general layout of the system. R A Y T R A C E is then used to fine tune it. However, the clamshell dipole can not be modelled accurately with T R A N S P O R T so R A Y T R A C E was used for the entire optical design of the S A S P . 3.1.1 Describing Dipoles in R A Y T R A C E Figure 3.1 shows the major parameters used to describe a magnetic dipole. Originally R A Y T R A C E assumed that dipoles had homogeneous magnetic fields. Th is assumption has determined how the subsequent versions of the program have been developed. A particle travelling along the optical axis of the system with a momentum equal to the CHAPTER 3. SASPDLPOLE 27 central momentum of the dipole P = P0 (6 = 0) would travel in a circular path of radius R. This path (from B to C in figure 3.1) is called the central trajectory or central radius of the magnet. The particle traverses an arc <f> along this path. R A Y T R A C E uses four special coordinate systems to describe the dipole. A l l are right-handed with their origins on the optical axis. A ) I n c o m i n g , z is parallel to the optical axis, pointing in the direction of motion, x is parallel to the radius vector from the centre of curvature of the magnet to the point where the entrance field boundary passes through the central radius of the dipole. B ) E n t r a n c e , z points out from the pole edge, x is tangent to the pole edge, pointing in the direction of decreasing dipole radius. C ) E x i t , z points out from the pole edge, x is tangent to the pole edge pointing in the direction of increasing dipole radius. D ) O u t g o i n g , z is parallel to the optical axis, pointing in the direction of motion, x is parallel to the dipole radius vector from the centre of curvature of the magnet to the point where the exit field boundary passes through the central radius of the dipole. T h e edges of a dipole magnet are called its Vir tual Field Boundaries ( V F B ) . The position of the V F B ' s are such that a magnet with a Sharp Cutoff Fringe Field ( S C O F F ) would end exactly there. The V F B ' s can be modified in several different ways (separately or in combination): • T h e entrance/exit aperture can be tilted by an angle a/p° with respect to the central trajectory. • A V F B can be made part of the arc of a circle of arbitrary radius. CHAPTER 3. SASP DIPOLE 28 • A V F B can be described by an eighth order polynomial. T h e symmetry plane of the dipole is called the median plane. B y symmetry arguments, it can be seen that the field lines always pass perpendicularly through this plane: that is, the field has only a y component, B = (0, By, 0), on the median plane. R A Y T R A C E assumes that the field in an nonhomogeneous dipole varies radially from the center of curvature of the magnet (figure 3.1): BP B ' { D R ) = TTndxWb (31> where RB is the central radius, DR is the distance from that radius (positive DR being further from the center of curvature) and NDX is a dimensionless parameter determining the rate of drop off of the field. For a derivation of this formula, see section 3.3. For inhomogeneous field magnets such as the S A S P dipole, the magnetic field is calculated analytically only on the median plane. Off the median plane, a Taylor expansion is used. _ dBx y 3 PBX Bx = y — h dy 3! dy3 y2dBy y*d*By  y ~ y + 2! dy2 4! dy4 _ dBz y*c?Bz  Z ~ V dy + 3! dy3 The derivatives are evaluated in the median plane. The y partial derivatives must be calculated without knowing Bx, By, or Bz. This can be done by looking at Maxwell's CHAPTER 3. SASP DIPOLE 29 equations for a magnetostatic field, - - ( f - t ) - ( t - f ) ^ ( t - f ) ^ - <«> V . B = ^ + ^ + ^ = 0 (3.3) ox oy oz From equation 3.2, dBz dBy dy dz dBx dBz dz dx dBx dBy dy dx (3.4) (3.5) (3.6) A n y of the y derivatives can be calculated by converting it to a sum of partial derivatives in x and z, r)nR dm+pR . dyn ^ dxmdzf> ' where, n = m + p and i = x,y,z. The derivatives of By in the median plane are then calculated numerically using a thirteen point grid. As an example of converting the y partial derivative to an x, z derivative, take ^ p 1 , dzBx d2 dBx dy3 dy2 dy Substituting in equation 3.6, d2 dBy d2 dBy dy2 dx dxdy dy CHAPTER 3. SASP DIPOLE 30 since the order of differentiation does not matter and then using equation 3.3, dxdy dBx dBz dx dz d2 dBx d2 dBz dx2 dy dxdz dy and using equations 3.4 and 3.6, d3BX d3By d3By dy3 dx3 dxdz2 The other derivatives can be handled similarly. 3.1.2 Fringe Fields in R A Y T R A C E When calculating the fringe field of a magnet, R A Y T R A C E describes the field on the median plane as a function of the width of the magnet's gap (called the gap width), By = Bif(s) (3.8) where s = z/D, D is the gap width of the magnet, z the distance from the V F B (positive is out), and B , is the field inside the magnet. T h e shape of the fringe field is specified with the coefficients in table 3.1. To calculate the field at a point in the aperture ("in" meaning the space just outside the magnet bounded by the curved pole edge), R A Y T R A C E calculates the distance s to the nearest point on the V F B in units of the dipole gap width. This distance is put into equation 3.8 along with B{. For a wedge magnet like the S A S P dipole, the gap at the central radius is used. This is surprising since it would seem more logical to use the gap width at the point on the V F B used in the field calculation. The R A Y T R A C E code was modified to use the local gap width with almost no effect on the spectrometer performance. Th is is probably because the gap width changes only a small amount over the S A S P dipole apertures. If CHAPTER 3. SASP DIPOLE 31 the change had been larger, the effect on the optics may have been significant. Fringe fields off the median plane are calculated with the same Taylor expansions used for the dipole interior. T h e method used in R A Y T R A C E for calculating fringe fields should be considered approximate. Provided the aperture geometry is well behaved this approximation is adequate. "Well behaved" means the air gap changes slowly and the V F B has a gentle curvature. The effects of a curved aperture on the fringe field are considered in section 4.1. 3.2 Dipole Layout The optics designed with R A Y T R A C E completely specify the magnetic fields every-where in the spectrometer. It is necessary to translate these specifications into the physical hardware that will be built. R A Y T R A C E is designed so that the parameters used to specify the magnetic fields of a dipole can be readily translated into the major physical dimensions of the magnet's pole pieces. The pole pieces are the two pieces of iron that shape the useful field region of a magnet. The smaller the gap between the pole pieces, the more intense the magnetic field. The S A S P dipole's variable magnetic field is produced by flat pole pieces set at an angle to each other forming a wedge shaped gap (figure 3.2). T h e curvature of the pole edge determines the magnet's virtual field boundary. A bevel on the pole edge affects the shape and extent of the fringe field. Once the pole piece is specified, the dipole's return yokes, side plates and coils must be designed. Referring to figure 3.1 the parameters describing the dipole are listed in table 3.1, producing the layout shown in figure 3.3. The short dashed line in figure 3.3 shows the approximate outline of the pole piece. CHAPTER 3. SASP DIPOLE P o l e P i e c e R 220,000.0 cm <i> 2 x (0 .0298°) = 0.0596° a - 7 5 ° 0 - 7 5 ° 1/R1 0 1/R2 0 In te rna l F i e l d BF 1.6073 T NDX 1300.0 F r i n g e F i e l d s D 10.0 cm Entrance Exit Co 0.388258 0.388258 Cl 1.563493 1.563493 c 2 -0.483961 -0.483961 c 3 0.544297 0.544297 c4 -0.169069 -0.169069 c 5 0.015043 0.015143 V F B C u r v a t u r e s Entrance Exit s2 -684.58 -654.38 s3 -5.2751 x 10 5 -4.9721 x 10 5 s4 -2.0709 x 10 9 1.7044 x 10 9 s5 -2.1053 x 10 1 3 -8.6203 x 10 1 2 Se 0.0 1.68105 x 10 1 6 s7 0.0 0.0 S8 0.0 0.0 Table 3.1: R A Y T R A C E parameters for the S A S P dipole. central ray Figure 3.2: A conceptual drawing of the S A S P Clamshell dipole showing the pole pieces and wedge shaped air gap. Not shown are the dipole's curved apertures and return yokes. Pole Piece T o describe the S A S P dipole geometry in R A Y T R A C E , it is necessary to use a trick. As seen in figure 3.1, R A Y T R A C E assumes that magnets have a cylin-drical symmetry with the central ray (travelling along the optical axis at the central momentum) making a circular trajectory of distance R from the centre of curvature. This trajectory is called the central radius or the central arc of the magnet. Although the field in the dipole can vary radially, the magnetic field along any circle centred on the centre of curvature is constant. The S A S P dipole, with its flat pole faces, does not have cylindrical symmetry. As discussed in section 3.3, the field is constant along lines of constant gap. T h e trick is to model the S A S P dipole in R A Y T R A C E , by increasing the "central" radius, R, of the magnet and decreasing the angle spanning the dipole, <f>, until the central arc looks approximately like a straight line segment. In the case of the S A S P dipole, R = 220,000cm and <j> = 0.0596°. Th is produces a 228.8cm long central arc that deviates from a straight line by less than 0.3mm (figure 3.3). T h e field is •CHAPTER 3. SASP DIPOLE 34 Figure 3.3: S A S P dipole pole piece geometry. The large value of R makes the central radius appear as a straight line. The short dashed line shows the approximate outline of the pole piece. CHAPTER 3. SASP DIPOLE 35 strongest at the small gap near the dipole apertures, decreasing with the increasing air gap. Unlike the normal R A Y T R A C E magnet where the central ray enters the dipole along the central radius, the central ray enters the clamshell dipole at a 45° angle to it. Inside the magnet, the central ray follows a trajectory that looks only approximately like a circle as it passes through regions of varying magnetic field. The parameter \/R\ (I/R2) is used to give the entrance (exit) aperture a circular curvature. The sign of the parameters make the circular apertures convex or concave. The S A S P dipole's pole edge curvature is described completely with the polynomial expression and these inverse radii are set to zero ( Ri(R2) set to infinity). I n t e r n a l F i e l d T h e internal field of the dipole is determined with the two parame-ters used in equation 3.1. BF is the magnetic field at the central radius, and NDX determines the rate at which the field drops off. T h e derivation of equation 3.1 and how the pole faces are set to produce the desired fields are described in section 3.3. F r i n g e F i e l d s The aperture fringe fields of the dipole are modelled in R A Y T R A C E using equation 4.1 where the distance from the pole edge is measured in terms of the dipole gap, D. T h e parameters c 0 , . . . in table 3.1 describe the field shape. The S A S P dipole uses the same field shape for both the entrance and exit apertures. The method used to determine the dipole fringe field is described in section 4.7. V F B C u r v a t u r e s The seven coefficients 5 2 , . . . are parameters in an eighth order polynomial (equation 4.3) which modifies the basic V F B ' s described with the ac(0) and 7^(7^) parameters. Because the inverse radii are set to zero the basic boundary modified by the polynomial is a straight line. T h e shapes of the V F B s and the fringe fields for the two apertures are shown in section 4.2. CHAPTER 3. SASP DIPOLE 36 3.3 Interior Fields 3.3.1 Magnetic Field of a Clamshell Dipole The magnetic boundary conditions between two materials of different permeability fi are, where /x = / i r / io and Bn(Bt) is normal(tangential) to the boundary. If material 1 is iron (fj,T & 500), material 2 air (fj,T = 1), and the B field in the iron is oriented approximately perpendicular to the interface, the tangential component in air goes to zero and the magnetic flux lines leave the iron perpendicular to the surface. (Actually, because of the large difference in permeabilities, the field lines will leave the iron perpendicular to the surface even if there is only a slight normal component to the field inside the iron). A magnetic scalar potential $ can be defined (see for example Hayt[10]), That is, the B field is the gradient of the scalar potential. Since a surface of constant potential has the B field everywhere normal to it, the interface between the iron and air forms an equipotential surface. The S A S P dipole has flat pole faces set in a wedge shape (figure 3.4), forming two equipotential planes with cylindrical symmetry. The magnetic scalar potential between the pole faces can be calculated using Laplace's equation in cylindrical coordinates, Bn2 '— n l tl (3.9) CHAPTER 3. SASP DIPOLE 37 Figure 3.4: The pole pieces of the S A S P form two equipotential planes with cylindrical symmetry. The equipotential planes are parallel to the z axis and not dependent on either p or z, therefore equation 3.9 reduces to, T h e solution for which is, $ = A<f> + B O n the median plane, $ = 0 and on a pole face $ = $ 0 so, <±> = —<f> a CHAPTER 3. SASP DIPOLE 38 were a is the angle between the median plane and the pole face. Calculating B, -* i ^ i (d$. i a $ . 5 $ „ \ b = — v $ = — K - « P + - ^ T a * + ^- a 0 /*o Mo \op p d(p Oz J only the d/d<f> term survives leaving, B = - ( ± ) h> Therefore, on the median plane of the clamshell dipole, the magnetic field is every where normal to the plane and inversely proportional to the distance from the apex o f the wedge. BY oc - (3.10) P 3.3.2 The SASP Dipole's Internal Field As discussed in section 3.2, R A Y T R A C E describes a magnet in terms of the central radius R. For the S A S P dipole R does not describe the trajectory along which a ray with the central momentum of the magnet, PQ, travels. Rather, the large value o f R (220,000cm) is used to produce a magnet with planer geometry. In addition to the pole piece geometry, R A Y T R A C E describes the internal field of the magnet in terms of the central radius (equation 3.1). To determine the slope for the dipole pole faces that will produce the desired field profile, the internal field must be described in the same notation. Rearranging equation 3.10, PQBQ My = ~rz Po + x CHAPTER 3. SASP DIPOLE 39 where B0 is the field at the central radius of the magnet (R), a distance p0 from the apex of the wedge, and x is the displacement from that central distance. In general, p0 will not equal R (the parameter used in R A Y T R A C E ) . For the S A S P , these numbers differ by three orders of magnitude. Simplifying, y ~ 1 + -s-Po Comparing this with the field used by R A Y T R A C E , R B F " > - l + NDX^ where RB is the central radius (also called R), NDX is a field shape parameter, and, NDX _ _1_ RB ~ po Assuming a 10cm gap at the central radius and using the S A S P field parameters from table 3.1, we get a pole face slope of, A>i NDX 1,300 cm m = — * - = D0> — — = 5cm—1-—— = 0.02954545— po 2 RB 220,000cm cm where DQi is one half the dipole air gap at the central radius. T h e distance between a 10cm gap and a 15cm gap is CHAPTER 3. SASP DIPOLE 40 E x t r e m e S A S P R a y s X 0 y <l> / 6 (cm) (mrad) (cm) (mrad) (cm) (%) -1.0 -85.0 0.0 0.0 0.0 -10.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 85.0 0.0 0.0 0.0 10.0 1.0 85.0 0.0 0.0 0.0 15.0 Table 3.2: Components used to generate the extreme rays sent through the S A S P . 3.3.3 Dipole Design The S A S P dipole engineering design was produced by Allen Otter [4]. T h e size of the pole piece is determined by the momentum acceptance ("bite") of the spectrome-ter. The original design specified ± 1 0 % , meaning that the spectrometer could accept rays with 8 ranging from —10% to +10%. This specification was later changed to a momentum bite of (—10%,+15%). R A Y T R A C E was used to draw the trajectories of "extreme" rays through the spectrometer (figure 3.5). The extreme rays deviate b y the greatest distance from the central ray used in the optical design. In fact, even more diverse rays can get through the spectrometer but they will not necessarily pass through "good" magnetic field. The rays shown in figure 3.5 were generated b y setting the components of the input rays to the values in table 3.2. T h e dipole pole pieces were laid out around the extreme ray trajectories leaving a width of pole face approximately equal to a dipole gap width beyond the trajectory to ensure the ray doesn't enter the dipole's fringe field. It is this safety margin that increases the spectrometer acceptance beyond its design specification. Because the optics were not designed with these additional rays, aberrations may make them useless for high resolution spectroscopy. T h e size of the dipole side plates and return yokes were estimated by modelling CHAPTER 3. SASP DIPOLE 41 Figure 3.5: R A Y T R A C E was used to draw the trajectories of extreme rays through the S A S R Shown is the central ray, the extreme —10% ray and the extreme +10% and + 15% rays. This drawing reproduced with the permission of TRIUMF. CHAPTER 3. SASP DIPOLE 42 a cross section through the middle of the dipole (figure 3.6a) using the magnetostatic modelling program P O I S S O N (appendix C) . Figure 3.6b shows the B field on the median plane of the dipole predicted by P O I S S O N compared to the analytic expression used by R A Y T R A C E . The dipole is at approximately half the maximum excitation. In the figure, R is the distance from the central radius. The input files used by P O I S S O N to model the dipole cross section are listed in appendix E . T h e solid line is the \ Jp field profile used by R A Y T R A C E while the points show the field predicted by P O I S S O N . The spacing of the points is the grid size used to describe the problem in P O I S S O N . While the P O I S S O N field at the central radius is not exactly equal to the required R A Y T R A C E field, this can be corrected by adjusting the excitation of the dipole in the model. For purposes of plotting, the field is scaled to be 1.0 at the central radius (R = 0). Note the field drops off at the sides of the pole piece. Figure 3.7 shows a plot of the ratio of the P O I S S O N field to the R A Y T R A C E field for the cross section through the dipole. The fields are normalized at the central radius (R = 0) and, ideally, the ratio should be exactly 1.0 across the dipole. Looking at the low field (large R) side of the pole piece, the ratio and therefore the P O I S S O N field is too large. T h e cause of this difference between the P O I S S O N model and the analytic expression is not clear. The dipole is at about half the full excitation and has not yet started to saturate. This difference (approximately 0.3%) might be due to edge effects of the dipole or it might be due to the digitizing process in P O I S S O N . Regardless, a small difference like this in the dipole's field shifts the spectrometer's focal surface but will not affect its performance. Since R A Y T R A C E shows that the spectrometer design is tolerant to a small error in the field profile, the P O I S S O N result is accepted as correct and the slope of the dipole pole faces are altered to match the P O I S S O N field profile to the R A Y T R A C E profile at the middle excitation. This changes the distance between the 10 and 15cm gaps CHAPTER 3. SASP DIPOLE 43 a) PROB. NAME = D I P 0 L E 4 D 0=84. 6 1 5 3 7 0 0 0 . D C Y C L E = 520 b) 1.2 -I 1 1 ' 1 I L R (cm) Figure 3.6: a) Cross-section of the S A S P dipole modelled with P O I S S O N . b)The inte-rior field of the dipole. The solid line is the analytic expression used by R A Y T R A C E , the points are the field predicted by P O I S S O N . R is the distance from the central radius of the dipole. CHAPTER 3. SASP DIPOLE 44 1.01 -1.00-0.99--4-* CD 0.98-I 0.97-CO 0.96-0.95-0.94-DIP0LE33 D=84.615 37000.0 B(r=0)=9030.506 maxdiff=0.264% at r=73.304 -20 0 20 40 60 R (cm) 80 100 Figure 3.7: The ratio of the P O I S S O N field to the R A Y T R A C E field BP/BRT. T h e Bp is scaled to exactly equal BRT at the central radius (R=0). from 84.615cm to 83.777cm (figure 3.8). The new pole face slope is, 7.5 - 5.0 m = — — — = 0.02984113 83.777 The spectrometer design should be unaffected if P O I S S O N turns out to be slightly off in its prediction of the dipole's internal field profile. 3.3.4 Dipole Saturation Saturation is the term used to describe the nonlinear response of a magnet's field to the exciting current in its coils. Below some critical value, the magnetic field will rise linearly with excitation. Above the critical current, the field stops rising linearly and levels off at a maximum value. When the excitation is above the critical value, the magnet is said to be saturating. Excitation is expressed as the current times the CHAPTER 3. SASP DIPOLE 45 10 c m 8 3 . 7 7 7 c m 1 5 c m / Figure 3.8: T h e new dipole air gap. The distance between the 10 and 15cm gap widths has been changed from the predicted 84.615cm to 83.777cm. number of turns of the coil — Amp-Turns (NA) . The saturation properties of a magnetic material are characterized by its B - H curve — a plot of the magnetic field B produced by a given excitation H (expressed in A m p Turns or oersteds 1). Figure C.3 shows the B - H curves of some of the materials considered for the S A S P dipole. For a uniform gap dipole, the entire pole piece enters the saturation region at the same time so a series of plots of the field profile for different excitations would look the same — flat across the pole piece; dropping off at the edges. Only the magnitude of the field profiles would change with excitation. The S A S P ' s clamshell dipole behaves quite differently. Because the field varies across the pole piece, the small gap (high field) side will go into saturation before the large gap (low field) side. Th is changes the field profile across the magnet which is undesirable since the dipole field shape is an important part of the spectrometer optics. Figure 3.9 shows a series of field profiles for the S A S P dipole at different excitations varying from 17000NA to 77000NA. The 1one oersted equals 4£ Amp Turns CHAPTER 3. SASP DIPOLE 46 maximum excitation the dipole is intended to be operated at is 67000NA, although both the coils and the power supplies will be able to go 15% higher. W h e n operating the S A S P above the design excitation, saturation will degrade the magnetic field and, therefore, the spectrometer performance. Figure 3.9 shows that at low excitation, the centre of the magnet is actually under excited. Th is is because the S A S P dipole is an H magnet and for low excitations, the flux stays in the iron near the coils. Similar studies of an existing C style clamshell dipole, the L A M P F Low Energy Pion Spectrometer, or L E P S , (described in appendix D) did not show this effect for two reasons. First , the L E P S dipole is smaller so the flux does not have as far to go. Second, the return yoke is on the large gap side of the dipole. T h e large gap has a greater magnetic reluctance so flux prefers to travel to the lower reluctance, smaller gap side of the dipole. These two effects will tend t o spread out the flux more uniformly than in the S A S P dipole. As the excitation in the S A S P dipole increases, the flux spreads out into the entire yoke producing the field at 37000NA. This was the excitation at which the pole face slope was adjusted to match the P O I S S O N field profile to the R A Y T R A C E profile. A s the excitation is increased beyond 37000NA, saturation begins to alter the field profile and is quite noticeable (0.979% deviation from ideal) at 67000NA — the maximum design excitation. The P O I S S O N field is scaled to be exactly equal to the R A Y T R A C E field at the central radius, which is on the high field, small gap side of the dipole. Th is makes the P O I S S O N field appear high at the large gap side when in fact the field is low at the small gap side. Th is saturation effect is potentially disastrous. Intuitively one would think that a 1% change in the dipole's field at 67000NA would destroy the spectrometer's resolution. P O I S S O N simulations using steels with better B - H curves were tried with only small improvements in the field profiles. The saturation effect is a fundamental problem CHAPTER 3. SASP DIPOLE 47 -20 (NPOLE41 0-84.615 17000.0 B(r-0)-4143.161-maxdrtf—0J35S ot r-83B22 20 40 60 R (em) BO n o toi too 0.99 £ 0.98-0.96 0.95 0.94 -20 DP0LE42 D-84.615 27000.0 B(r-0)-6593B96 mcacdiff—0.3SIX at r-831522 20 40 60 R (em) BO 100 101 too 0.99 I 0.98 &0.97-0.96-0.95-0.94 -20 WPOLE43 0-84.615 370tK>.0 B(r-0)-9026|760 nwxdrff—0.318% el r-B3B22 20 40 60 R (cm) BO 100 101-1.00-0.99-m 0.98-\ o a 0.97-m 0.96-0.95-0.94--20 DP01E44 D-B4.615 47000.0 B(r-0)-1M18.E46-mexdi f f—0.21« ol r-B3.B22 20 40 60 R (cm) 80 100 R (em) Figure 3.9: T h e S A S P dipole internal field profile as a function of magnet excita-tion. Starting from the upper left, the excitations are: 17000NA, 27000NA, 37000NA, 47000NA, 57000NA, 67000NA, and 77000NA. CHAPTER 3. SASP DIPOLE 48 with a clamshell dipole. The best that can be done is to adjust the pole face slope to minimize the distortion at one excitation which in turn limits the operating range of the spectrometer. To determine the effect of the field sag on resolution, it is parameterized as a distortion to the ideal field (changing the notation from equation 3.1), By = r ^ r x CF(xy 1+TlR where the correction factor CF is equal to 1 at x = 0, CF(x) = 1 + bx + cx 2 + dx z + ex 4 This distorts the field uniformly over the length of the dipole and does not take into account the various effects due to the position of the return yokes. The 67000NA curves were used to study the distortion effects. The only noticeable effect of the distortion on the spectrometer's characteristics was a shift of the focal surface a few centimeters further away from the dipole exit — the resolution was essentially unaffected. However, as discussed in section 2 .2 , part of the spectrometer's momentum determination is done using software corrections. The software corrections are very sensitive to the magnetic optics and the saturation effects will significantly alter them. This will have a major effect on the spectrometer's operating procedure. Because the optical properties and software corrections change with excitation, it will be necessary to have a new calibration for each setting of the spectrometer. Care is required to obtain these calibrations since the standard method of "stepping" the peak of some easily resolved reaction such as pp —> dTC+ across the focal surface by changing the spectrometer setting will change the optics and the focal surface being mapped. CHAPTER 3. SASP DIPOLE 49 As part of the study of the S A S P dipole's internal field, it would have been useful to generate excitation (B-H) curves. However, attempts to replicate the measured exci-tation curve of the L E P S dipole (appendix D) failed. A n excitation curve is a measure of both the local saturation of the dipole steel and of the flux "leaking" out of the yoke. T h e situation is analogous to a parallel resistance problem in an electric circuit. It becomes a question of whether the flux prefers to travel through the steel or the surrounding air. Both the S A S P and L E P S dipoles are complicated three dimensional shapes with many possible paths for the flux. Trying to model these dipoles with two dimensional cross sections does not capture the nature of the problem. Simulations with the L E P S showed that one could replicate the true excitation curve of the magnet by enlarging the return yokes of the model but this arbitrary procedure cannot then be applied to an unknown magnet such as the S A S P dipole. Chapter 4 Dipole Apertures In section 3.1 it was explained that the dipole aperture fields are used to correct various aberrations in the S A S P optical design. If the aperture fields are not shaped correctly, the performance of the spectrometer will suffer. As with the interior of the dipole, the R A Y T R A C E optics design only produced a specification of how the aperture fields look — not the steel geometry. The specifications are given as sets of parameters in table 3.1 which describe the Vir tual Field Boundary ( V F B ) and fringe field shapes for each aperture. These descriptions of the fringe fields are used to create the following engineering specifications: 1. The Pole Edge Boundary ( P E B ) . 2. The Pole Edge Bevel. 3. The Field Clamp Boundary ( F C B ) . Although these items are not completely independent of one another, their effects on the aperture fringe fields can be roughly separated as follows. T h e pole edge boundary ( P E B ) is the curve the dipole pole piece follows. It determines the shape of the V F B which it is similar in shape to. The pole edge bevel is the rounding of the edge of the pole piece and shapes the upper part of the fringe field. T h e bevel also helps determine the behaviour of the fringe field with changing dipole excitation. T h e field 50 CHAPTER 4. DIPOLE APERTURES 51 clamp boundary ( F C B ) describes the shape of the field clamp, which is a device placed near the aperture of the magnet to help shape the fringe field. While it can effect the field shape in several different ways, the most important function of a field clamp is to limit the extent of the fringe field; it "clamps" the field in. The main parameter for limiting the extent of the fringe field is the gap between the clamp and the P E B . Like the P E B , the F C B will depend on the V F B and on the P E B as well. A n engineering design has more requirements than meeting the R A Y T R A C E fringe field specification. The system must be feasible to build. For example, the best position for the field clamp conflicts with the dipole coils on one side and the dipole vacuum box on the other. The final relationship between the clamp, coil, and vacuum box is a compromise. The final specification for the dipole apertures and fringe fields may not be ideal, but it is the best practical design. Th is work does not provide an engineering design for the S A S P dipole apertures, but rather an engineering specification (appendix F ) . There are some aspects of the geometry that must be made exactly as specified (such as the three items listed above), there are other aspects that can be altered as necessary to develop a practical design. For example, constriction of the field clamp by the coil and vacuum box is resolved by having the clamp sandwiched between the coil and box with no clearance. This is impractical to build and either the coil and vacuum box will have to be moved slightly or the clamp made thinner. 4.1 Fringe Fields Consider a dipole as a magnetic circuit. This is analogous to an electrical circuit: the magnet coils create a magnetomotive force (voltage) that induces flux (current) to flow through through the magnet's yoke and air gap which have a certain reluctance (resistance) that impedes the flow of flux (current). Like current in the electric circuit, CHAPTER 4. DIPOLE APERTURES 52 the flux is conserved around the magnetic circuit, but unlike the electric circuit, the flux can flow through the air surrounding the magnet as well as through the yoke. Also, unlike the resistance in the electric circuit, the reluctance of the steel is nonlinear. Generally the flux prefers to flow through the iron in the magnetic circuit but is forced to jump across the dipole's air gap. Crossing the air gap, the flux tends to bulge out producing a fringe field that drops off further from the magnet. This is the fringe field associated with a dipole aperture. The shape and extent of this field is determined b y the shape of the pole edge, the size of the air gap, and the excitation of the magnet. Enge [9] produced a function that can describe the fringe field on the median plane where B = (0,BY,0). B> = TT?m w with, S(s) = c 0 + c x 5 + c2s2 + c3s3 + c4s4 + c5s5 and s = z/D, D is the width of the air gap, and z is the distance from the Vir tual F ie ld Boundary ( V F B ) . For a fictitious magnet with a homogeneous internal field and no fringe field (it drops abruptly from the interior field value to zero at the V F B ) the virtual field boundary lies at the field drop off. Such a magnet is said to have a Sharp Cut Off Fringe Fie ld ( S C O F F ) . A n actual magnet has an Extended Fringe Field ( E F F ) and the V F B lies at the point where the integral of a S C O F F B field equals the total integral of the E F F B field, /°° txVFB BEFF(S)(1S = / BSCOFF -oo J—oo ds where oo is far outside and —oo far inside the magnet. The angle a ray (particle) is deflected by a bending magnet is equal to its angular CHAPTER 4. DIPOLE APERTURES 53 Figure 4.1: T h e effect of a fringe field is to displace the trajectory and alter the bend angle of a ray. CHAPTER 4. DIPOLE APERTURES 54 velocity (caused by the B field) integrated over time, 6= t ujdt Jo Converting from a time integral to a position integral, <U __ dt~V and, dl dt = — v since the speed of the particle is constant in a purely magnetic field (i.e. no electric field). T h e angular velocity is given by the cyclotron frequency, qB{l) UJ — m where B(l) is perpendicular to the bend plane of the particle and varies with position. The angular deflection of the particle/ray then reduces to: mv or, 6= 2 fB(l)dl (4.2) p Jo P with p being the momentum of the particle. Thus the angular deflection of the ray is proportional to the integral of Bdl. Suppose a bending magnet was designed assuming a sharp cut off fringe field ( S C O F F ) , a homogeneous internal field and curved virtual field boundaries (figure 4.1). Tracing a particle (whose trajectory is called the S C O F F CHAPTER 4. DIPOLE APERTURES 55 ray) through the magnet, the path indicated with the solid line in figure 4.1 is made. U p to the V F B , the particle trajectory is straight. Entering the magnet, the particle travels along a circular trajectory of radius R with centre CSCOFF- Reaching the exit V F B , the particle resumes a straight trajectory. The total angle the particle (ray) is deflected through is &SCOFF ~ BohcOFF where I SCOFF is the length of the arc inside the magnet. Now suppose the magnet has an extended fringe field ( E F F ) extending a distance ±d from the V F B . The E F F ray enters the fringe field d before the V F B and starts to deflect. T h e rate of deflection increases with increasing B field until the particle reaches the homogeneous field where it has the same deflection (or radius of curvature) as the S C O F F case. Because the E F F ray started deflecting before the S C O F F ray, it is displaced from the S C O F F ray and will have a different centre of curvature, CEFF- Reaching the exit, the E F F ray passes through the fringe field with gradually decreasing deflection and finally leaves the magnet displaced from the S C O F F ray. T h e curvatures of the magnet V F B s are such that the E F F ray has travelled a shorter distance through the magnet with a smaller Bdl and therefore, QEFF < QSCOFF This result is dependent on the position and angle of the ray entering the magnet. A n E F F ray entering the magnet on the opposite side of the aperture in figure 4.1 could leave with a greater deflection than the S C O F F ray. The result is the fringe field introduces aberrations to the optical system of the form, (x /x '0 J ) CHAPTER 4. DIPOLE APERTURES 56 where i,j axe 0,1,2, If the magneto optics are designed without consideration of these aberrations, the performance of the system will likely be adversely effected. In practice the fringe fields are included in the calculation; R A Y T R A C E describes fringe fields with parameters the user inputs to the program (see section 3.2). However, if the fringe field shape assumed in the optical design does not agree with real shape, aberrations will again appear. Some magnetic systems will be more sensitive to the fringe fields than others. For instance, the S A S P ' s resolution proved insensitive to small changes in the dipole fringe field shape. Such a tolerance to error will make the S A S P easier to construct. T h e R A Y T R A C E fringe field model assumes that the field at each point in the aperture is determined by the Enge function which is a function of the distance to the closest point on the aperture V F B . This model basically assumes that the field is unaffected by the aperture's curvature and if this is not accurate, effects similar to the ones discussed in the S C O F F v.s. E F F magnet will be encountered. T h e fringe fields of a real magnet can differ from the model in one of two ways: as constructed, the magnet's aperture fringe fields do not conform to the desired shape; or the fringe fields change shape with magnet excitation. Both effects can be present. T h e aperture field not conforming to the desired shape will affect the spectrometer performance regardless of the excitation it is being operated at while an excitation dependent fringe field will cause the spectrometer's characteristics to change with excitation. It is already known that the S A S P ' s characteristics will have an excitation dependence caused by the saturation of the dipole's interior field (section 3.3.4), but the addition of aperture effects would aggravate the situation. Both virtual field boundaries on the S A S P dipole are concave and are thus sus-ceptible to an effect called aperture filling. Consider a homogeneous field magnet with CHAPTER 4. DIPOLE APERTURES 57 an aperture whose V F B is a circular segment. Ideally the lines of constant field in the aperture should be circular and concentric with the V F B (and pole edge) as shown in figure 4.2a. For the circular pole edge however, flux moving straight out from the pole edge will interact with flux from other parts of the pole causing the fringe field to bulge (figure 4.2b). T o understand this, take a straight V F B where, by symmetry, the lines of constant B field are parallel to the pole edge. If the straight V F B is bent into a circular arc, the length of the boundary and the total amount of fringe field flux associated with it remains constant but the space the flux has to pass through outside the magnet decreases. The flux "bunches up" which increases the local magnetic field. The contour lines will bulge out from the pole edge, the aperture appears to "fill in" with flux, and the V F B no longer follows the pole edge. The flattening of the contour lines decreases the effective curvature (i.e. increases the radius of curvature) of the V F B (figure 4.2b). T h e ratio of the extent of the fringe field and the radius of curvature of the aperture is given as a measure of this effect. If the ratio is small, the aperture filling is also negligible while if the ratio is large, the filling is significant. This is because the field at a point in the aperture ("in the aperture" meaning the space just outside the magnet) is effected only by the pole edge within about one fringe field extent of it. As the ratio of extent to radius decreases, a point in the aperture "sees" less of the pole edge and the fringe field becomes more like that of a straight boundary. A related type of aperture filling is caused by the pole piece's finite size. Consider the fringe fields of a homogeneous magnet with a rectangular pole piece. If the magnet had no fringe fields, a contour plot of the magnetic field would parallel the pole edge, the field would jump from zero outside the magnet to the internal field value at the pole edge. However, an E F F magnet behaves differently. Near the centre of each face of the magnet, the field contours follow the pole edge, but at the corners of the magnet, the contour lines must make a 90° turn. The contour lines do not form a sharp right CHAPTER 4. DIPOLE APERTURES Figure 4.2: Aperture Fil l ing, a) A n ideal fringe field, b) Local aperture filling Magnet edge effects, d) Edge effects on the aperture field. CHAPTER 4. DIPOLE APERTURES 59 angle, rather they make a gentle curve about the same size as the extent of the fringe field itself. Thus the corner of the magnet will perturb the fringe field along the pole edges. T h e larger the fringe field, the further from the corners the effect will be seen. If the fringe field extent is comparable in size to the sides of the magnet, the fringe field will appear to bulge without following the pole edge (figure 4.2c). In reality the lines of constant field axe curving in at the corners rather than out at the centre of the pole edge. T h e effect of the finite dipole on a concave aperture is shown in figure 4.2d. Th is type of problem is always present to a degree since the aperture curvature cannot continue indefinitely. It is necessary to design the magnet apertures sufficiently wide to accept the particle beam. This keeps the edge of the aperture with its poor field cleax of the particle. Both types of aperture filling can be lessened by minimizing the extent of the fringe field. T h e problems discussed above are caused by the geometry of the magnet; the effects of magnet excitation are also important. Suppose the aperture has an acceptable fringe field at some low excitation for which none of the magnet steel is saturating. If the excitation is lowered, the field shape remains constant. But when the excitation is increased to put the magnet steel into saturation, the fringe field will change if the aperture is not correctly designed. If the magnet's pole edge bevel is improperly designed, the fringe field and the V F B will shift outwards as the magnet goes into saturation (see section 4.4). If the extent of the fringe field is a sizable fraction of the aperture radius, the aperture will fill in (figure 4.3a). Alternately, if the ratio of field extent to radius of curvature is small, the V F B will shift uniformly out from the aperture. Th is decreases the radius of curvature of the V F B (figure 4.3b). The edge effects of the finite sized dipole will compound the problem. CHAPTER 4. DIPOLE APERTURES 60 Figure 4.3: Two different effects on the V F B caused by saturation: a) The aperture fills in decreasing the effective curvature of the V F B . b) T h e fringe field bulges out increasing the effective curvature. 4.2 The SASP Dipole's Apertures Before specifying the steel geometry for the dipole apertures, the exact field shapes assumed by R A Y T R A C E must be known. R A Y T R A C E uses two attributes to describe the fringe field region of the dipole: the V F B curvature and the fringe field shape. The fringe field shape is described with the Enge function, 4.1, using the six coefficients found in table 3.1 (note both the entrance and exit apertures have the same fringe field shape). T h e V F B is described with an eighth order polynomial modifying the straight boundaries shown in figure 3.3 (^- = 0 = i.e. the unmodified boundaries are straight): * ~ * t « . ( ! ) " (4-3) R = 220,000.0cm CHAPTER 4. DIPOLE APERTURES 61 where z and x refer to the local coordinate systems at the entrance (B) and exit (C) shown in figure 3.3. The Sn values used for the S A S P dipole are given in table 3.1. Different V F B curvatures are used for the entrance and exit apertures. It is important to understand the regions where these polynomials are valid. Aberrations in the spectrometer optics are corrected for by using a "ray bundle" consisting of 198 particles injected randomly into the S A S P ' s aperture. T h e extreme trajectories of the bundle in the dipole bend plane are shown in figure 3.5. T h e rays cross the dipole's entrance aperture between x = —28cm and x = +32cm using coordinate system B (figure 3.3). T h e rays cross the exit aperture in the region x = —45cm to x = +65cm. T h e S A S P ' s optics are thus only optimized for rays that enter and leave the dipole over these regions. Rays crossing the dipole V F B s outside the "good" regions will probably experience aberrations that will reduce the spectrometer's resolution. T h e term high resolution rays refers to rays that enter the S A S P dipole over the good V F B regions. Rays that enter the dipole outside these regions are called low resolution rays. The polynomials describing the V F B s are only valid for the high resolution ray regions. For values of x outside the high resolution regions, the polynomials become unrealistic and predict V F B s that are impossible to construct. This can be seen for the exit V F B shown in figure 4.4: the low end of the valid range is x — —60cm and at —100cm the ZVFB = —40cm while at x = —200cm, it drops to ZVFB — —3484.5cm. The concept of a virtual field boundary is obvious for a constant gap magnet with its homogeneous field (section 4.1) but not so obvious for the S A S P ' s clamshell. Knowing the V F B definition, the Enge function (equation 4.1) is constructed so that 6 = 0 lies on the V F B of the homogeneous field magnet. T o describe the fringe fields of a constant gap magnet, R A Y T R A C E calculates the position of the V F B and then uses the Enge function. R A Y T R A C E follows the same procedure for a nonhomogeneous field magnet. Ignoring aperture filling, it might be expected that the V F B is still the Figure 4.4: Dipole Aperture Fringe Fields assumed by R A Y T R A C E . T h e lines are contours of constant B field while the points show the V F B . The fringe fields near the V F B give way to the straight internal field contours, a) Entrance b) Exit CHAPTER 4. DIPOLE APERTURES 63 point where the total area under a S C O F F curve equals the area under the E F F but this is not so. The Enge function can be interpreted as a factor, 1 + e s( J) (varying from unity inside the magnet to zero outside) that modifies what the magnetic field would be at the point, s, if the pole face extended indefinitely without interruption by the aperture, * M = i f i f e ( 4 - 4 ) where Bo(s) is a constant for a homogeneous field magnet. Figure 3.3 shows that ap-proaching either of the S A S P dipole apertures from inside the magnet along a direction perpendicular to the V F B , the air gap narrows causing the internal field to rise. If the aperture was not there, the field would continue to rise "outside" the V F B . Thus both the denominator and the numerator of equation 4.4 are increasing. T h e exponential in the denominator will eventually dominate but the fringe field of the sloped pole face magnet will drop off more slowly than the fringe field of the constant gap magnet. The integral of the resulting fringe field is greater than that of a S C O F F curve ending at the " V F B " . Thus the defined V F B lies further from the magnet than the specified V F B . Another subtlety of the fringe fields of an nonhomogeneous magnet is that based on the assumption the Enge function is correct, the extent of the fringe field varies as a function of the dipole gap width. Equation 4.1 assumes that the fringe field extent is independent of the internal field, Bo, and only depends on gap width, D. This means that for a homogeneous magnet, the width of the fringe field region is the same regardless of the excitation. Th is should hold true for a magnet operating below saturation. T h e nonhomogeneous field magnet has a changing air gap over the extent of the aperture and since the extent of the fringe field is determined by s = z/D, the CHAPTER 4. DIPOLE APERTURES 64 fringe field region will be wider for the large gap (low field) side of the aperture. For the SASP, the gap varies from approximately 9cm to 11cm over the apertures and is 10cm where the optical axis crosses the V F B . Figure 4.4 shows contour plots of the R A Y T R A C E aperture fields. The plots are in the local aperture coordinate systems B and C (figure 3.3). For the entrance plot, the increasing gap (decreasing field) is to the left; for the exit plot, the increasing gap is to the right. These plots show the bulging of the fringe field caused by the changing gap width. The straight contours at the bottom of each plot show the internal field whose contours are parallel to the lines of constant gap. Also marked as a set of points are the VFBs . 4.3 Pole Edge Boundary Given the VFBs for the SASP, the Pole Edge Boundaries (PEBs) that produce them must be specified. Consider a beveled pole edge on a constant gap magnet. The flat pole face gives way to a bevel consisting of one or more angled cuts which give way to the usually vertical side of the pole piece. It is arbitrary which point is defined as the Pole Edge (PE) although it is convenient to use some easily recognized point such as the beginning of the bevel or the side of the pole piece. F o r th is w o r k the vertical side o f the p o l e p iece w i l l be ca l led the p o l e edge . Depending on the the dipole gap width and the shape of the bevel, the V F B can lie inside, outside, or at the pole edge. The distance from the V F B to the P E is called V F B P E ; it is positive if P E B is outside V F B . For a constant gap magnet, the fringe field, and therefore V F B P E , is constant for the entire aperture (ignoring any interactions between parts of the aperture discussed in section 4.1). Thus once the virtual field boundary shape and V F B P E are known, the pole edge boundary can be calculated. The procedure is to take points on the CHAPTER 4. DIPOLE APERTURES 65 VFB, draw perpendiculars to the tangent, move the distance VFBPE, and mark the points on the PEB. Repeating this, a line can be fitted through the PEB points and an analytic function produced for the PEB like the one used for the VFB (equation 4.3). For the clamshell magnet, VFBPE is not constant over the aperture. This is because, even though the bevel is the same over the entire aperture, the air gap width and the apparent slope of the pole face along a perpendicular to the VFB changes with position on the aperture. The change in the apparent slope of the pole face can be understood by examining figure 4.4. The greatest apparent slope is where the line of travel is perpendicular to the gap lines — this is the true slope of dipole pole faces (m=0.0298 for the SASP). Alternately, if the line of travel is parallel to the lines of constant gap, the gap remains constant and the apparent slope is zero. As can be seen from figure 4.4, the angle the line of travel makes with the lines of constant gap is dependent on the where it crosses the VFB and thus the apparent pole face slope will change over the aperture. The effect of the changing pole face slope is a changing VFBPE (the VFB - PE distance) over the extent of the aperture. To determine points on the pole edge bound-ary, VFBPE must be measured at several points on the aperture and parameterized with some function. This function can then be used like the constant VFBPE was for the homogeneous field magnet; points on the VFB are taken, normals to the VFB drawn, and the corresponding points on the PEB marked at the distance VFBPE along the normals. The PEB can be described with a polynomial similar to that used for the VFB. VFBPE is calculated at a point on the VFB using a POISSON model of a cross section of the magnet aperture perpendicular to the VFB. Taking a point on the VFB, the gap width and the pole face slope along the perpendicular to the VFB is fixed by the dipole geometry. Keeping the pole edge bevel constant and ignoring the presence CHAPTER 4. DIPOLE APERTURES 66 of a field clamp, the only parameter in the system is V F B P E (figure 4.5a). T h e field profile looks approximately like the curves shown in figure 4.5b. T h e S A S P geometry is such that the dipole gap decreases and the field rises as the apertures are approached from inside the magnet. Near the V F B the fringe field begins to dominate and the field reaches a peak value and then drops off outside the magnet. T h e shape of the peak and its position with respect to the V F B are determined by the pole edge bevel (discussed below). T h e peak is located between the pole edge and the V F B . V F B P E is used to match the internal fields and peaks of the P O I S S O N and R A Y T R A C E fields. The field clamp does not affect this process because it only alters the fringe field profile. The internal R A Y T R A C E field along the perpendicular to the V F B can be de-scribed by the expression, BRT = r - ( 4 . 5 ) 1 — cs where the field is normalized to 1 at the V F B , s is the distance from the V F B (positive is out from the magnet), and c is a positive constant. Th is equation is a simplified version of equation 3.1 which describes the internal field of the dipole along a line perpendicular to the lines of constant gap. Based on the V F B - P E distance being displaced from the correct value, V F B P E , by a small amount, d. The P O I S S O N field is given by, B p = l - c ( t - [ d + VFBPE}) where t is the distance from the dipole pole edge. The P O I S S O N field profile will only match the R A Y T R A C E profile if, t = s + VFBPE or d = 0. If d > 0, the pole edge extends too far into the dipole aperture and the CHAPTER 4. DIPOLE APERTURES 67 Figure 4.5: Determining V F B P E and thus the Pole Edge Boundary ( P E B ) . a) The V F B P E distance is changed so the P O I S S O N model's internal field matches the R A Y -T R A C E internal field, b) Varying V F B P E changes the internal field profile (these curves are illustrative only). CHAPTER 4. DIPOLE APERTURES 68 P O I S S O N field, Bp( f ) , will drop off faster than the R A Y T R A C E field, BRT(s), as shown in figure 4.5 case 3. If d < 0, then the V F B - pole edge distance is too small and the P O I S S O N field will drop off more slowly than the R A Y T R A C E field (figure 4.5 case 1). When d = 0, the P O I S S O N and R A Y T R A C E field profiles match. Note that the magnitudes of the fields at the V F B are unimportant. Because the fields are described by an inverse relationship, the field profile is completely specified by one point on the curve which in this case is the virtual field boundary. T h e actual comparison of the P O I S S O N and R A Y T R A C E curves is made by first rescaling the curves so that the peaks have field values of 1.0. Then the curves are shifted so that the peaks are superimposed and the internal fields compared. 4.4 Pole Edge Bevels T h e flux in a magnetic circuit tends to concentrate at sharp points such as the corner of a square edged pole piece. This concentration increases the local magnetic field in the iron causing it to saturate before the rest of the pole piece. As the excitation increases, flux will "leak" out from the saturated corner causing the field to bulge out from the pole edge. Braams [11] proposed a pole edge for dipoles that would prevent saturation effects. The so called Rogowski pole edge (suggested by Rogowski [12] for sparkless electrodes) is assumed to be an infinite cylinder which follows an equipotential surface of the magnetic scalar potential. For an air gap D the shape of the pole is given by CHAPTER 4. DIPOLE APERTURES 69 where z is perpendicular to the pole edge in the median plane and y is normal to the median plane. (j> can be eliminated, where the two solutions for y represent the two pole pieces of the dipole. Inside the pole piece, the flux density (B field) is constant. When the pole goes into saturation, the entire tip will saturate at the same rate and the field profile will remain unchanged, although the magnitude of the field will no longer increase linearly with excitation. The Rogowski bevel produces a more extended fringe field than a square edged magnet, but when a magnet is to be operated over a range of excitations, it is more important to ensure a consistent field profile than a sharp cut off at low excitations. Figure 4.6 shows the ideal Rogowski bevel compared with the bevel that will be used on the S A S P dipole. Since the S A S P dipole's gap changes over the extent of the aperture, the bevel should change as well. To machine such a complex, smooth bevel would be quite difficult and expensive. In fact, it is adequate to approximate the Rogowski bevel with two simple cuts that remain constant over the extent of the aperture (the solid line in figure 4.6). T h e exact procedure for shaping the bevel is described in appendix F . l . Figure 4.6: Rogowski pole edge bevel. Dashed line is the ideal bevel and the solid line is the actual bevel used on the S A S P . The axes show the coordinate system used in equation 4.6 CHAPTER 4. DIPOLE A P E R T U R E S 71 4.5 Field Clamps 4.5.1 Modifying Fringe Fields Often, the fringe field produced by the curvature and bevel of a dipole magnet is unsatisfactory. This could be due to the problems discussed in section 4.1 or perhaps the spectrometer optics behave differently than expected and the fringe fields must be modified to improve the spectrometer performance. To this end, there are two devices (having more or less opposite effects on the fringe field) that can be used to alter the aperture fields. A snake [13] consists of two laminated soft iron bars suspended near and following the curvature of the pole edge (figure 4.7a). It has the effect of bulging the fringe field outward (figure 4.7b) which brings the V F B out as well. The snake does this by acting as a "stepping stone" for flux to cross the air gap. F lux can also flow along the snake from one part of the aperture to another because the surface of the snake forms an equipotential, linking together the entire aperture. The main use for snakes is to fine tune the shape of a virtual field boundary. T h e snake is mounted approximately in place and then distorted to produce the correct V F B . The laminations have the dual role of reducing eddy currents and hysteresis in the snake and making it flexible. This flexibility greatly simplifies fitting the snake to the aperture. As discussed in section 4.1, concave apertures, such as the S A S P ' s , usually have the problem of fringe fields that are too extended. To correct for this, a device called a field clamp is employed. A field clamp (proposed in 1934 by Herzog [14]) consists of a rectangular iron tube which wraps around the magnet aperture (figure 4.8). T h e field clamp picks up flux that would otherwise extend into the aperture and then "short circuits" or shunts it around the edges of the aperture through the side plates, or return yokes, of the clamp. This decreases the extent of the fringe field and the effects Figure 4.7: A snake is an iron strip suspended near the aperture of a magnet (a). It has the effect of "pulling" out the fringe field and the V F B (b). CHAPTER 4. DIPOLE APERTURES 73 field clamp Figure 4.8: A field clamp is an iron tube that wraps around the aperture of a magnet, shunting flux around the aperture through the sides of the clamp. of saturation. T h e extent of the fringe field is approximately determined by the distance from the pole edge to the field clamp. B y adjusting this distance over the extent of the aperture, the V F B shape can be altered. Unlike a snake however, the shape of the clamp can not easily be changed making it difficult to "fine tune" the fringe field. A n undamped magnet can suffer from an effect called field reversal caused when the magnet coils are close to the aperture (such as on the S A S P ) . The field reversal bends the particles in the opposite direction to that in the dipole and thus reduces the total Bdl. T h e effect this has on the S A S P ' s resolution was not studied. Field reversal occurs as the fringe field caused by the pole piece drops off far from the pole edge and the coil field, which is opposite to the dipole field, dominates. This produces a reversed field that decays to zero further from the aperture. Simulations show that dependent on the geometry, this reversal can be 10% of the internal field and extend for several gap widths. F ie ld reversal is completely eliminated by a field clamp which shunts both CHAPTER 4. DIPOLE APERTURES 74 the pole piece and coil flux around the aperture. 4.5.2 Modelling a Field Clamp There are two basic methods of modelling a field clamp. The most accurate is to build a scale model of the clamp and the magnet whose field it is to modify. Hubner and Wollnik [15] performed a series of experiments with such a model studying the clamp characteristics. It would be difficult to build a useful model for the S A S P dipole. The probe used to map the fields must be able to get into the air gap of the model. It would probably be necessary to build a ^ or | scale model which is almost large enough to be used as a separate spectrometer. The second method of studying a field clamp is with the aid of a computer simulation which was done for this work. T h e advantage of a computer simulation over constructing a scale model is that it is considerably less expensive and the geometry can be altered quickly and easily. T h e main disadvantage of using a computer simulation to study the dipole apertures is that only a two dimensional simulation program was available to model the three dimensional apertures, thus it is not feasible to study effects like aperture filling. Ideally the field clamp should be modelled on the computer using a three di-mensional magnetostatic code. No suitable program was available so the 2 D code, P O I S S O N (appendix C ) , with which the interior of the S A S P dipole was studied (sec-tion 3.3), was used. A field clamp is a three dimensional object that fits into a curved asymmetric aperture, making it difficult to model using a two dimensional code like P O I S S O N . T h e model is limited to slices of the clamp-dipole system which do not show how different parts of the aperture interact. Nor does it show how the flux moves through the return yokes of the clamp when investigating saturation. A field clamp is modelled in two dimensions using a cross section with a "tail" on the end away from the dipole to connect the two halves of the clamp. This provides CHAPTER 4. DIPOLE APERTURES 75 the short circuit for the flux to cross the magnet air gap. The tail has to be situated far enough from the aperture to prevent it from influencing the magnetic field. A n actual field clamp shaped like this would be useless since it would block the aperture. Figure 4.9 shows a typical configuration modelled in P O I S S O N and the effect the clamp has on the fringe field. The dipole is modelled as a C magnet with the pole face sufficiently wide to ensure a homogeneous (flat) interior field. Modelling the clamp and dipole in this manner can be misleading. The presence of the clamp can appear to decrease the internal field of the dipole by 10% or more because the simulated field clamp provides a flux path which is a significant fraction of the size of the dipole return yoke. For an actual magnet, the clamp should not have a perceptible effect on the dipole's internal field. Therefore this model will not quantitatively predict what excitation the clamp and dipole saturate at. Figure 4.10 shows the relationship between the dipole, clamp and coil. T h e effect of the field clamp on the fringe field cannot be isolated from the effects of the dipole gap, the shape of the pole bevel, and the size and position of the coil. T h e main parameters in the system are: D The magnet gap width. Dp The field clamp gap width. Sp The field clamp - pole edge separation. Sc The coil - pole edge separation. SCF T h e coil - field clamp separation. tF The field clamp thickness. Numerous P O I S S O N simulations changing one or more parameters were run to deter-mine their effects on the fringe fields. Most of the simulations used a square edged magnet rather than a beveled magnet to simplify the system. CHAPTER 4. DIPOLE APERTURES 76 Figure 4.9: Modell ing a Field Clamp in P O I S S O N . a) The P O I S S O N geometry used to simulate a field clamp on a homogeneous field dipole — note the flux lines travelling through the field clamp rather than through the air gap. b) T h e magnetic field on the median plane of the dipole as calculated by P O I S S O N without (dashed) and with (solid) the field clamp. The distance is measured in terms of gap widths from the pole edge. CHAPTER 4. DIPOLE APERTURES 77 4.5.2.1 Clamp Position (Sp and Dp) The width of the field clamp, Dp, tends to change the shape of the lower half of the fringe field curve. Decreasing Dp raises the outer fringe field. This is because the clamp concentrates flux from the aperture at its tip. As the clamp is brought closer to the median plane, this flux increases the local B field. This effect is small and Dp can be set arbitrarily. In the S A S P a more important consideration than shaping the lower fringe field is that the clamp be above the plane of dipole pole face to allow rays to enter the dipole. A value for Dp that meets this requirement is: Sp is the main parameter for altering the field shape and varying the V F B posi-tion. For large values of SF, the field clamp does not interact with the dipole. As SF CHAPTER 4. DIPOLE APERTURES 78 Figure 4.11: Fringe field for different values of SF- There is a value of SF for which the field in maximally clamped (solid line). For smaller values of Sp a step field builds up under the clamp (broken lines). decreases, the clamp picks up flux from the air gap and the V F B shifts in. T h e field eventually reaches a point of "maximal clamping". For smaller values of Sp, a residual "step" field builds up under the clamp (figure 4.11). For a square edged magnet with Dp = IAD the maximal clamping is achieved with, £ - 0 . . Figure 4.12 shows the variation of the fringe field for a clamped magnet as a func-tion of the dipole gap width, D , with the ratios SF/D and Dp/D kept constant. The fringe field is adequately described by the Enge function (equation 4.1) which predicts that the field shape scales exactly as the dipole gap width, D. Th is indicates that the Dp/D ratio must be kept constant when investigating the S A S P dipole apertures with their changing gap widths. CHAPTER 4. DLPOLE APERTURES 1.0 H 1 L 79 0 . 8 -0 . 6 -o m \ CD 0.4 -0 . 2 -0.0 - 4 - 2 0 2 4 6 s=z/D Figure 4.12: The variation of the fringe field with D for constant ratios of DF/D and The coils for a magnet as large as the S A S P dipole are difficult to construct according to design. T h e copper windings are wrapped against a form and encased with an epoxy which acts as a binding agent and insulator. Portions of the coil have a tendency to spring when released from the form. Movements of a few millimetres are to be expected. Thus it is important to know if the magnet's fringe fields are significantly effected by the coil position and if so, how large an error in positioning can be tolerated. Simulations showed that for both the clamped and undamped magnet, the V F B varied linearly with Sc- A 1cm shift of the coil away from the pole edge, shifted the V F B out 1.3mm for the undamped magnet and 0.2mm for the clamped. For the vertical positioning of the coil, the field clamp coil separation, SCF does not significantly affect the fringe field. Shifting the coil from 1cm above the field clamp to 5cm above the field clamp, the V F B shifted 1.5mm outwards. Thus, coil positioning errors of the order of a few millimetres can be tolerated for the clamped magnet. SF/D. 4.5.2.2 C o i l P o s i t i o n (Sc a n d SCF) CHAPTER 4. DIPOLE APERTURES 80 4.5.2.3 C l a m p T h i c k n e s s (tp) a n d R e t u r n Y o k e s T h e more flux the clamp is required to carry, the thicker it must be to prevent satura-tion. Figure 4.13 shows the fields produced by a 2cm thick clamp, a 5cm thick clamp and two clamps that change thickness from 2cm to 5cm. As expected, the 2cm clamp does not clamp the field as well as the 5cm one. The clamps with the 2cm "tongues" extending under the coil produce almost the same field as the straight 2cm clamp. This indicates that the important part of the clamp is the tip under the coil. The tongue has to carry both the flux from the dipole and that which is produced by the proximity of the coil. It is driven close to saturation by the coil and acts as a bottle neck limiting the amount of flux that can be drawn off the dipole. Therefore the clamp should be as thick as is practical. Related to the clamp thickness is the size of the return yoke. Th is is difficult to model in two dimensions since unlike the P O I S S O N models with their "tails", the clamp yokes collect flux from over the entire aperture. F lux leaves the pole piece and enters the clamp perpendicular to the pole edge but then must turn to the sides and flow around the aperture through the yokes, back into the other half of the clamp and across the clamp - dipole gap into the other pole piece. The centre of the clamp has the smallest flux, but this accumulates towards the edges until the highest flux densities are reached at the return yoke. The return yoke must be thick enough to accept this flux without saturating. If it is important to keep the total amount of steel in the clamp to a minimum (cost or weight considerations say) but still maintain full clamping, the clamp should be made thin in the centre, thickening towards the edges. The final thickness of the clamp at the edge should be used for the return yoke as well. In practice the clamp weight is not a constraint and it is much simpler to make it all the same thickness. A simplistic method of estimating the clamp thickness is to determine the B field CHAPTER 4. DIPOLE APERTURES 81 n W « M I N l i t 140 100 tPJ » • t 10 40 M M 100 110 1 40 1 00 IM 211 s = z/D Figure 4.13: Several different clamp configurations. A 2cm thick clamp, a 5cm thick clamp, and two that are 5cm thick with a 2cm tongue under the coil. The points show the desired field contour and are the same for all four cases. CHAPTER 4. DIPOLE APERTURES 82 in the clamp model using P O I S S O N and then multiply this field by the total width of the aperture to determine the total flux that must pass through the two return yokes. T h e return yokes are then made wide enough to accept the flux while keeping the flux density (B field) below the saturation of the iron (approximately 15 kG) . If the return yokes are too small, the steel will saturate and flux will leak back into the aperture. This prediction is confirmed from the studies of the L E P S dipole in appendix D. There, a field clamp with only one return yoke produced a clamped field with a bulge in it. The inadequate return path produced an effect somewhere between a field clamp and a snake. If this problem is encountered on the S A S P , more material can be added to the clamp, so room to add additional material should be provided. One might suppose that rather than building a field clamp that shunts flux around the aperture, the clamp could be connected to the dipole return yoke (figure 4.14a). This would still collect flux from the aperture with the advantage of a shorter route to leave the clamp. Hence the clamp could be made thinner and occupy less space in the dipole aperture. However, this is equivalent to creating a second pole piece on the other side of the coil so that the flux flows in the opposite direction to that in the main dipole air gap producing a large field reversal (figure 4.14b). 4.5.3 Comparing Computer and Scale Models T h e P O I S S O N results can be compared qualitatively to the scale model studies of Hubner and Wollnik [15]. Th is comparison is qualitative because the scale model studies placed the clamp closer to the median plane (Dp « 0.625Z? compared to 1.4D for the computer studies) which seems to change the behavior of the clamp. Like the P O I S S O N studies, a residual field step was observed for small values of Sp although the value was much smaller (< 0.35Z? compared to « 0.6D for the P O I S S O N studies). Different from the P O I S S O N studies, Hubner and Wollnik observed a bui ld up of a CHAPTER 4. DIPOLE APERTURES 83 a) I I 1 1 I I 1 1 i l l i n 1 1 m l n I I I I I I ill In I I i m i l m i l i i i i i i i n l m i i M i i l i i i In m i n i II11 |l11 II III11IIII III II I7S 200 22E 2E0 b) x D E CD \ CD 0.8-0.6-0.4-0.2--0.0--0.2 ,LP2 SF=13.8339#72000.0 Bpmax-15007.623 at 138.000 Brtmax-H687.B37 at 138.000 -VFBPE[0.7128S] PE[ 153.000 tolo(100i000r200.0[X] loldiff-2.165478 xpole=C.OOO VFBPE FIT 100 120 140 160 z (cm) 180 200 Figure 4.14: Shorting the clamp to the dipole return yoke (a) turns the clamp into a "negative" pole face producing the field reversal shown as a solid line in (b) . The desired field is indicated by the dashed line. CHAPTER 4. DIPOLE APERTURES 84 residual field with increasing clamp thickness, tF- They explained this as the clamp collecting more flux which then saturated the return yokes. T h e residual step was observed for tp « 0.671? and SF = 0.35D. This geometry had the clamp much closer to the dipole and median plane than the P O I S S O N studies and does not seem to disagree with simulations. Also studied with the scale model were the clamp length and return yoke thickness — neither of which can be studied directly using the P O I S S O N model. Hubner and Wollnik concluded that a clamp with length about 3D and return yoke thickness 2D would perform well. They produced a specification for an optimized clamp for a dipole with rounded edges (their approximation of a Rogowski bevel). T h e clamp dimensions were: SP = 0.83.D, tp = 0.5D, Dp = 0.83D, length 3D, and return yoke thickness 2D. Note that the clamp is closer to the median plane than the dipole pole piece. Such a geometry is undesirable for the S A S P as the clamp would greatly restrict the spectrometer's particle acceptance. 4.6 Clamp Design From computer and scale model studies of the field clamp an ideal clamp for the S A S P can be described. The clamp plates should be 0.5D thick, set parallel to the pole faces of the dipole at a height of about 0.7D above the median plane. T h e length of the clamp should be at least three gap widths and based on Sp ~ 1.5D this makes the clamps project a total of 4.5D from the aperture. T h e exact clamp - pole edge separation, Sp, will be adjusted to make the fringe field follow the Enge function (i.e. scale with the gap width) over the aperture. The return yoke thickness will depend on the estimated flux through the field clamp which in turn depends on the width of the aperture and the excitation of the magnet. The clamp plates are set parallel to the pole face for two reasons. First , this keeps the geometry more or less constant over the CHAPTER 4. DIPOLE APERTURES 85 aperture. Second, it is simpler to construct a clamp made from flat plates. Figure 4.15 contains conceptual drawings of the field clamp. Figure 4.15a shows a cross section taken through pole edge, coil, clamp, and vacuum box perpendicular to the pole edge while figure 4.15b shows a plan view of the aperture and field clamp. The clamp plate is parallel to the pole face but further from the median plane allowing room for the vacuum box. The vacuum box itself is displaced from the pole face plane to allow room for rays to pass from Q2 to the dipole. The dipole coil also lies parallel to the pole face. T h e coil is displaced 5cm from the pole face. 3cm accommodate the vacuum box and the ray bundle, and 2cm provide for the "tongue" of the field clamp to project under the coil. As discussed earlier, this impedes the effectiveness of the clamp. Beyond the coil, the clamp expands to its full 5cm thickness. T h e clamp continues parallel to the pole face until the "bend line" where it flares away from the median plane to allow passage of the ray bundle from Q2. T h e position of the bend line varies over the width of the aperture. This is the line where the clamp can start flaring without hitting the coil as shown in figure 4.15b. While the spectrometer's effective aperture would be increased by flaring the clamp immediately after the coil, this would necessitate a bulging, curved field clamp and the vacuum box would be difficult to construct. T h e width (in the bend plane) of the clamp is indeterminate. Monte Carlo studies (section 6) indicate that both apertures are fully illuminated by rays. Not all of these may be desirable so determination of the final width of the vacuum box and clamp will be deferred until the engineering design of the S A S P is closer to completion. This field clamp design coupled with the Rogowski pole edge bevel produced a fringe field different from the one used in the original R A Y T R A C E optical design (the original fringe field was also for a clamped magnet). However, when the new fringe field was parameterized and fed into R A Y T R A C E , it had no effect on the spectrometer performance. Therefore the desired field was defined to be that produced by the aper-CHAPTER 4. DIPOLE APERTURES 86 a) pole piece field Qclamp vacuum box ^—-bend line median plane pole edge b) pole piece Figure 4.15: Conceptual field clamp design. Not to scale, a) Cross section perpendic-ular to the pole edge, b) P lan view of the aperture and field clamp. CHAPTER 4. DIPOLE APERTURES 87 ture design at XVFB = Ocm and SF = 14.0cm. Once the P E B (section 4.3) was fitted to the internal R A Y T R A C E field using the parameter V F B P E , the field clamp parameter Sf was adjusted to produce the desired fringe field over the aperture. SF was then described as a function of the V F B position. This function was used to produce points on the Field Clamp Boundary ( F C B ) which, like the P E B , was fitted with a polynomial curve. 4.7 Fitting the Clamp and Pole Edge 4.7.1 Fitting Procedure As discussed earlier, most of the parameters of the aperture design were fixed (e.g. bevel shape, clamp thickness and displacement from the median plane) leaving only two unknowns: the V F B — pole edge distance ( V F B P E ) ; and the field clamp — pole edge separation (Sp)- These parameters must be adjusted to produce the desired R A Y T R A C E fringe fields over the extent of the apertures. Since the exact widths of the field clamps have not been established, it is necessary to do this fitting process over a wider range than the field clamps are expected to extend. This results in the following ranges for XVFB (which are called xB (entrance) and xc (exit) in figure 3.3): Entrance (-50cm,+50cm) Exit (-80cm,+100cm) The fitting process for each aperture is divided into four steps. 1. F i t V F B P E at several locations on the V F B . 2. Using the above V F B P E values, fit SF at the same locations. CHAPTER 4. DIPOLE APERTURES 88 3. F i t polynomial curves to the V F B P E and SF data and use them to draw points on the Pole Edge Boundary ( P E B ) and the Field C lamp Boundary ( F C B ) in the same coordinate system as the V F B . 4. Fi t polynomial curves to the P E B and the F C B . The polynomial fits to the P E B s and F C B s can be utilized to layout and machine the dipole pole pieces and the field clamps. P O I S S O N was used to fit the parameters V F B P E and SF by modelling cross sections taken through the field clamp — pole edge structure at various locations per-pendicular to the V F B . Figure 4.16 shows the P O I S S O N geometry used for the cross section at XVFB — 40.0cm for the entrance aperture. The P O I S S O N model was run without the field clamp to fit V F B P E and then with the field clamp to fit SF-4.7.1.1 Measuring V F B P E and SF Measuring V F B P E proceeds as follows: First , an initial estimate of the correct V F B P E distance is made, and the appropriate pole edge geometry described in an A U T O M E S H (the P O I S S O N input program — see appendix C) input file, appendix E contains an example. P O I S S O N is run and the resulting field profile compared to the expected R A Y T R A C E profile. To compare the curves, it is necessary to normalize them by setting their peak values to 1.0 and superimposing the peaks. Since V F B P E is being fitted, only the interior R A Y T R A C E and P O I S S O N fields are compared as discussed in section 4.3. The difference between the R A Y T R A C E and P O I S S O N fringe fields is quantified by the area between the two curves, calculated using a numerical integration. T h e fitting process is repeated until the value for V F B P E is within ± 0 . 0 5 c m of the value that produces a minimum area between the R A Y T R A C E and P O I S S O N curves. Once the V F B P E is fitted, a similar procedure is used to fit the field clamp pa-rameter Sp using the optimal value for V F B P E . T h e difference between fitting V F B P E CHAPTER 4. DIPOLE APERTURES 89 z (cm) Figure 4.16: a) P O I S S O N model of the dipole aperture and field clamp. Position: Entrance, xvfb = 40.0cm b) The P O I S S O N field profile (points) compared to the desired R A Y T R A C E field profile. CHAPTER 4. DIPOLE APERTURES 90 and SF is that rather than only comparing the internal fields, the entire field profile is compared when fitting SF- Use of only the fringe field is precluded because the internal and fringe fields are not completely decoupled. B y fitting the entire field profile with Sp-, perturbations to the internal field by the clamp are minimized. Sp, like V F B P E is fitted to an accuracy of -b0.05cm. Each iteration of fitting V F B P E and SF require new P O I S S O N problem files with subtly different geometries. Furthermore, each location on the V F B has a different geometry since the clamp shape, pole face slope, and gap width change over the aperture (section 4.6). T o create the P O I S S O N problem files by hand would be very tedious and prone to error. Instead, the program M A C A U T O ( M A C r o processor for A U T O m e s h ) was written to act as a preprocessor for A U T O M E S H (which in turn is a preprocessor for P O I S S O N ) . Given as input XVFB, V F B P E , and S>, M A C A U T O calculates the correct pole face slope, pole edge bevel, coil position, and field clamp cross section for either S A S P dipole aperture. Analysis of the P O I S S O N predicted field profile was made by the program F R A N A (FRinge field ANAlys is ) which read the field shape from the P O I S S O N output file, cal-culated the R A Y T R A C E field shape for the appropriate V F B location, normalized the fields, and calculated the area between them over the desired range. T h e algorithm F R A N A uses to calculate the R A Y T R A C E fringe field is not exactly the same as that used by R A Y T R A C E . To calculate the field at a point in the fringe field region, R A Y -T R A C E finds (approximately) the closest point on the V F B , determines the internal field at this point and then modifies this field by the Enge function (equation 4.1). The Enge function depends on the dipole gap at the V F B point and the distance of the desired point from the V F B in units of the gap width. Th is means that the field profile along a perpendicular to the V F B could be determined by various points on the V F B , depending on how convoluted the V F B is. Since the S A S P V F B s are not convoluted, CHAPTER 4. DIPOLE APERTURES 91 the R A Y T R A C E algorithm was approximated by using the V F B point through which the perpendicular passes for all the field calculations. This should not have a significant effect on the results. The fitting process was fully automated with the program F I T T E R which, given the range to try fitting the parameter over, would perform a binary search looking for the minimum of the area between the curves. Each iteration of the fit generated a new value of the parameter ( V F B P E or SF) which M A C A U T O used to generate a new P O I S S O N problem file. Next P O I S S O N was run and the output analyzed with F R A N A which fed its results back into F I T T E R . Each iteration took typically 5 to 8 C P U minutes on the V A X 8600 computer. Since on the average it takes eight iterations per fit, the process is lengthy and it is desirable to fit as few points as possible. The resulting fits to V F B P E and SF for both entrance and exit apertures are shown in figures 4.17 and 4.18 and tabulated in table 4.1. A total of seven points on the entrance V F B and nine points on the exit V F B were used for the fit. T h e behaviours of V F B P E and SF as functions of XVFB a r e quite different for the entrance and exit apertures. Part of this is because in the entrance coordinate system, dipole gap width decreases with increasing XVFB while in the exit coordinate system, gap width increases with increasing XVFB- The sudden change in curvature of both V F B P E and SF for the exit at large negative XVFB is because of the change in curvature of the V F B which starts to decrease at about -60cm (compare with figure 4.4b). 4.7.1.2 F i t t i n g P E B a n d F C B T h e discrete points calculated for V F B P E and SF must be turned into polynomial descriptions of the Pole Edge Boundary ( P E B ) and the Field C lamp Boundary ( F C B ) . These polynomials should look similar to the polynomials used to describe the V F B ' s in section 4.2 (but without the dependence on dipole central radius, R). T h e conversion CHAPTER 4. DIPOLE APERTURES 92 Aperture a v F B ( cm) V F B P E ( c m ) Sf-(cm) ± 0 . 0 5 ± 0 . 0 5 Entrance -50.0 -0.05 15.70 -40.0 0.10 15.53 -20.0 0.15 15.13 0.0 0.51 14.00 20.0 0.70 13.25 40.0 1.62 10.44 50.0 2.45 8.80 Exit -80.0 0.10 14.70 -60.0 1.38 10.84 -40.0 1.18 11.50 0.0 0.51 14.00 20.0 0.10 15.00 40.0 0.15 15.20 60.0 0.25 15.44 80.0 0.30 14.70 100.0 0.35 14.13 Table 4.1: V F B P E and Sf fits to various locations on the entrance and exit V F B s . CHAPTER 4. DLPOLE APERTURES 93 a) - 6 0 - 4 0 i 1 r -20 0 2 0 x ( c m ) b) - 1 0 0 Figure 4.17: V F B P E as a function of XVFB for a) Entrance and b) Exit apertures. Figure 4.18: SF as a function of XVFB for a) Entrance and b) Exit apertures. CHAPTER 4. DIPOLE APERTURES 95 of the V F B P E and SF points into the P E B and F C B polynomials is accomplished in two steps. First polynomials are fitted to the data points creating analytic expressions for V F B P E and SF as functions of V F B location. These expressions are used along with the V F B curves to generate points on the P E B and F C B . Finally, polynomials can be fitted to the P E B and F C B points. The choice of polynomial curves to fit to the V F B P E and SF data is limited. The highest order for the polynomial is the number of data points minus one. O n the other hand because the data points do not vary smoothly, as high an order polynomial as possible is needed to fit the data reasonably. There is no such constraint when fitting the P E B and F C B curves however. The original V F B curves (which will affect the P E B and F C B curves) were of fifth order for the entrance aperture and sixth order for the exit. Thus, the P E B and F C B curves are expected to be at least this order. Because R A Y T R A C E uses eighth order polynomials in its calculations it was decided to do the same for the fits. Th is was done and the P E B ' s and F C B ' s generated. However, when investigating how dipole saturation (discussed later) shifts the V F B , a 2 0 % loss in spectrometer resolution was apparent. Using the eighth order curves to fit the P E B and F C B introduced high order distortions to the fifth and sixth order V F B s . As a remedy, the P E B and F C B data were refitted with the same order curves as the V F B s for each aperture. This smoothed out the high order distortions. To determine the effect this had on the V F B , the dipole - clamp geometry specified by the lower order curves was used with P O I S S O N to calculate the V F B P E distance. These values for V F B P E are different from those calculated previously since the low order P E B and F C B curves do not follow the original data exactly. V F B curves of the same order as the original ones are then fitted to these data and fed back into R A Y T R A C E . Th is step prevents any of the high order distortions from showing up. T h e R A Y T R A C E studies showed that using the lower order P E B and F C B , the spectrometer resolution CHAPTER 4. DIPOLE APERTURES 96 is 10% less than the resolution the ideal optics produced — which is 10% better than that produced by the high order polynomials. As of the writing of this thesis, no further work has been done to decrease the distortions caused by the pole edge and field clamp boundaries. T h e specifications presented in this work produce a 10% loss in the spectrometer resolution. Further work will be carried out later to attempt to improve the resolution. Specifically, the ideal V F B and the P O I S S O N produced V F B will be compared. A function relating a shift in V F B to a shift in the P E B and F C B will be developed and used to apply a correction to the P E B and F C B curves. Hopefully, this will produce an improved specification for the dipole apertures. The polynomial curves and coefficients used to describe the P E B and F C B are listed in appendix F along with the aperture and field clamp specification. 4.7.2 P E B and FCB Tolerances The loss in spectrometer resolution observed above helps estimate the effect of an inaccurately machined dipole aperture or field clamp. O n the basis that loss of spec-trometer resolution is approximately a linear phenomena, the V F B curve with a 10% loss is subtracted from the ideal V F B curve (figure 4.19), and the difference is said to cause the 10% loss in resolution. Then , since the P E B and F C B are roughly the same shape as the V F B , it can be said that a machining error the size of the V F B difference will produce the 10% resolution loss. The assertion that the V F B shift is equal to the corresponding shift in the P E B or F C B is a conservative estimate; the P O I S S O N stud-ies show that the V F B always shifts less than the steel geometry. Therefore a one to one relationship overestimates the sensitivity to the machining. Th is method concludes that a 1mm shift in the machined curve from the specified curve over a distance of 30cm will produce a 10% loss in spectrometer resolution. The actual machining tolerances CHAPTER 4. DIPOLE APERTURES 97 Figure 4.19: Subtracting the 10% loss V F B from the ideal V F B curve a) Entrance, b) Exit . CHAPTER 4. DIPOLE APERTURES 98 are expected to be much smaller than this and, therefore, will not be a problem. 4.7.3 Saturation Effects The resolution loss discussed above was observed when examining how the V F B shifts with magnet excitation. Ideally, the V F B should be unaffected by excitation. The aperture and field clamps were designed for a magnet operating at approximately half the maximum excitation — well into the linear range of the dipole. To see if there are significant saturation effects, the geometry specified by the P E B and F C B curves was used in a P O I S S O N simulation using a magnet at the S A S P ' s maximum operating excitation. T h e shift in the V F B was calculated and a new V F B curve constructed. This was then fed into R A Y T R A C E . Comparing the high and low excitation V F B curves visually, only a slight shift in the V F B is apparent. T h e R A Y T R A C E studies found no significant change in the spectrometer resolution caused by these small shifts (note that there is still the 10% resolution loss in both low and high excitation curves). Thus, the aperture design meets its goal of a fringe field unaffected by magnet setting. 4.7.4 Quality of the Calculation T h e quality of the specifications presented are difficult to quantify. Ideally the results from the P O I S S O N model should be compared with the field of an actual magnet of similar design to the S A S P dipole. This was explored with the data from the L A M P F L E P S dipole (appendix D) . Unfortunately, the field maps for the L E P S are not as extensive as desired and more importantly, the L E P S fringe fields differ in two major ways from the S A S P fringe fields. First, the pole edge bevel of the L E P S is a simple 60° slope, while the S A S P will have a Rogowski bevel. The P O I S S O N models indicate that these two types of bevel behave differently as a function of excitation. Second and more important, the L E P S is an undamped magnet with a correspondingly more extended fringe field. It is precisely S F ( c m ) Figure 4.20: A plot of the area between the R A Y T R A C E fringe field curve and the P O I S S O N fringe field curve as a function of the distance of the field clamp from the pole edge (Sp) for two different P O I S S O N convergence criterion e = 1 0 - 3 and e = 10~ 4. The units for area is fraction of the total R A Y T R A C E fringe field. an extended fringe field that P O I S S O N is expected to model poorly. Internal consistency in the P O I S S O N model can be checked. Figure 4.20 shows a plot of the area between the R A Y T R A C E and P O I S S O N fringe field curves called the residue as a function of Sp for two values of e, the P O I S S O N convergence criterion. Plotted as a fraction of the total curve being fitted, r is on the order of 1% and the change in r with Sp on the order of 0.1%. A l l the field clamp and pole edge calculations were done with e = 1 0 - 3 . The oscillations of e are disturbing, r was expected to vary smoothly, dropping with changing Sp, passing through a minimum, and then increasing again. The fact that the oscillation region decreases in width when the more stringent convergence criterion is used indicates that the oscillation is an artifact of P O I S S O N and not the magnetic system being simulated. The oscillations were not anticipated; similar studies of the dipole were carried out with a simpler geometry (flat pole face, CHAPTER 4. DIPOLE APERTURES 100 coil and clamp parallel to the median plane) with no such oscillation observed. T h e over-relaxation process of fitting the differential equations may have become unstable and the solutions start to oscillate. These studies show that while SF and V F B P E were fitted to ± 0 . 0 5 c m , a more realistic estimate of the error might be ten times this. However, not all the points fitted experienced the oscillations and a look at the plots of V F B P E and SF clearly show that the data is too smooth to have an error of ± 0 . 5 c m . Another problem encountered when fitting V F B P E and SF was that for some positions on the pole edge, P O I S S O N was incapable of creating the steel geometry from the input file. This happens when a complex geometry is described with too coarse a mesh. P O I S S O N ' s mesh generator, A U T O M E S H cannot fit all the features of the geometry with the mesh points available and gets stuck. A solution is to use a smaller mesh but this increases the execution time for the program and more important, there is a limit to the number of points that can be described in a problem. The current mesh size used for the S A S P aperture simulations is approximately at this limit. The result of this reluctance of P O I S S O N to fit some points on the V F B is that the V F B P E and Sp functions are not mapped out as thoroughly as is desired. For example, it would have been useful to have more points around the peak in V F B P E for the exit aperture. These problems with P O I S S O N indicate that the program has been stretched to its limit. P O I S S O N was never intended for such detailed studies of small changes in the field. A t this point it should be remembered that a three dimensional system is being approximated in two dimensions. Regardless of the accuracy of P O I S S O N , the effects associated with a 3D aperture discussed in section 4.1 can not be modelled and may completely invalidate the results presented here. There is a distinct possibility that the aperture specifications given in this work will not produce a satisfactory field. Thus it is important to consider methods for altering the dipole fringe fields after construction CHAPTER 4. DIPOLE APERTURES 101 of the dipole. Possible methods of doing this are, altering the pole edge boundary, and altering the field clamp. Altering the dipole pole edge would be a complicated task. The pole piece is made from a single piece of steel so it would be necessary to send both pole pieces out for machining. Alternately, the field clamp studies indicate that the V F B can be shifted by a centimetre or more by changing SF- If the tongues of the field clamps are made detachable, modifying them or machining new ones could be readily accomplished. However, altering the dipole V F B ' s would be time consuming if it is necessary to remachine the field clamp tongues for each measurement. It would be useful to be able to do fast temporary modifications of the F C B . One way of doing this is by building up the F C B with thin iron strips attached to the field clamp with nonmagnetic fasteners. Once a satisfactory F C B was found with this method, the field clamp could then be sent out for machining. Part II Simulating the SASP 102 Chapter 5 Monte Carlo Method There are several reasons for simulating the D A S S / S A S P system. As is discussed in chapter 6, simulations were used during the design of the S A S P to calculate the profiles of the ray bundles passing through the spectrometer, helping to locate the vacuum vessels and detector arrays. Simulations can also predict the solid angle and data rates of the spectrometer allowing an estimate of the required run times of an experiment. If the kinematics of a reaction are included, a simulation can predict the distributions of the properties of the particles to be detected. This guides the experimenter in the design of detectors and the positioning of the experimental apparatus. B y including the characteristics of the detectors in the simulation it is possible to study the performance and resolution of the detector systems. These studies can be used to track down problems in the real apparatus. Any simulation of the D A S S / S A S P system must describe the properties of the incident particle beam, the target and the nuclear reactions within it, the scattering of the various reaction products towards the detectors, and the detectors themselves. The proton beam entering the target chamber at 4 B T 2 is created by accelerating negative hydrogen ions in the T R I U M F cyclotron, stripping them of electrons and injecting the remaining protons into beam line 4B. The energies of protons leaving the cyclotron are not identical, they have gaussian distributions. Typical ly a beam at 500MeV energy 103 CHAPTER 5. MONTE CARLO METHOD 104 will be made up of particles with energies varying 0.1% or more from this value. T h e beam enters a series of bending, focussing, and dispersive optics which correlate many of its properties. For instance a set of optical elements collectively called the twister is used to rotate the beam from a horizontally smeared spot to a vertical one and to dispersion match it (appendix B) with the result that the vertical position of the beam is linearly related to the proton momentum. In the scattering chamber the proton beam hits the target which is usually a rectangular block oriented at an angle to the beam. Most of the protons will pass through the target without interaction and continue on "downstream". A small fraction of the beam particles do interact with the target atoms. The interaction can be a simple scattering of the beam particle off the target atom or a complex nuclear reaction where the beam and target nuclei are destroyed and new particles created. As the beam particle travels through the target, it may undergo multiple scattering and energy loss. Similarly, the reaction products may be deflected by the intervening material as they leave the target. T h e energies and trajectories of the reaction products are governed by the reaction kinematics which are in turn affected by the energy and trajectory of the incident proton. O f all the possible trajectories of the scattered particles, only those that encounter the detectors are of interest. T h e detectors can be magnetic spectrometers such as the S A S P and M R S , counter telescopes consisting of wire chambers and scintillators, or anything else that will reg-ister the passage of particles. A l l detectors have some common features however. They have a limited aperture through which the particles must pass, quantified as the detec-tor's solid angle. The solid angle is the fraction of a sphere (47r steradians) surrounding the target that the detector occupies and is usually measured in millisteradians. A l l detectors have an efficiency less than 100% which varies depending on the type of de-tector and the number of sensing elements in the detector system. A l l detectors have an CHAPTER 5. MONTE CARLO METHOD 105 inherent resolution meaning they do not measure particle properties exactly. Lastly, most detectors alter the properties (such as the energy and trajectory) of particles traversing them. This may be significant if a particle passes through several detectors in series. Based on the purpose of the simulation, there are two basic simulation meth-ods, analytic and Monte Carlo. A n example of an analytic simulation is the program T R A N S P O R T [6] used to study beam optics. A particle beam injected into a system of optics is described by various parameters such as the centroid of the beam and its maximum extents. The optical system is described by mathematical formulas approx-imating how each optical element effects the beam. The effect of the entire optical system on the beam is determined by performing the mathematical transformation of each element in succession. This approach to modelling the system is adequate for studying beam lines where the behaviour of the particles in each part of the system is precisely known. A n advantage of a program such as T R A N S P O R T is that it is fast and gives an accurate prediction of the general properties of optical systems. How-ever, the analytic approach breaks down when investigating the kinematics of nuclear reactions in the target or the resolution of a detector. Random processes occur, for ex-ample where incident particles are injected into a target more than once with the same trajectory and energy and different reaction products are produced. Where reaction products are the same for each trial, they can have different trajectories or energies. To ascertain these effects with an analytic model, it is necessary to integrate the kine-matics of the reaction over all possible beam profiles. Such an integration would have to be done numerically, and its limits and variables would be difficult to formulate. The Monte Carlo type of simulation eliminates the need to describe difficult in-tegrals. Monte Carlo is the name given to any technique that uses random numbers to solve a problem. For nuclear experiment, a Monte Carlo can be either a direct simu-CHAPTER 5. MONTE CARLO METHOD 106 lation or an integration. Direct simulation consists of generating beam particles with properties following the beam's known distributions at the target and having these par-ticles interact with the target particles. The random aspects of scattered particles are selected from random populations with the proper distributions. These reaction prod-ucts are then transported through the detector system; setting off the various sensors and undergoing interactions in the detector materials. Where appropriate, numbers from a random distribution are generated and the effect calculated. After numerous runs, distributions of the states of the particles and the outputs of the detectors are devieloped and the average and extreme behaviours of the various parts of the system can be deduced. Where the momentum resolution of a spectrometer is being examined a Monte Carlo simulation of the spectrometer is run where the momenta of the par-ticles injected are kept constant while the trajectory is varied. A n ideal spectrometer with perfect detectors would produce a sharp momentum peak. However, the nonideal spectrometer and the imperfect detectors, will smear the momentum peak and the res-olution of the spectrometer is then measured by the peak's width. To determine how much of the error in momentum is caused by the spectrometer optics and how much by the detector resolution, the simulation would be run with perfect detector resolution and the resultant resolution would be a function of the optics only. Whi le direct simulation of the system by a Monte Carlo shows general properties and trends, answers to specific questions such as the data rates for a specific experiment or the solid angle of a detector are often required. Such questions can be answered with a Monte Carlo integration. To see how a Monte Carlo can be used to evaluate an integral consider the following. Suppose the area within the region described by some known function / is to be found and that / is either impossible or very tedious to integrate. One familiar solution to this problem is to integrate the function with a numerical technique like Simpson's Rule [8]. A less familiar technique is to randomly generate CHAPTER 5. MONTE CARLO METHOD 107 points uniformly distributed within a known region (call it Ao) that encompasses the region with the unknown area. The known region could be a simple box or it could be some elaborate shape. It is only necessary that it's area is known and that it completely engulfs the unknown region. For each randomly generated point, it is noted whether the point lies inside or outside the unknown area. For N tries, say there are n„ points that lie inside the area. Then as iV gets very large, the area enclosed by / is, where U{ is a uniform random number generated on the interval (a, b) and /(it,) the function evaluated at that point. The technique can be extended to multi-dimensional integrals. Th is method of solving an integral is called a Monte Carlo. T h e Monte Carlo approximates the correct solution to an integral. T h e error in the Monte Carlo summation is estimated as the standard error in the mean value of / . This is called the law of large numbers and can be generalized [16] to where / is the mean value of / , / = 4 E/(«••) CHAPTER 5. MONTE CARLO METHOD 108 and e = s/y/N. s is the sample deviation, So the Monte Carlo evaluation of the integral is, jf/(«)d« = ^ E * ± ( * - a ) As the number of points increase, the Monte Carlo approximation improves as ap-proximately l/y/N. Provided the function / is well behaved (always finite, piecewise continuous, finite number of discontinuities), a Monte Carlo integration will converge to the correct solution. However, it will not necessarily converge quickly. There is an art to setting up a Monte Carlo integral and selecting biased sampling populations to speed convergence [17]. For a particle beam experiment, the primary advantage of a Monte Carlo inte-gration is that it is unnecessary to explicitly state the limits of integration. Apply ing Monte Carlo integration to the calculation of the solid angle of a detector, particles are generated at the experiment's target and aimed into a cone centred on but larger than the detector. A l l the particles that penetrate the detector are recorded as hits and the solid angle of the detector is given as, n* ^detector = where AQ is the solid angle of the cone. Th is is equivalent to integrating the differential solid angle of the detector as a function of the incident ray, N(N - 1) CHAPTER 5. MONTE CARLO METHOD 109 where r\ has four components, x i , #i, j / i , <^ >i, (6 is fixed). The limits of integration are complex functions dependent on the optics and the apertures of the detector system. If this integration was carried out with a numerical technique, these limits would have to be stated explicitly. The Monte Carlo implicitly considers the integration limits by simulating the behaviour of the particle traversing the detector. Similarly, the data rate for an experiment could be calculated by injecting N beam particles into the system for which n* products are detected. T h e data rate of the system is then, where Rbeam is the rate at which the beam particles hit the target. It would be naive to just inject beam particles into the simulation and count the number of successful events in the detectors since only one in 10 6 or fewer beam particles interact with the target — let alone pass through the detector system. Instead, the experimenter would force every beam particle to interact with the target and multiply the resultant data rate by the probability of the beam particle interacting with the target, R =  WintJ~f Rbeam where u;, n t is the probability of an interaction and is called a weighting factor or just weighting. It is equivalent to changing the limits of the Monte Carlo integration. Weighting is also used when aiming a scattered particle towards a detector. Suppose a reaction scattered particles homogeneously in all directions. If the Monte Carlo faithfully simulated this, most of the particles generated would miss the detector which may have an acceptance of only a few millisteradians. Instead, the Monte Carlo would force the particles into a cone centred on and fully encompassing the detector's aperture. Most of the reaction products would enter the detector but similar to the beam hitting CHAPTER 5. MONTE CARLO METHOD 110 the target, the success rate would be improperly high. Another weighting factor, ^cone WSCat — ~ . 47T must be used in the calculation of the data rate. R — IVintWgcat—Rbeam If the Monte Carlo accurately models the geometry, kinematics, and probabilities of the experiment, no weighting factors are required. Where short cuts are taken to speed up the simulation, the probabilities are affected and a weighting factor must be used. Another way of looking at this is to imagine the particle starting the simulation with a weighting of unity. A t each point in the simulation, a local weighting factor in multiplied to the particle's weighting factor. For most points, this local weighting factor is unity, but occasionally, such as for the scattering towards the target, the local weighting is a number less than one (the weighting factor must be in the range 0 - 1 ) . If the particle spawns other particles during a reaction, then these daughter particles inherit the parent's weighting factor. 5.1 EASY, a Monte Carlo Simulator There are several objectives for the design of the S A S P ' s computer simulation. First it has to accurately model the transport of particles through the spectrometer to allow the examination of the effects of the various apertures in the system on the spectrometer's acceptance and solid angle. Using studies of the ray profiles at various locations in the spectrometer, the vacuum system could be designed to have a minimal effect on the solid angle. Many surfaces in a spectrometer can cause particles to scatter and still penetrate the spectrometer. Such particles will have their momentum and trajectories slightly CHAPTER 5. MONTE CARLO METHOD 111 altered, and will decrease the resolution of the spectrometer. Sometimes these bad rays can be recognized by their trajectories through the detector array. Scattering from the pole face near the exit could be recognized by projecting the ray back from the detectors to the dipole exit. However, if the scattering occurs deeper in the spectrometer, it may be impossible to identify. A solution to this is to place "active collimators" (i.e. scintillators) at critical locations to identify bad rays passing through the system. The simulator can identify the optimum locations for the collimators and the event rates they will experience. The second intent of the simulation is to study the performance of the S A S P as a spectrometer. Specifically, such things as the position and shape of the focal surface, the momentum resolution of the spectrometer, and its ability to trace rays back to the target. T h e resolution and traceback properties would be studied in a variety of detector configurations such as with and without a front end chamber. The simulation must consider the effects of multiple scattering in the detectors and vacuum windows and position detector resolution. This information determines the optimal location for detectors and their necessary resolution. This is significant since the cost of wire chambers goes up with the increased resolution. It is pointless to build chambers ten times more accurate than the resolution of the spectrometer optics warrants. B y studying the performance of the S A S P with a computer model, a feel for the real system can be obtained which will speed up the commissioning process. Finally, the computer simulation will be used to model experimental arrangements of the D A S S / S A S P system. A simulator permits data rates and the requirements for detector sizes and positions to be estimated. A n y experimental arrangement is effected by the kinematics of the reaction being studied. For a simple case, such as in a two body reaction (section 5.3) the kinematics (but not the angular dependence of the cross section) can be determined analytically. For three body and more complex reactions, CHAPTER 5. MONTE CARLO METHOD 112 approximate models have to be utilized. Even crude approximations to the desired reaction can provide much useful information. In addition to modelling the kinematics at the target, the simulator must reproduce the profile of the beam incident on the target and the resolution and efficiencies of the detectors. T h e different aspects of an experiment can be thought of as a four level hierarchy: 1) Equipment 2) Acquisition Electronics 3) Analysis 4) Control The equipment in an experiment includes all the hardware including vacuum systems, targets, and detectors. The acquisition electronics takes the raw data from the detec-tors, processes it, performs simple processing such as ensuring various detectors fire in coincidence, and then passes the data on to be analyzed and stored. Analysis consists of examining the data, determining which is valid and then reconstructing the desired information about the system. Analysis can be simple "on line" analysis to check that all the systems are performing properly, or the thorough "off line" analysis that exam-ines the entire system. Finally, control represents the acts of the experimenter. The experimenter directly controls the particle beam, the position of a spectrometer about a target, detector settings and, beam characteristics. E A S Y (Equipment and Aquisi t ion SYstem) attempts to mimic the experimental hierarchy. The first aspect of the experiment (equipment) is simulated by constructing the experimental apparatus out of standardized pieces of equipment (called devices). The effects of these devices on particles passing through them are simulated using a direct Monte Carlo simulation. The second aspect of the experiment (Acquisition Elec-tronics) is modelled partly analytically and partly with a Monte Carlo. T h e analytic CHAPTER 5. MONTE CARLO METHOD 113 aspect of the acquisition electronics is the description of the "logic" used in the exper-iment. Namely, which detectors must fire in combination to produce a valid signal. A Monte Carlo method is used to create the errors in the signals from the position sensors. Analysis, the third aspect of an experiment, is completely analytic. Data taken from the simulation of the acquisition electronics can be analyzed by standard functions built into E A S Y or recorded for processing by other programs. T h e last aspect, Control of the experiment, is also completely analytic. A n experiment, or problem as it is called in E A S Y , is described in three parts using three separate files. First all the equipment (devices) used in the experiment are described in the problem's device file. Devices include targets and detectors, drift regions (air and vacuum) and the spectrometer optics. The simulations of some of devices are abstracted. For example, the beam optics simulated by the device b e a m . The beam optics describes what happens to the proton from its initial excitation at the ion source, its passage through the cyclotron and beamline optics up to the tar-get chamber (for the D A S S / S A S P system the target is 4BT2) . During its travels, the proton's characteristics (position, angular divergence, energy, etc.) form inter-dependences, correlations and limits. A n example of a limit is that no beam particle will be found outside the beam pipe. The device b e a m simulates the beam optics by giving the particle the distribution of properties it should have just after leaving the cyclotron and then using a first order transport matrix to move the particle from the cyclotron to the target where it is checked to ensure it is still in the beam pipe. This simplification of the beam optics is not perfect. The proton could leave the beam pipe between the cyclotron and the target and re-enter before reaching the target without detection. Th is would produce an unrealistic beam profile on target. A more accurate model of the beam optics would calculate the particle's position at several locations in the beam line. Such a model requires massive computation and is not necessary for CHAPTER 5. MONTE CARLO METHOD 114 simulation of the D A S S / S A S P system. Devices like the optics of the S A S P and M R S spectrometers are modelled with various transport matrices and a series of "aperture cuts" that block rays at strategic locations within the devices. As well as devices that mimic hardware in the experi-mental system, E A S Y has devices with no counterpart in the real world. For example the device b a c k t r a c k drifts a ray backwards through space. Th is could be used to put a front end chamber before a spectrometer which expects the ray to start at the target. The ray would first be transported through the front end chamber within which it would undergo multiple scattering altering it's trajectory. T h e n the ray would be back tracked to the target and passed into the spectrometer with its new trajectory. Once the experimental hardware is described, E A S Y has to be told where data is to be collected and how to process it — analogous to the acquisition electronics and analysis components of an experiment. This is done in the data acquisition file (dac file for short) using E D A C ( E A S Y D a t a Acqu is i t ion ) . E D A C is a language that allows the user to specify unique points in the experimental system and tell E A S Y to collect and process information from those points. If modelling a real experiment, the points data would be collected from are the outputs of the detectors. In the simulation however there is the freedom to collect data anywhere. This could be used to measure the trajectory of a ray with wire chambers and then compare the reconstruction with the ray's true trajectory. The data acquisition electronics in an experiment usually perform logical operations on the detector data, ensuring that the proper combination of detectors have/haven't fired before sending the information on to be processed fur-ther (this is called a hardware trigger). E D A C allows simple logical operations to be performed on the data collected from the system before further processing by E A S Y . Basic processing by E A S Y consists of displaying data using histograms and scat-ter plots or storing the data "to tape" (actually to a file) for post processing by another CHAPTER 5. MONTE CARLO METHOD 115 program. In addition to this, E D A C has constructs called functions that can perform calculations within E A S Y . Each function is based on a generic function which would perform a task, such as taking the output of two Vertical Drift Chambers and recon-structing the trajectory of a ray. The function would consist of a call to the generic function with parameters describing the geometry of the two V D C ' s and with argu-ments which would be the outputs of the various wire planes on the chambers. The advantage of such a system is that frequently encountered problems can be described in a general way using a generic function and then the specific details easily added for each new experiment. Furthermore, different functions can be set up using the same generic function with different arguments and parameters. Thus both the S A S P and M R S V D C ' s could use the same trajectory reconstruction algorithm even though they have different geometries. The final aspect of E A S Y is control. This takes the form of a series of commands found in the command file. Commands can mimic the control found in a real experi-ment such as changing the beam energy or setting a spectrometer's central momentum. Commands can also turn off all the multiple scattering in the system or turn off the errors in position sensors. Some useful commands are m a p s o l i d and m a p f o c a l which map the solid angle and focal plane respectively of a device. Commands such as these can do the job of an entire elaborate experiment. T h e hierarchy of E A S Y is as follows: first a system is described, then the data acquisition specified, and finally it is controlled. As in a real experiment, the control has no effect on the equipment and the electronics. The electronics and analysis can be altered without effecting the equipment. If the equipment is altered (detectors added, distances changed, etc.) the acquisition and analysis may have to be altered as well. A complete description of E A S Y is found in the manual EASY, A Monte Carlo Simulator[19]. CHAPTER 5. MONTE CARLO METHOD 116 5.2 Modeling the SASP Optics As discussed in chapter 2, the S A S P is modelled using the conventions of the program T R A N S P O R T , in which a projectile is represented as a six element ray propagating in the positive z direction. x 8 - . _ y l . 8 . where (x, y) is the position of the ray in the plane perpendicular to the nominal direction of travel; 8 and <f> are the angles the ray makes with the z axis in the x—z and y—z planes (figure 2.1); / is the difference in path travelled by the ray and the central ray; and 8 is the ratio of the particle's momentum to the central momentum of the spectrometer, P/Po- The S A S P is described with first and second order transfer matrices and many higher order terms. ft2 = R1fi+R2f[2 + H(fi) Individual terms are represented with the notation, r'2 = ... + (r'/r3rk . . .) x rj X r f x . . . + . . . For example, (x/y8) is the coefficient that modifies the product yi#i when calculating x 2 . CHAPTER 5. MONTE CARLO METHOD 117 5.2.1 Global Matrices As part of the S A S P ' s optical design, R A Y T R A C E generated the sets of transfer coef-ficients to transport a ray from the target to the focal surface. R}, i 2 2 , H where H is a set of eighteen high order terms. Each set of coefficients were for a specific ray momentum (6= -10%, -5%, 0%, +5%, +10%, +15%). T h e first order matrix for 0% is, •^o% — •0.59782 3.94159 0 0 -1.45125 0 0 -1.67637 0 0 0.47013 0 0 0 -3.99876 -12.65651 0 0 0 0 -0.20371 -0.88448 0 0 0 0 0 0 1 0 2.80849 5.75814 0 0 1.04494 1 (5.1) The coefficients were generated by tracing a standard bundle of 46 rays through the spectrometer and are only good for rays close (0.1%) to the momentum they were generated for. T h e quality of the transfer coefficients was tested by sending a bundle of 198 randomly generated rays through the spectrometer using both R A Y T R A C E and transfer coefficients. Taking the final trajectory of the ray calculated by R A Y T R A C E , ^RAYTRACE^ a g c o r r e c ^ the accuracy of the coefficients is measured by subtracting components of the matrix final trajectory, f*2matTtx from the R A Y T R A C E components, t, error i.RAYTRACE i.matrix = r - r, For example, the quantity, Qerror QRAYTRACE /matrix 02 — ^2 — f/2 CHAPTER 5. MONTE CARLO METHOD 118 is the difference between the angle the ray makes in the x — z plane after the S A S P as calculated by R A Y T R A C E and by the transfer coefficients. T h e dependency of these errors on the initial ray (r~l) can be determined by plotting a component error against each of the initial ray components (x i , 8t, yt, <j>i, 8). A n example of this is shown in figure 5.1a where #| r r o r is plotted against 6\. The errors in the other components showed similar dependences on the initial trajectory. This presented a problem. It would be impractical for E A S Y to use the accurate but extremely slow R A Y T R A C E code to model the S A S P . It has to be done with matrices to get a reasonable speed but the matrices generated by R A Y T R A C E are inaccurate. T h e situation is worse when considering how well a set of matrices model rays of differing momentums. The solution was to fit additional terms to one set of coefficients to produce a good fit to the true trajectories. T h e fitting process was done completely by hand. Different order terms and coefficients were added by trial and error to produce a reasonable agreement between R A Y T R A C E and the matrix predicted trajectories of the 198 ray bundles at each of the momentums -10%, -5%, 0%, +5%, +10%, and +15%. When finished,.a total of 105 extra terms had been added to the original 0% matrices. The major additions were, (*/*") (6/9n) (y/<An) (H<f>n) {i/8n)i=x,e,y,<}, CHAPTER 5. MONTE CARLO METHOD 119 a) SCATTEI PLOT: T « U EMMX VI TMETA OX 5.0OO0: 4.8000: 4.6000: 4.4000: 4.2000: 4.0000: 3.8000: 3.6000: 1.4000: 3.2000: 3.0000: 2.8000: 2.6000: 2.4000: 2.2000: 2.0000: 1.8000: 1.6000: 1.4000: 1.2000: 1.0000: 0.8000: 0.6000: 0.4000: 0.2000: 0.0000: •0.2000: •0.4000: -0.6000: 11112 2 11 212 11111 2 212 1 1 1 1 1 2 1 22311S23112 11 2 2 1 1 1 11 2 1 1 11 21311 23 1 3 141 1 311 1 131 1 2 1 11 12 •1.0000: •1.2000: -1.4000: -1.6000: -1.8000: •2.0000: •2.2000: -2.4000: •2.6000; •2.8000: -3.0000: -3.2000: -3.4000: •3.6000: •3.8000: -4.0000: •4.200C: •4.4000: •4.6000: •4.8C I I I t I I I I 1 I •85.0000 -68.0000 -31.0000 -34.0000 -17.0000 0.0000 17.0000 34.0000 51.0000 68.0000 85.0000 LARGEST UN 3 1 CVEMTI IN tANGE 1M IVENTS CUT or 1AHCE: SAMPLE 0WAI1ANCE -21.88678 b) fCATTtt PLOT; TKTA DdtOB VS WtTA OX 3.0000: 4.8000 4.6000 4.4000 4.2000 4.0000 3.8000 3.6000 3.4000 3.2OO0 3.0000 2.8000 2.6000 2.4000 2.2000 2.0000 1.8000 1.6000 1.4000 1.2000 1.0000 0.8000 0.6000 0.4000 0.2000 0.0000 •0.2000 -0.4000 •0.6000 •0.8000: •1.0000: •1.2000: -1.4000: -1.6000: -1.8000: -2.0000 -2.2000 -2.4000 -2.6000 •2.8000 -3.0000: •3.2000: -3.4000: •3.6000: -3.8000: -4.0000: •4.2000. -4.4000 •4.6O00 •4.8000 1 1 2 1 1 1 1 13 1 1 1 1 11 ' 1 1 111 15 21113 3 3131 5131 1 2121 111 1 2 1 1 3 111 1111 1211 112 2 2 1 1 21 1 132 1 1311 3 133 21 1122 2 -85.0000 -68.0000 -31.0000 -14.0000 -17.0000 0.0000 17.0000 14.0000 31.0000 68.0000 85.0000 LUCE ST BIN S 8CALIM0 1.0000 EVENTS IN 8ANCE 198 EVENTS OUT Of 8ANGE: LEFT 0 SIGHT SAMPLE COVUIAkTE 0 480VE -0.63589 0 BELOW MEAN SO [ -4.5989 40.5070 0.1229 0.3982 Figure 5.1: 9lTTOr = eRAYTRACE - By**"* plotted against Bx. T h e momentum of the rays are within 0.1% of P0. a) The original R A Y T R A C E transfer matrices, b) After additional terms have been added to reduce the errors. CHAPTER 5. MONTE CARLO METHOD 120 R M S E r r o r o f S A S P M a t r i c e s 6 <7e i(cm) CTe^mrad) ^e v(cm) (mrad) -10% 0.0840 0.7851 0.1293 0.6277 -5% 0.0496 0.4260 0.1098 0.2537 0% 0.0316 0.3982 0.0901 0.2328 +5% 0.0405 0.5472 0.0905 0.3647 +10% 0.0681 0.8241 0.1308 1.0840 +15% 0.1209 1.4579 0.1896 0.6769 Table 5.1: T h e Standard deviations of the errors in the fits of the S A S P matrices to the R A Y T R A C E trajectories where n varied from one to seven. There were also some two and three variable terms such as (x/9*62) (y/<f>96) Some dependencies are left but are increasingly difficult to eliminate. T h e final fit for 9 is shown in figure 5.1b; the standard deviation of the error in 9 was reduced from 0.9mrad to 0.4mrad — a factor of 2. These matrices are only good for rays in the range \8\ <85mrad, \<j>\ <43mrad, and —10% < 8 < +10%. Outside this range the high order terms blow up. A measure of the quality of the matrices is the standard deviation of the errors shown in table 5.1. T h e position errors in table 5.1 are at the focal surface positions for each ray bundle. These position errors will increase at increased distances from the focal surface because of errors in the angles. A 1 mrad angle error will produce a 1 m m shift in the ray at a distance of 1 metre. For purposes of calculating aperture cuts and detector positions, this magnitude of error is insignificant. For momentum calculations and CHAPTER 5. MONTE CARLO METHOD 121 track reconstruction, it is difficult to predict the effect the errors. The spot sizes of matrix bundles are approximately the same size as the R A Y T R A C E bundles. Thus the matrices get similar momentum resolutions to R A Y T R A C E . However, the error between the matrix and R A Y T R A C E rays is also about the same size as the spot size. Thus, while the spot sizes are similar, the matrix rays are basically uncorrelated to the R A Y T R A C E rays. Th is will effect the software corrections to the rays. Regardless of whether the matrices produce a better or worse resolution than R A Y T R A C E after software corrections, the matrices will require different corrections. 5.2.2 Aperture Cuts A n ideal simulation of the S A S P would use a routine like R A Y T R A C E to trace each particle through the system examining its position at each step to see if it collided with a surface. Such a simulation, although slow, would give an accurate estimate of the spectrometer's solid angle and acceptance. As a shortcut, E A S Y only examines the position of the particle at nine key apertures in the system. This leads to an overestimate of the solid angle if a particle is travelling in a curved orbit that would strike a surface between apertures. To ascertain the important apertures in the S A S P , the focussing properties of the optical elements and the geometry of the vacuum vessels must be considered. A ray approaching the S A S P must first travel the 70cm from the target to Q l . T h e first aperture it encounters is the entrance to the quadrupole, the aperture of which is assumed to be a circle of diameter 20cm. Inside Q l , the ray is focussed in the bend plane and defocused in the nonbend plane. Leaving Q l , it travels the through the vacuum pipe connecting Q l to Q2. Inside Q2, the ray is defocused in the bend plane and focussed in the nonbend plane. The net effect of the two quadrupoles is a defocusing in the bend plane and a focussing in the nonbend plane. T h e ray's entrance CHAPTER 5. MONTE CARLO METHOD 122 to the dipole is a slit varying in width from about 9 to 11cm. Without focussing, part of the ray bundle would be blocked by the dipole's pole piece. Wi th in the dipole, the rays are bent approximately 90° towards the vertical and in the process are momentum dispersed and focussed in the bend plane. Leaving the dipole the rays pass through the exit vacuum box and emerge from the vacuum system at the focal surface. The dipole does not focus in the nonbend plane. The ray bundle comes to a waist in the nonbend plane at or after the middle of the dipole depending on the ray momentum. The becomes about 30cm wide when it reaches the focal surface. T h e trajectories of a ray bundle transported through the S A S P by R A Y T R A C E are shown in figure 5.2. Figure 5.2a shows the bend plane of the spectrometer while figure 5.2b shows the nonbend plane. In the nonbend plane, the rays are projected onto a curved plane which follows the central trajectory through the spectrometer. T h e nine apertures used in E A S Y ' s simulation of the S A S P are shown in figure 5.3. They are in order from the target: the entrance and exit of each quadrupole; the constriction of the entrance vacuum box; the entrance of the dipole; the middle of the dipole; the exit of the dipole; and the constriction of the exit vacuum box. T h e method used to calculate the position of the ray depends on the aperture. For several of the apertures, R A Y T R A C E was used to generate transfer matrices from the target similar to those used for the complete S A S P optics. For other apertures (such as aperture #5, the entrance vacuum box constriction) the ray is drifted from the previous aperture. These locations were chosen to find the limiting apertures of the spectrometer as is discussed in section 6.2. Only a few of the apertures are l imiting and the balance can be removed to speed up calculations. The sizes and shapes of the apertures used by E A S Y are given in table 5.2. The meaning of the parameters given are explained in [19]. Figure 5.2: A 8 = 0% ray bundle traced through the S A S P by R A Y T R A C E . a) Bend plane. Shows the quadrupoles, and the entrance and exit of the dipole. b) Nonbend plane. T h e rays are projected onto a curved surface that follows the central trajectory. CHAPTER 5. MONTE CARLO METHOD 124 Figure 5.3: The S A S P Apertures used by E A S Y CHAPTER 5. MONTE CARLO METHOD 125 S A S P A p e r t u r e s A p e r t u r e L o c a t i o n T y p e P a r a m e t e r s #1 Entrance Q l quad d 14.0 d2 2.0 W 8.0 R 14.0 x0 17.0 y 0 17-0 #2 Exit Q l quad Same as #1 #3 Entrance Q2 quad Same as #1 #4 Exit Q2 quad Same as #1 #5 Entrance Vacuum Box Constriction trap xmin -45.0 xmax 38.5 slope 8.844 x 1 0 - 3 intercept 5.376 #6 Entrance of Dipole trap xmin -47.5 xmax 47.5 slope -7.6123248 x 10~ 3 intercept 5.0 #7 Middle of Dipole trap xmin -20.0 xmax 100.0 slope 29.4117 x l 0 ~ 3 intercept 5.0 #s Exit of Dipole trap xmin -65.0 xmax 88.0 slope 7.6123248 x 10~ 3 intercept 5.0 #9 Exit Vacuum Box Constriction trap xmin -67.0 xmax 82.5 slope 2.308 x 1 0 - 3 intercept 5.1993 Table 5.2: The S A S P Aperture Cuts Figure 5.4: S A S P Quadrupole Beam Pipe 5.2.2.1 Q u a d r u p o l e s A quadrupole focussing element has four pole pieces (spaced regularly about the op-tical axis) which run the length of the magnet and are shaped to produce the desired field. A sketch of a cross section of a quadrupole is shown in figure 5.4. While in a pure quadrupole magnet, the pole faces would have hyperbolic curvatures, the SASP "quadrupoles" have various higher order field terms which are created by slight distor-tions of the pole faces. As with a dipole, the magnetic field tends to bulge out from the ends of a quadrupole, increasing its effective length. The quadrupoles were originally designed to hold a standard eight inch diameter ( 10.16cm radius) circular vacuum pipe. Subsequent studies with the Monte Carlo showed that the solid angle of the spectrometer could be increased if a "cruciform" shaped vacuum pipe was used (figure 5.4). A cruciform vacuum pipe increases the solid angle but many of the added rays are of poor quality. Only the region around the optical axis has a good quality field. The field between the poles is distorted by the coils and the limited extent of pole faces. However, the additional solid angle achieved CHAPTER 5. MONTE CARLO METHOD 127 by the cruciform vacuum pipe will be useful for work where resolution is less important than data rates. For high resolution work, the unwanted rays can be "vetoed" by active collimators or ray tracing from the focal surface back through the spectrometer. The entrance and exit of both Q l and Q2 were used as apertures for the S A S P using the cruciform vacuum pipe. Apertures are placed at the effective edges of the quadrupoles. T h e effective edges were used because transfer matrices to these locations could be readily generated by R A Y T R A C E while transfer matrices to the pole edges would be more difficult. This geometry should produce slightly low solid angles. Straight line trajectories in the drift spaces imply that the acceptance of the entrance apertures can be accurately determined. T h e exits of the quadrupoles differ because the trajectories of the rays change. If a ray is diverging when it reaches the aperture, it will be removed, even though the ray would have hit a surface before the aperture. If the ray is converging an error can occur. In a quadrupole, the ray follows a curved trajectory. In the focussing plane, a ray initially diverging will reach a maximum displacement from the optical axis and then turn back. It is possible for the trajectory to pass through the surface of the vacuum vessel, and still clear the exit aperture. Thus rays that should be stopped will pass through the magnet. For Q l , which focuses in the bend plane, aperture #2 provides a satisfactory estimate of the aperture in the nonbend plane and a not so good estimate in the bend plane. T h e opposite is true for the exit of Q2 (aperture #4). The size of the errors will be small and not likely to effect the solid angle calculations. 5.2.2.2 D i p o l e Three apertures are used to describe the dipole: the entrance (#6), the middle (#7), and the exit (#8). Aperture #7 in the middle of the dipole has simple trapezoidal cross section following the pole faces of the dipole. The width of the aperture in the CHAPTER 5. MONTE CARLO METHOD 128 bend plane is set at the approximate extent of the "good field" region of the magnet. T h e entrance and exit apertures (#6 and #8) approximate the curvature of the dipole pole edges. T h e ray is transported to the z = 0 plane of the local aperture coordinate systems (figure 3.1). T h e distance from the z = 0 plane to the pole edge parallel to the z axis is calculated and the ray drifted this distance. The ray is then compared to a trapezoidal aperture which follows the projection of the pole faces into the local coordinate system. T h e widths of the apertures are set at approximately the true widths of the dipole apertures. This method is only exact if the ray is travelling parallel to the 2 axis. If the ray is at an angle, then it will not be drifted back to the exact crossing of the aperture. Th is proceedure was taken to avoid the time consuming calculation of the intersection of two arbitrary lines and should not have a significant effect on the solid angle calculation. Lastly, the pole edge of the magnet is used for the aperture rather than the point where the bevel cuts the pole face. Th is will produce a slightly underestimated value for the solid angle. Rays entering the dipole are diverging in the bend plane and converging in the nonbend plane. T h e dipole has little effect on the rays in the nonbend plane where they continue to converge. Low momentum rays reach a waist or focal point in the centre of the dipole after which they diverge. As the momentum increases, the focal point shifts downstream out of the dipole. Thus for low momentum rays, the bundle is diverging at the dipole exit while for high momentum rays, the bundle is converging. Th is property makes the dipole apertures reasonably accurate in the nonbend plane. High momentum rays leave the dipole still converging, any not blocked by the entrance will pass through the rest of the dipole. Low momentum rays hitting the pole face of the magnet will be blocked by the middle or exit apertures. In the bend plane, rays of all momenta are focussed and leave the dipole con-verging. Rays can pass the entrance aperture, balloon out inside the dipole and then CHAPTER 5. MONTE CARLO METHOD 129 converge passing the exit aperture. It is possible for extreme rays to leave the region of good field in the dipole without being detected. The middle dipole aperture (#7) lessens the chance of this by sampling the ray bundle at approximately its widest point. 5.2.2.3 V a c u u m B o x e s The S A S P dipole vacuum boxes are described in sections 4.7 and appendix F . Referring to figure 4.15, the vacuum box must extend outward from the pole edge beyond the coil after which it can flare upwards away from the ray bundle. T h e plate of the vacuum box is parallel to the dipole pole face but spaced further from the dipole's median plane. Because of the curvature of the dipole apertures, the vacuum box must extend some distance from the coil before flaring. The relation between the vacuum boxes, Q2, and the dipole is shown in figure 5.5. Ideally, the vacuum boxes should be far enough away from the median plane that they do not interfere with the ray bundle and form limiting apertures for the spectrometer. Since the coil and vacuum box thicknesses are fixed, raising the vacuum box up necessitates adding more steel to the dipole pole pieces and return yokes to maintain the same gap between the pole faces. A 1cm increase in the width of the box would require approximately an additional 800 kg of steel. This would also increase the minimum angle the S A S P could make with the beamline and the M R S . T h e dimensions settled on are a compromise that result in the vacuum boxes blocking a small part of the high momentum ray bundles. 5.2.3 EASY's sasp_optics Device The complete S A S P optics (global matrices and aperture cuts) are simulated in E A S Y with the device s a s p - o p t i c s . This device takes a ray at the target plane of 4 B T 2 and transports it to the exit of the S A S P dipole. If the ray is blocked at any of the nine apertures in the spectrometer, it is lost. The exit coordinate system of sasp_opt ics has CHAPTER 5. MONTE CARLO METHOD 130 Figure 5.5: T h e relationship between Q2, the dipole, and the vacuum boxes. The way the figure is drawn, rays appear to originate inside the Q2 vacuum pipe material. Because the rays are focussing in the nonbend plane, it is possible for rays that should be blocked by the quadrupole pole tips to pass unnoticed through the quadrupole. Similarly, rays could scatter off the pole tip without detection. T h e study of such scattering effects would best be studied with R E V M O C or perhaps a combination of R E V M O C and E A S Y . CHAPTER 5. MONTE CARLO METHOD 131 z pointing vertically up and x in the direction of increasing dipole radius (see section 8.1). Whi le E A S Y ' s sasp_opt ics device reasonably models the R A Y T R A C E predicted optics, because of the saturation effects of the S A S P dipole, the optics on the real spectrometer will change. Therefore, E A S Y ' s model will behave less and less like the S A S P as it is run at higher central momenta. 5.3 Two Body Kinematics A two body nuclear reaction is when two particles collide, interact, and two particles (not necessarily the incident ones) are emitted. m i + m 2 ~~*  m3 + m4 A n example is the reaction p(p, ir+)d where two protons collide and a pion and deuteron are emitted. T h e reaction is said to be isotropic if the reaction products are emitted with no preferred direction in the Centre of Mass ( C M ) Frame. T h e centre of mass frame is defined as the frame in which the total momentum of the incident and emitted particles is zero. Most reactions are not isotropic, rather the products have preferred directions determined by the reaction. This differs from the kinematics which specify the possible trajectories and energies of the reaction products and is dependent on the trajectories and energies of the incident particles. For example, the kinematics of the p(d,7r +)d reaction with an incident proton energy of 500MeV, constrain the trajectory of the product deuteron to within a few degrees of the beam line. Most reactions are not isotropic, rather the reaction products have preferred directions determined by the reaction. Apart from being the simplest reaction, a useful property of a two body reaction is that once the trajectory of one of the reaction products is determined, the momen-CHAPTER 5. MONTE CARLO METHOD 132 turn and trajectory of both are fixed. To calculate the two body kinematics, consider two incident particles with known masses, energies, and momentums: (mi,Ei,Pi) and ( m 2 , ^ 2 , ^ ) - Also known are the masses of the reaction products: 1713 and m 4 . First calculate the Lorentz transformation that takes one from the Lab Frame to the Centre of Mass frame. If the total momentum in the Lab Frame is along the z axis, then the transformation is described by the two quantities 7 * and /?*. p; = P , P; = Py P; = Y(P2-P*E) E* = 7 * ( £ - / T P , ) where the * denotes a C M value. Next the total energy available to the reaction in the C M frame, W, is calculated. The total momentum of the reaction products in the C M frame is zero, giving the relations, E; + E* = w P; + P; = 6 Thus the reaction products travel in opposite directions in the C M frame. The method for calculating P 3 and P4* is given in appendix H . l . For an isotropic reaction, a prod-uct can be emitted in any direction in the C M frame. T h e Monte Carlo simulates the reaction by randomly selecting a trajectory for one of the products. Th is fixes the tra-jectory of the other. In practice, only particles that hit the detectors are of interest. A CHAPTER 5. MONTE CARLO METHOD 133 truly isotropic reaction would spray particles about and most would miss the detectors. This is avoided by scattering one of the reaction products into a cone centred on, and fully covering, the aperture of one of the detectors. This way, most of the particles generated at the target enter the detector, with only a few around the edges of the aperture missing. For example, in a (p, 7r +) reaction, where the pion is to be detected in the S A S P spectrometer, a cone totally covering the aperture of the S A S P in the centre of mass frame is specified. A l l the pions will be scattered isotropically into this cone. If the cone is chosen so that it is only slightly larger that the S A S P , most of the particles will enter the spectrometer and the simulation will be efficient. B y scattering the reaction product into a cone rather than a sphere, the symmetry of the reaction is destroyed. As discussed earlier, to make the reaction isotropic, a weighting factor must be carried with each reaction product that states the probability of it being scattered into the cone. T h e weighting factor is given by, where is the solid angle of the cone in steradians. Thus if the cone is a full sphere, the weighting factor reduces to unity. This weighting factor is carried with the particle through the remainder of the simulation and is used in solid angle calculations and in plotting. T o point the reaction product at the detector, it is necessary to calculate the angle it lies at in the C M frame. This angle is different than the lab frame angle due to Lorentz contraction. The method of calculating the C M detector angle is described in appendix H.2. Once the C M trajectories of the reaction products are determined, the products are shifted to the lab frame, CHAPTER 5. MONTE CARLO METHOD 134 detector • \ Figure 5.6: E A S Y ' s two body reaction. The scattered particle is pointed towards a T h e reaction products are then handed to the rest of the simulation to be transported through the remainder of the system. T h e reaction kinematics and the aiming of the products at a detector is handled in E A S Y by the devices target and s w i t c h y a r d . T h e incident beam interacts with a target particle some where in a rectangular target block described by ta rge t . T h e target routines calculate the kinematics and aim the reaction products at one of the detectors described in the s w i t c h y a r d device (figure 5.6). Currently, E A S Y can only handle isotropic two body kinematics but the routines are written to make modification relatively simple. For example, to make the reaction anisotropic, an angle dependent weighting factor is added to the reaction product. This would be multiplied with the scattering cone's weighting factor to produce the final weighting factor for the particle. detector. P = P* PZ = 7*(P; + /3*E*) E = J-(E- + P*P;) Chapter 6 Monte Carlo Studies of the SASP Five basic Monte Carlo studies were completed for the S A S P : the solid angle of the spectrometer as a function of momentum, the total acceptance of rays from the target plane, the limiting apertures of the spectrometer system, the ray profiles downstream of the dipole, and mapping the focal surface of the spectrometer. T h e E A S Y problem files used to generate the solid angle studies are listed in appendix I. A l l the solid angle studies shown used the spectrometer's limiting apertures. Only blockage of the rays by the detectors and by vacuum vessels mentioned in section 5.2 were considered. T h e S A S P ' s solid angle studies used several target spots. The target spot refers to the area in the target plane from which particles originate. For "high resolution" solid angles, a 2cm by 1cm target spot was used (the larger dimension is always x, the smaller y). For limiting apertures and the downstream ray bundles, a 10cm by 4cm target spot was used. The total acceptance of the S A S P was checked with a 40cm by 20cm grid. The first two target spots had rays generated randomly over their areas while the third had rays generated at discrete points. A l l studies utilize the same method of pointing rays at the spectrometer. A ray generated at the target would be pointed randomly into a cone centred on the middle of Q l ' s aperture. The cone was sufficiently large to fully illuminate the aperture. 135 CHAPTER 6. MONTE CARLO STUDIES OF THE SASP 136 6.1 High Resolution Solid Angle The solid angle of a detector is a measure of the rate at which it detects particles. Suppose an imaginary sphere surrounds a point source radiating in all directions. The solid angle of a detector is the size of the cone whose base is the effective aperture of the device. If the detector accepted all the particles radiating from the source, it would have a solid angle of 47T steradians. A realistic solid angle for a spectrometer is on the order of a few millisteradians. The larger the solid angle of the detector, the more particles it accepts and the higher the data rates. T h e effective aperture of a spectrometer is the cumulative effect of all the aper-tures the particles pass through. This can include the detectors if they do not accept all the particles that pass through the optics. Because there are dispersive elements in a spectrometer, the effective aperture and solid angle will be a function of momen-tum, usually having a peak near the central momentum. T h e momentum acceptance of a spectrometer is called it's momentum bite and is usually measured as a percent-age of the central momentum. The S A S P was designed to have a momentum bite of (-10%,+15%). A Monte Carlo simulation process is used to calculate the solid angle of a device by randomly sending rays into a cone that fully covers the device's entrance aperture. The solid angle is then approximated by, ^device = where N is the number of particles sent into a cone of size A f i and n» the number that get through. Th is is the Monte Carlo integration discussed in chapter 5. T h e error in CHAPTER 6. MONTE CARLO STUDIES OF THE SASP 137 the calculated solid angle (from chapter 5) is: N(N-l) where / ; = 1 if the particle makes it through the system and /,• = 0 if it does not. Thus, 1  N n J iV ~ N 1=1 and N t=i since ff is also either one or zero. The error is then, A f i iNn. - nl e n = ^ T V N-l and for N >^ 1 and N >^ n» this reduces to, en = ^ A f > Typically, for N = 100,000 and n . = 10,000, the error is about 1%. Rather than calculating the solid angle of the S A S P for a point source, a distributed source that approximated the target spot size was used. This produces an averaged solid angle for that spot size and is a more reasonable measure of the data rates expected for the spectrometer. T h e spot size (based on a realistic beam spot at 4BT2) was 2cm vertical (x) and 1cm horizontal (y). This small size spot is called a "high resolution" spot because the aberrations that affect the momentum resolution of the spectrometer CHAPTER 6. MONTE CARLO STUDIES OF THE SASP 138 S A S P S o l i d A n g l e s 6 &SASP (msr ) i O . l m s r C r u c i f o r m C i r c u l a r -10% 13.149 10.155 -5% 14.535 11.611 0% 15.404 12.507 +5% 14.744 11.894 + 10% 13.938 11.112 +15% 13.210 10.372 Table 6.1: Solid Angles for S A S P calculated by E A S Y for the high resolution target spot using the cruciform and circular quadrupole vacuum pipes. increase with the target size. Larger target spots can be used where lower resolutions are acceptable. Rays in the study were generated uniformly over the target spot. Th is is not realistic because most beam profiles are Gaussian with most of the particles at the centre of the spot. If the acceptance of the spectrometer drops off radially from the optical axis (and it does) a uniform spot size will under estimate the true solid angle. Using the apertures described in section 5.2.2 with the high resolution spot size and 100,000 rays, the solid angles in table 6.1 were calculated. The E A S Y problem files utilized in this calculation are provided in appendix I. Table 6.1 contains solid angles for both the cruciform and circular vacuum pipes. For high resolution studies, the data for the circular vacuum pipe are more realistic since the cruciform pipe permits rays to pass through the poor field region of quadrupoles. 6.2 Limiting Apertures T h e limiting apertures of the spectrometer determine if and where slit scattering may occur as well as determining the maximum solid angle of the system. Slit scattering occurs when particles graze a surface with a small change in momentum and trajectory. CHAPTER 6. MONTE CARLO STUDIES OF THE SASP 139 # 2 # 3 # 4 # 5 # 6 # 7 # 8 # 9 6 X y x y X y X y X y X y X y X y -10% +- +- +- +- +- +- +- + +- +--5% +- +- +- +- +- +- +- + +- +-0% +- +- +- +- +- +- +- + +- +-+5% +- +- +- +- +- +- +- + +- +-+10% +- +- +- +- +- +- +- + +- +- +-+15% +- +- +- +- +- +- +- + +- +- + +- + +-Table 6.2: Apertures hit by rays. The + indicates the top was hit and a - the bottom. A +- means that both top and bottom were hit. The apertures are shown in figure 5.3. Looking downstream, +x is vertically down and +y horizontally to the left. Such grazing degrades the momentum resolution of the spectrometer and the ability to trace the ray back to the target plane. Where sources of slit scattering are identified, passive and active collimators can be inserted to prevent it. A passive collimator is a thick aperture designed to prevent grazing. For high energy particles, such a collimator has to be quite thick. A n active collimator is a scintillator that defines the aperture. When a particle passes through the scintillator, the unit fires and the event is vetoed by the data acquisition electronics. Too many particles hitting the active collimator saturates it and good events will be vetoed. While E A S Y is not designed to study the effect of particles grazing a surface (a program such as R E V M O C [20] would have to be used), it can show the limiting apertures of the system and thus expose areas that may require further study. T h e limiting apertures in the S A S P were determined using the large 10cm by 4cm target spot although the high resolution spot (2cm by 1cm) produces similar results. Table 6.2 shows the apertures that rays hit. A (+) indicates that the top of the aperture was hit a (-) the bottom, and a (+-) both top and bottom. Aperture #1 is not included in the table since it is intentionally fully illuminated. A l l the apertures of the quadrupoles are illuminated and the constrictions of both vacuum boxes are CHAPTER 6. MONTE CARLO STUDIES OF THE SASP 140 6 x m i n x m a x y m i n y m a x (cm) (cm) (cm) (cm) -10% -12.19 12.45 -2.54 2.55 -5% -11.94 12.45 -2.88 2.90 0% -11.76 12.40 -3.44 3.45 +5% -11.41 11.87 -3.81 3.79 +10% -11.32 11.70 -3.93 3.99 +15% -10.99 11.27 -3.73 3.72 Table 6.3: Effective Apertures of S A S P . Rays that do not pass through the approxi-mately rectangular region (dimensions indicated in the this table) at the entrance to Q I will not pass through the spectrometer. illuminated in the nonbend high x side of the dipole entrance. Although (#6) is illuminated, the number of rays that hit this aperture is insignificant. Essentially the limiting aperture in x is the exit of Q2 (#4). Similarly the hits in apertures 7, 8, and 9 for the +15% rays are also very small. What is being observed is the inability of the dipole to deflect the extreme high momentum rays. A s the momentum increases beyond +15%, these apertures dominate. In the nonbend plane the limiting aperture is the constriction of the exit vacuum box for all momenta. Figure 5.5 is a schematic representation of the vacuum system in the nonbend plane. T h e vacuum boxes intrude slightly on the ray bundle that would otherwise pass the dipole. The effect is on the order of a few percent and is unfortunate since ideally the dipole pole edges should form the aperture. These vacuum boxes are a potential source of slit scattering and active collimators will probably be installed at the entrance constriction. Scattering from the exit constriction can be detected by projecting a ray back from the focal surface detectors to the aperture making a collimator unnecessary. T h e effective apertures at the entrance to Q l for the different momenta are pro-vided in table 6.3. The effective aperture tends to narrow in the x direction and widen in the y direction with increasing momentum. The aperture widens in the y direction CHAPTER 6. MONTE CARLO STUDIES OF THE SASP 141 because at higher momenta, the rays are deflected less in the y plane and fewer cross over the median plane and hit aperture #9, the exit vacuum box constriction. The aperture shrinks in the x direction because Q l does not focus the rays as much and more hit Q2. 6.3 Target Plane Acceptance T h e target -plane acceptance describes how the solid angle of the spectrometer is a function of target plane position. For most experiments, which have a small beam spot on the target, the acceptance is not important. There are some experiments, however, which have large target spots and need a large acceptance. Two classes of these are (p, n) and (n,p) experiments. A (p,n) experiment hits the target with a proton beam and detects the emitted neutrons. A n uncharged neutron cannot be detected directly by the spectrometer so it is converted into a proton in a hydrocarbon cell. T h e proton knocked out of the conversion cell by the neutron is detected in the spectrometer. A (n,p) experiment first converts the incident proton beam to a neutron beam with a conversion cell. T h e neutron beam hits the target and a proton is given off which is detected by the spectrometer. During either of these processes, the initially small beam will spread out becoming quite large. The larger the spectrometer acceptance, the greater the data rates for these experiments. Maps of the S A S P ' s acceptance for different momenta are shown in figure 6.1 as contour plots (a two dimensional histogram of the 0% acceptance is shown in figure 6.2). T h e maps were created by generating 2000 rays at each point on a 2cm square grid of size 40cm (x) by 20cm (y) centred on the origin of the target plane. T h e solid angle at each point on the grid is shown on the contour plots as a function of the target plane position. Irregularities are caused by the poor statistics gathered from such small ray bundles. For all momenta, the S A S P acceptance is approximately a rectangle with size CHAPTER 6. MONTE CARLO STUDIES OF THE SASP 142 Figure 6.1: Target plane acceptance contour plots for -10%, -5%, 0%, +5%, +10%, and +15%. T h e target plane is located with its normal pointing along the spectrometer's optical axis. Positive x points vertically downward and positive y horizontally to the left (when looking downstream); the origin is at the optical axis. T h e irregularities are due to the low statistics used in the calculation. Figure 6.2: Target plane acceptance histogram for 0%. ± 1 6 c m in x and ± 6 c m in y. At low momentum, the acceptance is peaked at the origin (the optical axis) and falls off uniformly as distances increase. A t high momentum, the acceptance becomes natter and narrower (in y) with a small residual step around the edge. Changes in acceptance can be explained by the apertures in the system and how focussing varies with momentum. The acceptance narrows with increasing momentum because the higher momentum rays are less focussed in the nonbend plane by the quadrupoles and tend to hit the dipole entrance pole edge. In the x direction, the limiting aperture for all momenta is the exit of Q2. Th is aperture does not change significantly with momentum and the x acceptance of the S A S P remains approximately constant. T h e acceptances shown here can be misleading. They were generated assuming only the S A S P with the apertures described in section 6 were blocking the rays. As well, nothing is said about the quality of these rays. The S A S P was optimized for rays from a small target spot and the matrices used to simulate the S A S P were fitted to CHAPTER 6. MONTE CARLO STUDIES OF THE SASP 144 this region. Rays from the edges of the acceptance will be of poor quality. 6.4 Downstream Ray Profiles Design of the vacuum boxes and detectors of the spectrometer requires knowledge of the exact profiles of the ray bundle to be detected after the S A S P . Th is is accomplished by drifting the ray bundles "downstream" (vertically upward in the laboratory) from the exit of the spectrometer. Figure 6.3 shows the -15%, -10%, +15% and +20% ray bundles drifted four metres downstream from the dipole exit in the bend and nonbend planes. Also shown on the diagram are the dipole exit and the focal surface (see section 6.5). These calculations used the large target spot (10cm by 4cm). The S A S P design specifies a momentum range of -10% to +15% but figure 6.3 shows that if the detectors are made just large enough to accommodate these ray bundles, a significant number of rays outside this range will pass through. T h e detec-tors will act as limiting apertures, reducing the effective solid angle for these extreme momenta. 6.5 Mapping the Focal Surface For an ideal spectrometer the focal surface is the surface on which rays of constant momentum focus to a line (see figure 6.4a). One coordinate of the surface would have a one to one mapping with the momentum of the rays while the other coordinate would relate information about the ray's initial trajectory. Idealy, the focal surface should be a plane, then the momentum of the ray can be read off as the output of a wire chamber positioned on the plane. More likely, the focal surface will be curved but provided the rays still come to a sharp focus on the surface, the momentum can be determined simply by ascertaining the intersection of the ray with the surface. It is not perfectly monochromatic rays that come to a focus. Consider a flat focal CHAPTER 6. MONTE CARLO STUDIES OF THE SASP 145 y (cm) Figure 6.3: Downstream Ray Profiles after the dipole exit for the large target spot, a) Bend Plane, b) Nonbend Plane. The bundles shown are: -15%(long dash), -10% (solid), +15% (dot-dash), and +20% (short dash). The focal plane is shown in the bend plane figure. Note in a) that the waists of the ray bundles lie above the focal surface. Th is is because the focal surface was calcuated using the smaller high resolution spot size which produces ray bundles that focus closer to the dipole. CHAPTER 6. MONTE CARLO STUDIES OF THE SASP 146 Figure 6.4: a) A n ideal focal surface. One coordinate of the surface maps directly into the momentum of the ray while the other depends on the initial trajectory before the spectrometer in some way. b) A true spectrometer will not bring the rays to a sharp focus, instead it will be blurred. surface (plane) with the x coordinate (call it x2) directly proportional to the momentum of the ray. T h e n from appendix B , equation B.8, x 2 = M x j +D(6S + A8B) where M is the magnification of the system, D, the dispersion, x i the target plane position, 8s the momentum of the particle being measured in the spectrometer, 8B the momentum of the beam particle, and, PodP, PB and Po are the central momentums of the beam and scattered particles respectively while Pi and P 3 are the absolute momentums of the beam and scattered particles respectively. Suppose that to map the focal plane, the beam is injected directly into CHAPTER 6. MONTE CARLO STUDIES OF THE SASP 147 the spectrometer so PB dP3 = 1 aft and A = 1. Thus , x2 = Mxi + D(6S + SB) Now suppose the beam was perfectly monochromatic (6B = 0). One still wants x2 to be proportional to £5. But there is still the x2 dependence on xx due to the M term. This is resolved by having a dispersed beam meeting the condition, 0 = Mxx + D8B That is, once the x position in the target plane is chosen, an additional term, is added to the original momentum, 8$ that is being measured. In an actual spectrometer, monochromatic rays will not come to a line focus, rather, they will be smeared out (figure 6.4b) making the focal point difficult to define. For purposes of mapping the S A S P ' s focal surface, the focal point of a ray bundle is defined as the location where the bundle has a "waist" in x. A n additional complication in mapping the focal surface is that the nonmomentum coordinate of the surface (y) is not uniquely determined by one component of the initial ray. That is, the first order x2 = Mxi + D6S CHAPTER 6. MONTE CARLO STUDIES OF THE SASP 148 equation for y is, V2 = R33Vi + Rzi<f>\ so y is a function of both y\ and <f>\. To map the y extent of the focal surface, y\ and <f>i are incremented in discrete steps. Figure 6.5 shows the method for mapping the focal surface. Keeping yi fixed at zero (since y is usually small), (j> is shifted across the spectrometer aperture in several steps for each momentum. For the S A S P , D = -0.59892 and M = 2.81323 giving a g = - 4 . 6 9 7 c m / % . Figure 6.6a shows a plot of the positions of the waists of each ray bundle sent through the S A S P in the (x, y) plane. Figure 6.6b shows a perspective view of the focal surface generated by fitting a two dimensional polynomial to the points on the surface. T h e ripples are an artifact of the fitting routine. The edge of the focal surface is defined to be where the solid angle falls to approximately one third of the maximum. The method used here to define the focal surface differs from that used by R A Y T R A C E . For a given momentum, R A Y T R A C E defines the focal point as the intersection of two rays injected into the spectrometer: one pointed along the optical axis and the other pointed slightly off the axis. A comparison of these methods is shown in figure 6.7. The solid line shows a cross section of the Monte Carlo defined focal surface in the median plane of spectrometer (y = 0). The different focal surfaces produced by the two methods are caused partly by the different ways of defining the surface and partly by the inaccuracies in the model of the optics used by E A S Y . Whi le basically flat in the nonbend (y) plane, the curvature of the S A S P ' s focal surface in the bend plane presents a problem. Ideally, the exit window of the dipole vacuum box would be laid directly on the focal surface. This minimizes the effect of multiple scattering in the exit window on track reconstruction since regardless of the change of ray's trajectory by the window, it will still be traced back to the point where it crosses the focal surface. It is probably not possible to lay a thin exit window along the CHAPTER 6. MONTE CARLO STUDIES OF THE SASP 149 Figure 6.5: Mapping the focal surface. A fan of rays with a contant momentum and that varies continuously in x and 0 (vertical dimensions) but constrained in y and $ is "injected" into the spectrometer simulation. The ray bundle is focussed to a waist by the spectrometer optics and the location of this waist is defined as a point on the focal surface. B y repeating this process for many different values of y and <f>, a map of the focal surface is built up. CHAPTER 6. MONTE CARLO STUDIES OF THE SASP 150 a) T 1 1 1 1 1 1 1 1 1 r - 6 0 - 5 0 - 4 0 - 3 0 - 2 0 - 1 0 0 10 20 30 40 50 60 x(cm) b) Figure 6.6: S A S P Focal Surface, a) A projection of the focal surface onto the (x, y) plane (horizontal in the lab with x in the spectrometer's bend plane). Each point is the focal point of a ray bundle (500 rays) with a different 6s and <j>i. 6s was stepped across the focal surface (x direction) in 2% steps from -20% to +24%. For each value of 6s 19 different values of (j>\ were used, b) A perspective view of the focal surface. The ripples are an artifact of the algorithm used to fit a polynomial surface to the points in (a). CHAPTER 6. MONTE CARLO STUDIES OF THE SASP 151 250 200-50-U I 1 r— -150 -100 -50 0 x(cm) 50 100 150 Figure 6.7: The R A Y T R A C E focal surface (points) compared with the focal surface defined here (line). The R A Y T R A C E points are (from left to right): -10%, -5%, 0%, +5%, +10%, +15%. curved focal surface since air pressure would cause it to buckle and kink. A n acceptable compromise is struck by laying the window tangent to the focal surface at the point where the central ray crosses it. The exact position of this will be determined by field mapping of the completed spectrometer rather than with the Monte Carlo simulation. Chapter 7 SASP Detectors T h e S A S P design is optimized for a momentum bite of (-10%,+15%) but these are not the extreme momenta that can pass through the system; higher and lower momentum rays will get through but with a reduced solid angle. After the dipole exit, these rays spread out over a larger range than the design (-10%,+15%). To detect all rays leaving the spectrometer would require an enormous detector array and costs would scale roughly linearly with size. The S A S P ' s detector system will capture the rays within the designed momentum bite. Outside this range, the detector array will become one of the spectrometer's limiting apertures. Figure 7.1 shows the proposed S A S P detector array in relation to the focal surface and the dipole exit. The detector stack consists of three Vertical Drift Chambers ( V D C s ) set at an angle of 46° to the horizontal followed by a six segment scintillator hodoscope parallel to the V D C s acting as the main trigger for the stack. Mounted further downstream is a horizontal scintillator followed by either a Cerenkov counter or second horizontal scintillator. T h e detectors and electronics are described more fully in [21]. Not shown in figure 7.1 is the S A S P ' s Front E n d Chamber ( F E C ) located between the target and the first quadrupole. T h e three identical V D C s are stacked tightly against each other and are set right against the exit window of the vacuum system. The exact exit window position will be determined after the finished spectrometer has its field mapped and the position of the 152 CHAPTER 7. SASP DETECTORS 153 ^PROTON HALL ROOF ~ 5 . l m FROM CLAM SHELL EXIT TOP SCINTILLATOR OR CERENKOV Figure 7.1: T h e S A S P detector array shown with respect to the dipole exit, the focal surface, and the Proton Hall roof CHAPTER 7. SASP DETECTORS 154 Figure 7.2: V D C wire planes. The open circles represent sense wires while the small solid points are guard wires. focal surface determined. T h e intent is to have the exit window of the vacuum tank lie on the focal surface. This will minimize the effect of multiple scattering in the window on tracing the rays back to the focal surface. In practice it is difficult to curve the exit window to conform to the curvature of the S A S P ' s focal surface (figure 7.1) so a flat window will be laid tangent to the intersection of the central ray with the focal surface. 7.1 Vertical Drift Chambers A Vertical Drift Chamber [22] is a type of wire chamber that detects both the position and angle of a particle track. Set at approximately 45° to the nominal particle track, the V D C wire plane is composed of sense and guard wires sandwiched between cathode planes (figure 7.2) producing drift cells having uniform fields except in the immediate vicinity of the sense wires. When a particle passes through three adjacent cells, it leaves a trail of electrons that drift to the sense wires at a constant velocity (because of the CHAPTER 7. SASP DETECTORS 155 uniform fields). The position and angle of the particle track through the wire plane can then be reconstructed from the drift times. Because the particle must pass through at least three cells, there is a minimum track angle to the normal that the V D C can accept. - l w c'm.n = tan — where w is the width of a cell and h is the height. The V D C cell geometry and tilt angle (OVDC) are selected so that all the desired rays can be detected, for the S A S P , the minimum angle was determined with the Monte Carlo simulation. The worst case is for -10% rays, and 0 = 28.38°. The V D C s were thus designed for a minimum angle of 28° and the chamber tilt (0VDC) increased from the nominal 45° to 46° This gives an error margin of approximately a degree. The S A S P V D C s are almost identical in design to the ones currently employed on the M R S spectrometer [23]. T h e differences being that the S A S P V D C s are larger to cover the S A S P ' s bigger focal plane and the drift cells are taller to allow a smaller 0min. 7.1.1 V D C Construction E a c h of the S A S P ' s three V D C s has two wire planes. The x plane has wires running in the y direction and gives a position in the chamber's x direction. The u plane wires are tilted at an angle of 30° to the x plane wires. Using the u and x coordinates the y coordinate of the ray can be reconstructed. Figure 7.3 shows a cross section of a V D C . It consists of a mylar gas window followed by an aluminized mylar cathode plane, the x wire plane, a cathode plane, the u wire plane, a cathode plane, and another gas window. T h e windows and cathode planes are made out of 0.001" (0.00254cm) mylar. Between each plane is a 0.625" (1.5875cm) gas layer. The gas is under a slight pressure, causing the windows to bulge out approximately 0.5" (1.27cm) making the outer gas layers 1.125" (2.8575cm). The CHAPTER 7. SASP DETECTORS 156 j window cathode i u plane cathode * • x plane cathode window 1 1 Figure 7.3: Cross section of V D C ' s . gas mixture used is 50% isobutane (C4HW) 50% argon. The position resolution of the V D C ' s is 0.15mm and the angular resolution is 0.5mrad. The minimum detectable angle to the normal is 28°. 7.1.2 V D C Modelling Each layer of the V D C is modelled in E A S Y as a drift region with an appropriate radiation length which determines the extent of multiple scattering of the ray. The simulation only scatters the ray and does not consider energy loss. T h e wire planes are modelled with the v d c option of E A S Y ' s pos_sensor device. Th is option outputs both a position and angle for the ray passing through the wire plane. If the angle of the ray to the normal through the V D C is less than the 0 m t n parameter, the pos_sensor considers it a bad event and does not fire. A n actual wire chamber would output wire numbers and timing information from which the position and angle of the track is reconstructed. T h e pos_sensor device skips all this and directly outputs a position CHAPTER 7. SASP DETECTORS 157 in cm and an angle in mrad. The position and angle reported by E A S Y will not be the exact values for the track. Rather, there will be random errors following normal distributions added to the detector outputs. T h e sizes of the position and angle errors are given as parameters for the pos_sensor . The position errors can be "turned off" using the command @ p o s e r r o r . For the level of simulation that E A S Y is intended for, these outputs are. adequate. (It is possible that a more advanced simulation would produce signals closer to the true outputs of the detectors. These could then be fed into a simulation of the data acquisition electronics.) The ray is completely unaffected by the pos_sensor unlike a true wire plane where there is a small chance that the ray can hit the wire and be deflected. Finally, the user can specify the efficiency of the wire chamber which determines the probability that it will fire for each event. 7.2 Front End Chamber The S A S P has a Front E n d Chamber ( F E C ) between the target chamber and the first quadrupole. Having a F E C improves the trace back of the trajectory to the target and makes calibration of the system easier. However, an F E C degrades the momentum resolution of the spectrometer and can not take a high flux of particles. Th is severely limits the data rate for the system. Unlike the M R S , the S A S P can avoid this problem by retracting the F E C and the operating without it. The S A S P optics allow good operation with or without an F E C . 7.2.1 F E C Construction The S A S P F E C will be similar to the M R S F E C and consists of a sandwich of mylar, gas, and wire planes. A mylar window is followed by a gas layer followed by an alu-minized mylar cathode, another gas layer, an x wire plane, gas, a cathode, gas, an x' plane, gas, a cathode, gas, a y plane, gas, a cathode, gas, a y' plane, gas, a cathode, gas, CHAPTER 7. SASP DETECTORS 158 and an exit window. Each layer of gas is 0.5" (0.127cm) thick but the windows bulge outward under pressure adding an extra 0.5". The windows and cathodes are made out of 0.001" (0.00254cm) mylar. The gas used is isobutane at one third atmosphere. The position resolution of the wire planes is something less than 0.5mm. The chamber is inserted into the vacuum pipe between the target and the quadrupole (hence the windows bulge outward). The M R S chamber can handle data rates of some one to two million events per second before saturating. 7.2.2 F E C Modelling T h e F E C is modelled similarly to the V D C s , each region represented by a drift space with the appropriate radiation length. The wire planes are simulated with the p c option of pos_sensor which outputs the position, but not the angle of the ray. 7.3 Track Reconstruction The trajectory of a ray after the dipole can be reconstructed using the x and u positions from any pair of the three V D C s in the detector stack. Three V D C s are used for redundancy and to provide information on detector efficiencies. E A S Y has built into it a generic function called v d c 2 which will automatically calculate the trajectory of the ray at an arbitrary reference plane called the canonical plane. Th is calculation uses the outputs of the detectors and so includes the position errors when reconstructing the trajectory. T h e algorithm for v d c 2 is given in appendix J . Chapter 8 Spectrometer Models This chapter (along with appendices K and L) describes how the S A S P and M R S spec-trometers are modelled using E A S Y . For a full description of E A S Y and an explaination of the various constructs found in this chapter and the appendices see EASY, A Monte Carlo Simulator [19]. T h e models of the spectrometers presented are intended to be run as "black boxes", where details such as the exact dimensions and compositions of detectors are preset. The S A S P and M R S definitions are stored in files that are referenced by the simulation using the models. To update the models only one set of files must be changed rather than every E A S Y problem using them. T h e models of the S A S P and M R S are complex and consequently slow to execute. Many problems do not need as thorough a simulation of the spectrometers as provided here and would benifit from simplification. T h e S A S P and M R S models contain the different regions that the particles pass through, but not necessarily with the correct geometry. Whi le the model may describe the separation between wire chambers as, say, 5.7cm, the actual value may be different. The S A S P model is not truely representative because the spectrometer has not been built and various features of the design may change. For the M R S , several dimensions were estimated. F ina l measurements of the spectrometer will be made in the future. Whi le the models are known to be inaccurate, this does not make them useless. For 159 CHAPTER 8. SPECTROMETER MODELS 160 example, provided the geometry of the models are used, rays can still be traced back to the focal surfaces of the S A S P and M R S . Inaccurate geometry will to a small degree adversly affect results when studying the effects of multiple scattering and detector errors on ray reconstruction. 8.1 SASP Model E A S Y ' s S A S P model is based upon the device sasp_opt ics , described in section 5.2, which transports a particle from the target plane to the exit of the spectrometer. The other components of a complete S A S P model are the F E C and the focal plane detectors (also called the detector stack). The F E C is a small wire chamber located between the target chamber and the first quadrupole. The chamber contains four wire planes named xO, xOp, yO, and yOp ('p' is for 'prime'). The detector stack consists of three V D C s placed directly against the exit window of the dipole vacuum system which is located tangent to the focal surface. Adjacent to the wire chambers is a six segment set of "trigger paddle" scintillators and located horizontally above (downstream of the dipole) are two other scintillators, a Cerenkov counter is not used in the model. T h e V D C s are labelled in order of distance from the vacuum exit window, V D C l , V D C 2 , and V D C 3 . The V D C s have x and u wire planes called x l , u l , x2, u2, x3, and u3. E A S Y ' s S A S P model is contained in the two files S A S P _ D E F . D E V and S A S P - D E F . D A C which are listed in appendix K. As indicated in section 5.2, while s a s p . o p t i c s provides a reasonable approxima-tion of the S A S P ' s R A Y T R A C E optics design, it will differ significantly from the true S A S P optics when the spectrometer is run at high magnet settings. Due to saturation distortions of the dipole's magnetic field, the actual spectrometer will have a focal sur-face that shifts position with excitation. Thus E A S Y ' s model of the S A S P should be thought of as a reasonable approximation that is much better behaved than the real CHAPTER 8. SPECTROMETER MODELS 161 spectrometer. 8.1.1 SASP Device Definitions In E A S Y , complex pieces of equipment (called devices) can be built up out of sim-pler devices. Commencing with basic devices such as dr i f t , sc in t , and pos_sensor , eleborate detector arrays can be constructed. For example, the device sasp is defined as, d e f i n e sasp{ sasp_ fec s a s p _ o p t i c s sasp_s tack > This definition of the S A S P is placed in the user's device file T h e three devices used in the definition of the S A S P (sasp.fee, sasp .optics, and sasp.stack) are pre defined in the file S A S P J D E F . D E V (see appendix K for a listing), and are included in the user's problem description using the ^ i n c l u d e command. Of the three components of s a s p , only sasp_opt ics is a basic device built into E A S Y . The device s a s p - s t a c k describes the spectrometer system from the exit of the dipole through the vacuum win-dow, the three wire chambers, and the three scintillators. Each wire chamber is a device called s a s p . v d c made out of simpler devices; and the six segment trigger paddles are approximated as a single large scintillator. A series of complex rotations and transla-tions (using the "devices" ro ta te and t rans la te ) are used to change the coordinate systems to produce the correct orientation of the wire chambers and scintillators. Modell ing the front end chamber may seem difficult at first glance since it must come between the target and the spectrometer, and sasp_opt ics requires the particle start at the target plane for the transfer matrices to work. This problem is solved by first drifting the particle to and through the F E C where it undergoes various multiple CHAPTER 8. SPECTROMETER MODELS 162 scattering and then projecting the new trajectory back to the target plane using the device b a c k t r a c k . Because sasp_fec is modular, the simulation of the S A S P can also be run without a F E C by deleting it from the S A S P description. 8.1.2 SASP E D A C Definitions In E A S Y , data is gathered and processed in an experiment using E D A C ( E A S Y Data Acquis i t ion system) instructions in the problem's data acquisition file. E D A C in-structions are of two types, definitions and actions. Definitions can be of three types, addresses, booleans, and functions. Addresses tell E A S Y which locations in the system to gather data from. Booleans are logical expressions that determine if a specific com-bination of events happen in the system. For example, a boolean could be defined to be true only if all the scintillators in the S A S P detector stack fire. Functions process data from the system after each particle has passed through it. For example, a function can be defined to take the data from two V D C s in the detector stack and calculate the trajectory of the ray at the focal plane. Actions are used to display and store data collected from the experiment; there are four basic types. The actions hist and scat will histogram and scatter plot data from any point in the system. The location to gather data can be specified with either a device path or with an address. The address is really just a short hand for naming points in the system to collect data from. T h e action tape records data to a file where it can be examined by different programs after the simulation is completed. T h e last type of action is called a conditional action. This is like the if then else construct found in most programing languages. It checks to see if a condition is true and if so, processes the specified actions. The condition can be made up of booleans defined prior to the conditional action or of references to points in the system. Like the address, the boolean is a short hand for long expressions. Conditional actions allow the user CHAPTER 8. SPECTROMETER MODELS 163 F E C : sasp_xO sasp_xOp sasp.yO sasp_yOp V D C #1: sasp-xl sasp.ul V D C #2: sasp_x2 sasp_u2 V D C #3: sasp_x3 sasp_u3 T P : sasp_tp SI: sasp_sl S2: sasp_s2 Table 8.1: S A S P Detector Addresses in E A S Y to set up simple logic in his data acquisition system. For example, he can instruct E D A C to only histogram the x readout of V D C 1 if both the F E C and the downstream scintillators fire. Th is way the user can study the efficiencies and performances of compound systems such as the S A S P detector stack. To simplify the user's data acquisition files, some commonly used addresses, booleans, and functions have been defined in the file S A S P - D E F . D A C (listed in appendix K ) . B y including this file in his dac file (using the # i n c l u d e command), the user can use these definitions. There are ten position sensors and three scintillators that signals can be taken from in the S A S P system (although there is nothing stopping you from taking data from anywhere else as well). The front end chamber has four wire planes; two x and two y. Each of the three V D C ' s has an x and a u plane. The three scintillators consist of the trigger paddles (TP ) next to the V D C ' s and the two downstream scintillators S i and S2. Table 8.1 lists the addresses of the various detec-tors in the S A S P system. To create a scatter plot of the x readout in V D C 1 versus the u readout in V D C 3 the following statement would be placed in the dac file. CHAPTER 8. SPECTROMETER MODELS 164 sasp_vdcl -fired sasp_vdc2_fired sasp_vdc3_fired sasp_fec_fired sasp_tp_fired sasp-scints_fired sasp.vdcs-fired sasp_trigger Table 8.2: S A S P detector booleans in E A S Y . scat sa sp_x l / s min=-100.0 max=100.0 : sasp_u3/s min= -50.0 max= 50.0 ; The s following an address tell's E D A C the location is a sensor; the ranges of the scatter plot axes are specified with the min and max statements. Because the detectors are not one hundred percent efficient, a particle traversing the S A S P ' s detector array will not necessarily trigger all the detectors. In a real experiment, the acquisition electronics would be wired to look for certain combinations of detectors firing before processing the data. This can be simulated in E A S Y using booleans and logical expressions. Using conditional actions in E D A C , an experimenter can insist on any desired combination of detectors firing before processing the data. Listed in table 8.2 are the standard boolean expressions defined in S A S P - D E F . D A C . The booleans s a s p _ v d c l _ f i r e d . . . are true if both the x and the u planes of each V D C fire. T h e boolean sasp_fec_fired is true if at least one x plane and one y plane in the F E C fire. The boolean sasp_t r igger is used if the processing is to wait for all the detectors to fire; it is true if all the V D C s , the F E C , and all three scintillators have fired. CHAPTER 8. SPECTROMETER MODELS 165 <sasp_vdcl2(0) 9c <sasp_vdcl2(l) Vc <sasp_vdcl2(2) 4>C <sasp_vdcl2(3) Table 8.3: Reconstructing the S A S P canonical ray using the generic function, v d c 2 . T h e components of the canonical ray, vc are calculated by v d c 2 and made available to the user as outputs 0 — 3 of the function. 8.1.3 SASP Track Reconstruction As discussed in section 7.3, E A S Y has a built in generic function called v d c 2 which takes wire plane readouts from two vertical drift chambers and reconstructs the tra-jectory of a ray at a canonical plane. The S A S P dac definition file, S A S P J D E F . D A C has defined two functions using v d c 2 . The function s a s p _ v d c l 2 takes the outputs of V D C 1 and V D C 2 and reconstructs the trajectory of the canonical ray, r c at the horizontal plane at the exit of the S A S P dipole (zc points vertically upwards along the optical axis). Similarly the function s a s p _ v d c l 3 reconstructs the canonical ray at the same plane using the outputs of V D C l and V D C 3 . Table 8.3 describes the outputs of s a s p _ v d c l 2 ; the outputs of s a s p _ v d c l 3 are similar. 8.2 MRS Model E A S Y ' s M R S model is basically the same as the S A S P model except that the spec-trometer's geometry is different. The M R S is a Quadrupole - Dipole (QD) spectrometer bending the rays nominally 60° upwards from the horizontal into a detector array lo-cated approximately 4.2m from the dipole exit. The M R S is shown in figure 1.1. The M R S ' s focal surface is approximately a plane oriented at 45° to the the central ray. T h e focal plane detectors consist of two V D C s and a set of trigger paddles located im-mediately after the vacuum box window which is situated at and parallel to the focal CHAPTER 8. SPECTROMETER MODELS 166 surface. Downstream of the V D C s and triggger paddles can be either a focal plane polarimeter or a pair of scintillators. Like the S A S P , the M R S has a front end chamber located between the target and the quadrupole, although unlike the S A S P , the M R S cannot run without the device. T h e M R S optics are modelled in E A S Y with the device, m r s _ o p t i c s and is simu-lated with first and second order transfer matrices generated by the program T R A N S -P O R T (the M R S ' s optics are simpler than the S A S P ' s so second order transfer matrices are adequate). There are five apertures checked in the M R S optics: the entrance and exit of the quadrupole, the entrance and exit of the dipole, and the exit window of the vacuum box at the focal plane. The M R S ' s F E C is the same design as the S A S P ' s but the V D C s differ from the S A S P V D C s . Specifically, the layouts of the chambers are the same but the M R S V D C s have narrower gas spaces than the S A S P , are oriented at 45° to the optical axis, and are much smaller than the S A S P V D C s (100cm by 35cm compared to the S A S P ' s 200cm by 40cm). The E A S Y model of the M R S is structured much like that of the S A S P . For sim-plicity, the M R S model uses two downstream scintillators rather than the focal plane polarimeter. T h e M R S device definitions are in the file M R S _ D E F . D E V while the E D A C definitions are in M R S - D E F . D A C (both files are listed in appendix L) . The M R S detector addresses and detector booleans are listed in tables 8.4 and 8.5 respec-tively. T h e function m r s _ v d c l 2 calculates the canonical ray at a plane perpendicular to the optical axis at the exit of the M R S dipole (zc is oriented at 60° to the horizontal). The outputs of m r s _ v d c l 2 are listed in table 8.6. CHAPTER 8. SPECTROMETER MODELS 167 F E C : mrs_xO mrs_xOp mrs.yO mrs_yOp V D C #1: mrs_xl mrs_ul V D C #2: mrs_x2 mrs_u2 T P : mrs_tp SI: mrs_sl S2: mrs_s2 Table 8.4: M R S Detector Addresses in E A S Y mrs_vdcl-fired mrs_vdc2Jired mrs_fec_fired mrs-tp-fired mrs_scints_fired mrs-vdcsJired mrs.trigger Table 8.5: M R S detector booleans in E A S Y . <mrs_vdcl2(0) 9c <mrs_vdcl2(l) Vc <mrs_vdcl2(2) <f>c <mrs_vdcl2(3) Table 8.6: Reconstructing the M R S canonical ray using the generic function, v d c 2 . Part III Summary and Appendices 168 Chapter 9 Summary P a r t I o f th is thes is investigated various aspects of the magnetostatic design of the S A S P clamshell dipole. Using the magnetostatic code, P O I S S O N , the excitation behaviour of the dipole's internal field was studied and its effects on the spectrometer performance determined. The dipole's aperture fringe fields were characterized and a specification for the aperture steel shape and field clamps that produces the fringing fields specified in the optical design was presented. The accuracy and applicability of P O I S S O N to the S A S P dipole, was assessed by limited studies of the L A M P F Low Energy P ion Spectrometer. T h e wedge shaped air gap of a clamshell dipole produces a magnetic field that varies as the inverse of the distance from the apex of the wedge and as consequence the dipole does not enter saturation uniformly over the pole piece causing the field to sag on the high field side of the dipole at high excitations. This phenomena produces a 0.979% distortion of the S A S P dipole's field. While its primary effect on the spectrometer's behaviour is to shift the focal surface, the changing field profile necessitates recalibration of the spectrometer for each momentum setting. The dipole apertures are used to correct for aberrations in the spectrometer op-tics with virtual field boundary shapes described by fifth and sixth order polynomials. Although the S A S P ' s optical design allows for the effects of fringe field in the dipole's 169 CHAPTER 9. SUMMARY 170 apertures, it did not address the steel geometries required to produce the required V F B and fringe field shapes. Characteristics of the fringe fields were studied by modelling cross sections of the dipole apertures were modelled with the two dimensional magne-tostatic program P O I S S O N . T o produce the desired field shapes, a device called a field clamp was employed to remove excess flux from the aperture and limit the extent of the fringe field. Aided by a Rogowski pole edge bevel, the field clamp also prevents the aperture fringe fields from changing shape with magnet excitation. T h e P O I S S O N sim-ulations were used to fit the dipole steel and field clamp geometries to the R A Y T R A C E field specifications. The curves formed by the dipole pole edge and the leading edge of the field clamp (called the Pole Edge Boundary and Field Clamp Boundary) were fitted with polynomials similar to those used to describe the virtual field boundaries. These curves will be used to layout the dipole pole pieces and field clamps for machining. The fringe fields produced by the geometry described by the P E B and F C B do not produce V F B s identical to the ones specified in the S A S P optics design. T h e aperture geometry specified in this thesis produces a 10% loss in the spectrometer's resolution. Before dipole construction commences further investigation to reduce the resolution loss due to the apertures is required. Studies of the dipole utilized a two dimensional magnet simulation program to model a three dimensional object. Simulating cross sections through the dipole cannot capture such three dimensional behaviour in the clamshell as the distribution of flux in the return yokes or the way flux is distributed in the concave dipole apertures. POIS-SON's performance was assesed by studying the clamshell dipole used for L A M P F ' s Low Energy Pion Spectrometer. The L E P S dipole differed from the S A S P and had limited maps of the magnetic field so quantitative comparison with P O I S S O N was dif-ficult. T h e L E P S data showed that P O I S S O N reproduces the general behaviour of a clamshell's fringe fields, but did not indicate whether P O I S S O N is quantitatively pre-CHAPTER 9. SUMMARY 171 cise. P O I S S O N had a tendency to produce erratic results when small perturbations were made in the magnet geometry. P O I S S O N has been extended to its limit fitting the S A S P dipole aperture geometries. Study of subtle effects would probably produce meaningless results. Design of the S A S P field clamps must allow for alterations to permit the fringe fields to be fine tuned. A procedure for using maps of the dipole field must be developed to facilitate modification of the clamps. In P a r t II o f th is thes is a Monte Carlo simulation of the Second A r m Spec-trometer was developed. The simulation program, called E A S Y (Equipment and Ac-quisition SYstem simulator), was used to study the extent and profiles of rays transiting the spectrometer to help facilitate positioning vacuum boxes and detectors. After the commissioning of the spectrometer, E A S Y will be utilized to assess experimental ar-rangements of both the S A S P and the D A S S (Dual A r m Spectrometer System). Monte Carlo studies performed in this work included: solid angle of the spectrometer as a function of momentum, limiting apertures in the spectrometer system, target plane acceptance, downstream ray profiles, and mapping of the focal surface. Rays passing through the spectrometer are alternately focussed and defocused by the quadrupoles. The ray bundle leaves Q 2 diverge in the dipole bend plane and converge in the nonbend plane. After passing through the dipole, the rays are deflected and focussed in the bend plane and come to a focus at the spectrometer's focal surface. In the nonbend plane, the rays — only marginally effected by the dipole — cross over the median plane and diverge after the dipole. T h e point of cross over (which can be thought of as a focus in the nonbend plane) varies with momentum. The high momentum rays are deflected least and cross the median plane near the dipole exit while the low momentum rays cross near the entrance. T h e limiting aperture of the spectrometer in the bend plane is the exit of Q 2 which restricts the amount of divergence the ray bundle has after leaving the quadrupole. In the nonbend plane, the CHAPTER 9. SUMMARY 172 constriction of the dipole's entrance vacuum box (where it projects under the dipole coil) is the limiting aperture for the high momentum rays while the exit vacuum box constricts low momentum rays. Ideally, the limiting apertures in the nonbend plane should be the entrance and exit of the dipole but space constrictions have forced the vacuum boxes to protrude slightly into the ray bundle. The solid angle caused by the vacuum boxes is only a few percent smaller than what it would be if the dipole pole piece formed the limiting apertures. T h e solid angle of the spectrometer could be increased 20% by switching from the conventional circular vacuum pipe through the quadrupoles to a cruciform shaped pipe which follows the quadrupole's pole tips. The S A S P ' s solid angle using the cruciform vacuum box varies from 15.4 msr at the central momentum to about 13 msr at the extreme design momentums (8 = —10% and 6 = +15%). These numbers should be taken with caution since some of the rays traversing the edges of the cruciform vacuum box will pass through regions of poor magnetic field which leads to a degradation in the spectrometer's resolution. A more realistic solid angle for "high resolution" rays is 12.5 msr at the central momentum and about 10 msr at the extreme momenta. T h e target plane acceptance of the spectrometer is the region in the target plane from which rays will pass through the spectrometer. For the S A S P , this is a rectangle approximately 8cm horizontally and 16cm vertically, peaked at the centre and falling to zero at the edges. In this enormous spot, the centre is approximately flat and accommodates reasonably well the largest spot size (4cm by 10cm) likely to be used in the system. Spectrometer optics were optimized for a smaller spot size so the S A S P will not achieve its full resolution using such a spot. Whi le the M R S has a flat focal surface (plane), the S A S P ' s focal surface is akin to a paraboliod. Th is makes the process of momentum calculation for the S A S P more complicated than for the M R S . As well, it is not possible for the spectrometer's vacuum CHAPTER 9. SUMMARY 173 window to follow the focal surface (desirable to minimize the effects of multiple scat-tering on the spectrometer resolution). The focal surface encompasses a much larger momentum range than the S A S P optics were designed for — approximately -20% to +25%. Rays leaving the dipole exit illuminate much of the region above the spectrom-eter. It is not practical to construct a detector array that detects all these rays, so the array is designed to accept only the full ray bundle from the design momentum bite. Even this requires wire chambers 2m long and 40cm wide. E A S Y ' s model of the S A S P was based on first and second order transport matrices produced by R A Y T R A C E . These matrices proved inaccurate and many higher order terms were added. The final matrix model behaves similarly but not identically to the R A Y T R A C E optical design. A monochromatic ray bundle transported through E A S Y ' s S A S P model would focus to a position on the model's focal surface with a spot size similar to that of the R A Y T R A C E model. Beam spots on the focal surfaces of the E A S Y model and the R A Y T R A C E model look similar, but are only loosely correlated. A ray passing through the left side of the R A Y T R A C E focal surface spot could pass through the centre or right side of the E A S Y spot. The software corrections used to improve the resolution of the spectrometer will be different for the two cases. E A S Y will be able to predict the qualitative behaviour of the S A S P , but a procedure to optimize the momentum resolution developed with E A S Y can not be transferred directly to the real spectrometer — although it may indicate the method to be used. Because the S A S P dipole's properties change with excitation, the E A S Y model of the S A S P optics will only match the real optics at low magnet settings. The last chapter of this thesis presents complete E A S Y models of S A S P and M R S , simulating all the detectors, drift regions, vacuum windows, and such items found on the spectrometers. These models are maintained in special files which can be accessed by any simulation of the spectrometers. Neither model is numerically accurate, meaning CHAPTER 9. SUMMARY 174 that while all the objects found in the spectrometer systems are described in the models, the orientation of these objects with respect to one another is not correct. In the case of the M R S , this is due to an incomplete knowledge of the detector positions and will be corrected in the near future. For the S A S P , the detector arrays do not exist yet and are subject to change. Whi le these models are not exact replicas of the spectrometers, they are self consistent. For example, the S A S P model can still be used to calculate the reconstruction of a ray's trajectory after the spectrometer. The discrepancies in the models would become important when considering the effects of multiple scattering in windows and such. The S A S P and M R S models will be maintained and updated to make them as accurate as possible. While they are thorough, these models are also very slow to execute. They are intended to be prototypes for anyone running simulations of the D A S S / S A S P system. A user completely unfamiliar with the spectrometer system can use these models as a black box from which data is collected. As the user becomes more concerned with the details of the spectrometers, he can look into the particulars of the simulation. A n added advantage of keeping the prototypes of the spectrometer models is that any changes to the system only have to be made once rather than in every E A S Y simulation that uses the models. Appendix A Resolving Missing Mass Consider an experiment where the mass of a "recoil" particle is to be measured, mB + rriT —> mR + m i + m 2 + . . . where m# and my are the beam and target particle of known mass and mR is the recoil particle. Then scattered products m j , m 2 etc. are particles of known mass which have their momentums measured in various "detector arms" of the experimental apparatus. This appendix determines how the calculated resolution of the recoil (or "missing") mass depends on the measured momentums m i , m 2 , etc. Where the target particle is stationary, the momentum of the recoil particle can be calculated by, PR = PB-PS (A . l ) and the energy, Eft = EB + ET — Es (A.2) where P$ and Es are the total momentum and energy of all the scattered particles (mi , m 2 , etc.). T h e missing mass can then be calculated from, mR = J E R - P R ( A . 3 ) where PR = \PR\. T h e error in a missing mass, AMR is given by, (A.4) Often, the PR is small compared to the mass of the recoil particle and AmR reduces to 175 APPENDIX A. RESOLVING MISSING MASS 176 (since ER « mjj), A m R = A E R (A.5) where The target particle is at rest with a known mass so A £ r = 0 while the error in the beam energy is due to its momentum spread. T h e error in E$ is given by, Suppose only the momentum of each scattered particle is measured by the detectors. Then , since the masses of the scattered particles are known exactly, the error in the energy of each particle is, A * . = (A.6) where, A P , Pi is the fractional error in the momentum. This is the usual way of rating the resolution of a spectrometer. Then , In a single arm experiment (only one scattered particle, mi) and where the error in the beam momentum is negligible, the missing mass resolution is given by, &mR = ^fPl (A.8) while for a two body experiment the resolution is given by, A m * — ^ Consider the reaction, A(p, 2p)B and assume that the scattered protons each have approximately the same momentum, P (and hence the same energy, E). Then , A m H is APPENDIX A. RESOLVING MISSING MASS 177 given by, = -jsjfh+fh ( A . 1 0 ) where / p t and fp2 are the momentum resolutions of the two detector arms. Suppose one detector is a magnetic spectrometer with a resolution of fp = 1 0 - 4 ; the other a counter telescope with a resolution of fp = 10 3 . Then the missing mass resolution would be, The resolution of the system is completely dominated by the poorer detector. Basically the same holds true if the scattered particles have different masses and momentums although some improvement can be made by using the detector with the higher res-olution to measure the larger momentum. In conclusion, to achieve a high resolution measurement of the missing mass in a two arm experiment, it is necessary for both arms to have high resolution detectors. Appendix B Dispersion Matching In a particle beam - magnetic spectrometer system the momentum resolution of the spectrometer is limited by the spectrometer optics, the spot size of the particle beam on the target, and the energy spread of the particle beam. B y use of Dispersion Matching, the effects of the spot size and incident beam energy spread can be reduced or eliminated. Following the derivation of Blomqvist [24], consider the two body reaction, m i + m.2 —• ms + m 4 ( B . l ) where mj is a beam particle incident on the target particle m 2 with the reaction product m3 being detected in the spectrometer while m 4 is the undetected recoil particle. The incident (beam) particle, mx travels with a momentum P i which deviates by a small amount A P i from the central momentum of the beam, PB- The scattered and recoil particles have momentums P3 and P4 respectively while the target particle, m 2 is at rest. For convenience, all these momenta are assumed to be positive while the small deviation of P i from PB can be either positive or negative. The momentum of the detected particle, P 3 will depend on the kinematics of the reaction including the momentum of the beam particle, P\. P 3 = P 3 ( P i ) (B .2) Since P i deviates from PB by only a small amount, can be equation B.2 rewritten Taylor expansion, P3 = P S + g f A P , + 5 ! ^ ( A P 1 ) 2 + ... (B.3) where Ps is the momentum of the scattered particle if the incident beam was monochro-178 APPENDLX B. DISPERSION MATCHING 179 matic. Whi le P3 is the momentum of the particle that enters the spectrometer, it is Ps that one ultimately wishes to measure. Now expressing P 3 in terms of the central momentum of the spectrometer, P0, 6 = P z Z P ° x 100% Po For a monochromatic beam, Ss =  P s ~ P ° x 100% Po and the relative momentum of the incident beam, AP 6B = — - x 100% PB Equation B . 3 can now be written as, 6 = 6S + A8B + B6B + ... ( B . 4 ) where, , PBdP3 A — and, Blomqvist evaluates A as, P o d P i 2 Po 8P? PB^ZM±^11 P°Ps ( l + S)-^cosc? where, (B .5) and 6 is the angle between the incident and scattered particles (P x and P3). EB/ES is the energy of the incident/scattered particle for the monochromatic beam. Blomqvist derives a similar expression for B. APPENDIX B. DISPERSION MATCHING 180 To first order, the equation describing the focussing in the x direction is: x2 = RuXl + i2i 20i + RieS ( B . 6 ) Assuming a point to point focus (R\2 = 0) and calling Ru the Magnification of the system, M, and R\6 the Dispersion, D, x2 = Mxx + DS ( B . 7 ) Where X i is the position of the particle in the target plane and 8 is given by equation B.4. Ignoring the second and higher order terms in 83, x2 = Mx\ + D(8s + A8g) (B.8) In order to dispersion match the beam, the X i and 8B terms are made to cancel each other out, 0 = M x x + DA8B or substituting for A, X1 = _ ^ P (B.9)  1 MPodPi  K ' Thus by making x x a function of 8B, the first order focussing of the spectrometer is a simple function of 8s, the quantity to be measured, x 2 = D8S Since the dependence on the unknown quantities X i and 8B has been eliminated, the resolution of the spectrometer is improved. This calculation assumes that the spectrometer has no aberrations — i.e. there are no terms higher than first order for x2. In practice this is not true and if these terms or if the second and higher order terms in 8B in equation B.4 can not be ignored, the dispersion matching condition (equation B.9) becomes increasingly complicated and harder to achieve. The advantage of the dispersion matching method is that it eliminates the effects of the target spot size and the beam dispersion simultaneously. The disadvantage of the method, is that it can produce large extended spots on the target requiring large targets and spectrometers with large vertical acceptances. Appendix C Magnetostatic Field Modelling To design the field clamps and study excitation effects on the S A S P and L E P S , their magnetostatic fields had to be modelled. Ideally this should be done in three dimen-sions since both the S A S P and L E P S are decidedly 3D objects. Unfortunately the 3D program now in use, G F U N [25], is not suited for studying the fringe field regions on a magnet. A more likely candidate, T O S C A [26], has only recently become available to T R I U M F and is not ready for use. Both G F U N and T O S C A are limited in the number of points used to describe an object to be modelled, which is a problem with a complex geometry such as the S A S P ' s . So instead of modelling the dipoles in 3D, cross sections of the magnets were modelled in two dimensions. The computer code used to model these cross sections was P O I S S O N — a program in common use at T R I U M F . C . l Solving Magnetostatic Field Equations In two dimensions, the magnetic vector potential has only a component out of the plane of the problem. It can be shown[27] that solving for the vector potential in this case reduces to solving the two dimensional Poisson's equation, VV(x,y) = p(x,y) (C . l ) where <^(x,y) is some scalar function proportional to the vector potential and p(x, y) is the local current density. In two dimensions, the current J is either into or out of the plane. The Poisson equation can in turn be solved by a technique called point successive relaxation [28]. Consider a rectangular grid (figure C . l ) . Solving Poisson's equation over a con-181 APPENDIX C. MAGNETOSTATIC FIELD MODELLING 182 VT^r 0 Figure C . l : A rectangular grid over which the discrete Poisson's equation is solved tinuous region is equivalent to solving for every point on the grid, <p{i,J) = \-p(i,J) (C.2j That is at each point, <f> is equal to the average of its neighbours. If every point on the grid meets this condition, the region satisfies the discrete form of Poisson's equation. Note that as the grid size becomes arbitrarily small, equation C.2 converges to equation C . l . T h e input to a discrete Poisson problem is similar to the continuous case; regions at constant potential, <f>, are specified as points of fixed <f> on the grid. If a region has current, p(x,y), this is specified at each grid point. Boundary conditions must also be specified; the two types possible are Neumann and Dirichlet. A Neumann boundary condition is a constant potential on the boundary while the Dirichlet condition specifies that the first derivative of <f> is zero, on where n is the unit vector normal to the boundary. There are two main techniques for solving the discrete form of Poisson's equation for the region. Finite element analysis sets up a system of equations for all the points APPENDIX C. MAGNETOSTATIC FIELD MODELLING 183 on the grid and then solves them simultaneously using matrix algebra. If the equations are "well behaved" they can be solved exactly. If the equations are not, then an approximation to the correct solution can be made to any accuracy. T h e other method, called successive relaxation looks at a point on the grid, sets it to the average of its neighbours and then goes to the next point and repeats the process (using the new value of the last point in the calculation). If this is repeated over the entire grid many times, the values of (f> converge to the correct solution (indicated when values of <f> do not change from one iteration to the next). The more iterations used, the better the accuracy of the final value. Whi le the finite element method appears to be more rigorous than the relaxation method, under many conditions they require similar amounts of computing time. In addition, the successive relaxation method has the considerable advantage of being simple to implement. A slight modification to successive relaxation, is successive over-relaxation which speeds convergence by "over correcting" the potential at each point. First the difference between the old potential at the point and the average of its neighbours is calculated, A(f> = 4>1 + 4>2 + <f>3 + <t>4 . A 1" P told ( C . 3 ) where <f>\ ... are the four neighbours. In successive relaxation, A<j) would be added to <j>0id to produce (f>„ew. In over-relaxation, A4> is first multiplied by a constant greater than 1, <f>new = 4>old + rA(j) causing the potential to overshoot. As the values approach the solution, the relaxation parameter, r, is reduced to unity. This process can reduce the number of iterations required by a factor of two or better. A problem that does not appear with the continuous Poisson's problem is having the boundaries of the different regions conform to the grid. If a boundary is curved, the regular array of grid points can not follow it. The smooth curved boundary will be transformed into a "stepped" boundary that follows the grid. Th is changes the behaviour of the field and the effects of the boundary will not be observed. A solution is to distort the grid to conform to the boundary. It is difficult for a rectangular grid to be distorted to a curved boundary without introducing kinks and jumps. A triangular grid (or "mesh") does not have this problem and is more commonly used, the number of neighbours is changed from four to six. Because the grid is not regular, equation C.3 is no longer a simple average of the neighbouring points and weighting factors are APPENDIX C. MAGNETOSTATIC FIELD MODELLING 184 used to account for the relative distances of the neighbours, A<t> = Wi<j>i + W2(f>2 + W3(j>3 + W4<f>4 + W5<j)5 + W6<f>6 + PO — 4>old 6 C.2 POISSON the Program P O I S S O N is actually a group of programs designed to solve for the magnetic fields and forces in two dimensional arrangements of magnetic materials, conductors, and free space [29]. It does this by converting the continuous problem into a discrete problem on a triangular "logical mesh" and then solving for the magnetic vector potential on this mesh using successive over-relaxation [27]. P O I S S O N originated as T R I M in the late 1960's. T R I M required the user to manually translate the problem to the logical mesh, P O I S S O N now does that automatically. To solve a problem with P O I S S O N , the shapes of the various regions (steel, air, current) to be modelled are described in a file which a program called A U T O M E S H translates into the logical mesh. This mesh is an array of triangles contorted to follow the outlines of the different regions; the outlines of the problem's regions are assigned nodes on the mesh. Next a program called L A T T I C E distorts the logical mesh to con-form to the problem's geometry. Now the program P O I S S O N solves Poisson's equation over the mesh using successive over-relaxation. Boundary conditions at the edges of the modelling region (which is rectangular) have to be specified. T h e boundary con-ditions can be either Neumann (B = Bnh) or Dirichlet (Bn = 0). Dirichlet conditions do not allow the lines of magnetic flux to pass through the boundary while Neumann conditions cause the flux lines be perpendicular to the boundary. In P O I S S O N , the boundary conditions can be set by the user but the default values are for the top and sides of the "world box" (the modelling region) to be Dirichlet and the bottom Neu-mann. F lux can only enter or leave the box through the bottom boundary. T h e other boundaries confine the flux. This has the effect of mirroring the magnet geometry about the bottom of the box. Thus for symmetric problems, only half the geometry need be described. These boundary conditions make it difficult to model the extended fringe field of a magnet. The Dirichlet and Neumann conditions force all the magnetic flux in the system to be contained within the world box Most magnets have leakage, that is, the flux does not stay within the magnetic material or the air gaps. Instead it moves through free space outside the magnet. When the magnet is modelled with P O I S S O N , APPENDIX C. MAGNETOSTATIC FIELD MODELLING 185 a finite sized box must be used with all the flux contained within this region. Since B is flux per unit area, this has the effect of increasing the local B field. Thus , when modelling the fringe fields of a magnet, it is important to make the free space region outside the magnet large to minimize the effect on the field. This in turn requires a larger mesh for the problem and more computing time to find a solution. Figure C.2 shows an H magnet modelled with P O I S S O N . T h e default boundary conditions are used which has the effect of mirroring the magnet about the bottom of the box. Figure C.2c,d show the flux lines in the magnet and the B field along the x axis (centre of the magnet's air gap) calculated by P O I S S O N . C.3 Magnetic Materials Many of the regions modelled in a P O I S S O N problem are, of course, magnetic and respond nonlinearly to the applied magnetic field. This property is described with a material's B-H curve. P O I S S O N allows the B - H curve of the material being modelled to be specified as a table of values. P O I S S O N interpolates the points on the table to calculate the magnetic field produced in a material for a given excitation. W h e n modelling the S A S P and L E P S dipoles, several different steels with differing magnetic properties were examined. The B - H curves are shown in figure C.3. Apart from the effects of composition, the magnetic properties of the steels can also change quite dramatically due to different ways of production. APPENDIX C. MAGNETOSTATIC FIELD MODELLING 186 a) b) c) d) i.S 1.0 -. '.5 10 15 1.0 IS-. 1.0 i i n u m l n m n il i l l . • D 111II1111111U111 -13 16 | M1111111|| i n 1111111111111111111111111 jii 11111111 II 111 73 112 U l 160 183 PROS. NHT1E = H nRCNET "I I" 217 2«I 275 CYCLE -13 16 45 73 112 1 31 ICO 183 217 2«t 275 PROS. NfitlE = H nRCNET C T a t = 0 -13 16 «5 73 112 1 31 ISO 183 217 2«S 27S PROB. NfitlE = H tWCNET CYCLE = 240 14000 12000 10000 H 8000 6000 4000 2000 H 0 i _i_ 0 100 150 200 250 300 Figure C.2: A simple example of a H magnet modelled with P O I S S O N . (a) shows the original problem and (b) the logical mesh describing it. c) The flux line in the magnet, d) the B field (in Gauss) along the median plane of the air gap. APPENDIX C. MAGNETOSTATIC FIELD MODELLING 187 Figure C.3: Several B - H curves were used when modelling the SASP. Curve 2 is POIS-SON material #2 (1006 steel), curve 3 is material #3 (1010 steel) and curve 4 is material #4 (1020 steel). Appendix D The LEPS Study of the S A S P dipole with P O I S S O N had no precident. There was no estimate of the accuracy of the calculations. To facilitate understanding the modelling, the clamshell magnet used for the L A M P F Low Energy Pion Spectrometer ( L E P S ) was modelled and the P O I S S O N predicted results compared against actual field measure-ments. The L E P S consists of a vertical bend clamshell dipole magnet separated from the target by about 30cm. As the only magnet in the spectrometer, the dipole acts as both the bending and the focusing element. The dipole (figure D. l ) is a clamshell dipole with a single return yoke around the large gap (low field) side, this is quite different from the S A S P which has two return yokes. Other differences are that the L E P S is about half the size of the S A S P ( the S A S P dipole is about 110cm wide while the L E P S dipole is about 55cm), and has a steeper slope for the pole face (figure D . lb ) . Like the S A S P , both the entrance and exit apertures are concave but neither have field clamps. The central field at the 8.5cm gap is 17.647 k G compared with the S A S P ' s 16.073 k G at its 10.0cm gap. The R A Y T R A C E parameters for the L E P S ' s optics are given in table D . l (refer to figure 3.1). For comparision with the S A S P studies, two aspects of the L E P S design were investigated: the aperture fringe fields, and the saturation effects in the main body of the dipole. The pole edge curves of the L E P S do not follow the V F B curves used in its R A Y T R A C E design (table D . l ) , instead they are approximated by circles. T h e pole edges are beveled with a simple 60° cut as opposed to the Rogowski bevel the S A S P will have (see section 4.4). 188 APPENDIX D. THE LEPS 189 Figure D . l : L E P S dipole a) Layout showing the pole piece, coil return yoke and extreme rays, b) Cross section showing wedge shaped air gap. APPENDIX D. THE LEPS 190 P o l e P i e c e R 8500.0 cm <t> 0.2000° a -20 .0 ° -84 .0 ° I n te rna l F i e l d B F 1.7647 T N D X 100.0 F r i n g e F i e l d D 8.5 cm Entrance Exit c 0 0.3217 0.5391 Cl 1.6370 1,3376 c 2 -0.6408 -0.6185 c 3 0.3666 0.2542 c 4 0.0 0.0 c 5 0.0 0.0 V F B C u r v a t u r e Entrance Exit s2 -l.ooo x ro2 -3.000 x 101 5 3 0.0 -1.800 x 103 54 0.0 -3.300 x 105 s5 0.0 1.800 x 10s S6 2.250 x 1013 -1.900 x 109 s7 3.000 x 1016 -1.400 x 1012 Ss 1.000 x 1019 -1.700 x 1014 Table D . l : L E P S dipole R A Y T R A C E parameters. Compare with the S A S P parameters in table 3.1. APPENDIX D. THE LEPS 191 track x (in.) x (cm) 1 8 -4.1275 2 - 1 -2.5400 3 0 0 4 1 2.5400 5 2 5.0800 6 3 7.6200 Table D.2: L E P S track postions. x is shown in figure D.2. D . l Field Mapping The mapping of the L E P S ' s magnetic field was limited to a set of measurements taken at the entrance aperture (under different field settings and geometries) and an exci-tation curve taken with a retractable probe positioned at one point inside the dipole. This probe is normally used to set the dipole field during an experiment. It is unfor-tunate that the data are limited since a clamshell magnet is expected to behave quite differently from an ordinary dipole and a thorough mapping is required to get a proper understanding of its characteristics. The entrance aperture was mapped with a probe assembly that measured the field along the six tracks shown in figure D.2; the x pos-tions of the tracks are listed in table D.2. Six different combinations of configuration and excitation were mapped. 1. no clamp, mid excitation 2. no clamp, high excitation 3. clamp, no return yoke, mid excitation 4. clamp, no return yoke, 0.5 inches closer to dipole, mid excitation 5. clamp, no return yoke, 0.5 inches closer to dipole, high excitation 6. clamp, with return yoke, 0.5 inches closer to dipole, mid excitation T h e terms used (based on information provided from L A M P F ) are qualitative. High ex-citation presumably means the maximum design excitation (for which B F = 17.647kG) and mid excitation approximately 2/3 the maximum. The field clamp position is shown on figure D.2; the "0.5 inches closer" means along the line A B in figure D.2. Since the APPENDIX D. THE LEPS 192 Figure D .2: T h e six tracks along which the L E P S field was mapped. T h e pole edge boundary ( P E B ) is descibed by a circular arc centred at A . This is quite different from the S A S P P E B s which are described by eighth order polynomials. APPENDIX D. THE LEPS 193 results from the L E P S data are only qualitative, the exact excitations used and the exact positions of the field clamp are not critical. Note that the tracks for the field profiles are not perpendicular to the pole edge. This presented a problem when trying to model them in P O I S S O N which can only model the field in a plane perpendicular to an infinitely long straight boundary. Figure D.3 shows field profiles along the six tracks for three different mappings (1,4,6). The profiles for field map 1 (mid excitation, no clamp) are the solid lines. T h e dipole is approached from the left, with z as defined in figure D.2. These maps approximate those generated by P O I S S O N in section 4.5. To the left of the peak is the fringe field region, to the right is the 1/p internal field. The term "field clamp" used in the field map descriptions is misleading. The L E P S is close to the target scattering chamber, connected to it by an extension of the dipole vacuum box. A field clamp must fit inside this extension. A proper field clamp, with its two return yokes, would block the dipole aperture so the L A M P F group used only the top and bottom plates without the return yokes, therefore this is not a true field clamp — it is a snake (section 4.5). This geometry should produce a bulge in the field which is the opposite of what is required. The field profiles for the clamp with no side plates (map 4) are shown as the short dashed lines in figure D.3. The expected bulge is seen. Field map 6 was of a clamp with one side plate (attached at on the side of the clamp furthest from track 1). This configuration was not considered seriously since it blocked the dipole aperture. It is difficult to predict the effect of a single return yoke. The field profiles are shown in figure D.3 as the dot-dash line. If the clamp plates and the single return yoke are sufficiently large, the flux will be properly shunted across the gap and the field clamped. The field profiles along track #6 shown in figure D.3 look like a snake with some of the flux removed. Th is indicates that the return yoke is shunting some of the flux around the gap while the rest leaks across. This is equivalent to a regular clamp with saturating return yokes. The geometry of the clamp is such that the gap between the clamp and pole edge is largest for the low numbered tracks. Therefore, the effect of the clamp on the fringe field should decrease with decreasing track number; this is seen clearly in figure D.3. Another way of examining the field maps is by drawing contour plots. In a contour plot, evidence of the aperture filling in and a change in the field shape with excitation as discussed in section 4.1 would be investigated. Figure D.4 shows contour plots of map 1 (mid excitation) and map 2 (high excitation). T h e diamonds show the pole APPENDIX D. THE LEPS 194 Figure D.3: Magnetic field profiles along the six tracks for the mid excited dipole. Solid: M a p #1 no clamp. Dash: M a p #4 clamp with no yoke (snake). Dot-dash: M a p #6 clamp with one yoke. APPENDIX D. THE LEPS 195 - 8 H 1 1 — i — • 1 1 10 20 30 40 50 60 70 mid exc i ta t ion z (cm) b) 6 H ' u 1 L _ - 8 H 1 1 1—• 1 1 10 20 30 40 50 60 70 high exci tat ion z ( cm) Figure D . 4 : Contour plots of the L E P S entrance aperture fields. T h e diamonds show the pole edge. T h e x coordinate is plotted in reverse with x as defined in figure D.2 positive down, a ) Low excitation, b) High excitation. APPENDIX D. THE LEPS 196 edge. T h e x coordinate is plotted in reverse with positive x down as defined in figure D.2. If the two contour plots were superimposed they would appear almost identical except for magnitude. T h e only difference between them is the field begins to buldge out slightly more at the bottom (low field side) of the high excitation plot than the mid excitation plot. Thus there are no significant saturation effects. O n the other hand, both fringe fields change from concave at the pole edge to convex further out. Th is is more likely due to the proximity of the sides of the dipole than to aperture filling effect discussed in section 4.1. Another effect observed (more apparent on the low excitation plot) is a contour line crossing from outside the pole edge at the high field end of the aperture to inside at the low field end. This is qualitatively what is expected if the fringe field function (equation 4.1) is dependent on gap width. As the gap increases, the fringe field spreads out over a larger region with the result that the lines of constant B move towards the interior of the dipole. In addition to the field mapping of the entrance completed before final assemble of the spectrometer, data is available from a retractable probe that is inserted into the air gap to the position shown in figure D . l . This probe is used to set the field of the magnet. It produces the excitation curve shown as the solid line in figure D.S. D.2 Modelling the LEPS Fringe Fields Modell ing the L E P S fringe fields proved to be difficult. For comparing the true fields with the P O I S S O N predicted fields, the most convenient tracks for the field maps would have been along perpendiculars to the pole edge. This was not the case so it was necessary to project the P O I S S O N field profile onto the field map track. For this to be a valid procedure, the assumption is made that the fringe field along a perpendicular to the pole edge is essentially equal to that from a straight boundary (the bulged field in the contour plots show this is not strictly true). This projection is equivalent to a rotation about some axis or reference point. This is the point where the internal field, dominated by the pole face slope, gives way to the external field dominated by the pole edge geometry. The location of this point is not obvious. A reasonable first guess is the point where the bevel intersects the pole face. If the correct reference point is selected, the peaks of the field map profile and the P O I S S O N projection should coincide. If not, they will be shifted. B y shifting the peaks about, the correct reference point can be located. A n example of the P O I S S O N problem file used to model the L E P S aperture APPENDIX D. THE LEPS 197 is listed in appendix E . A n alternate approach would be to use the field maps to interpolate the field along the line that P O I S S O N models, but it would still suffer from the bulging fringe field. As well, there could be a problem deciding how to interpolate between the tracks. In the P O I S S O N model, the pole face slope was set equal to that seen by the field mapping probe along its track as opposed to the slope along the perpendicular to the pole edge. T h e projection is correct only outside the dipole. Inside, a different projection is required with a different reference point. Th is introduces a new param-eter to be fitted to the field maps and adds no information. The method should be acceptable if the interior and fringe fields are decoupled as expected. B y using the pole slopes along the profile tracks, direct agreement between P O I S S O N and the field maps for the dipole's internal field near the pole edge can be observed. Th is process for fitting the P O I S S O N data to the mapped fields is less than ideal as it involves several assumptions, a magic number (the reference point), and ignores the observed field bulge. It kills the hope of getting quantitative numbers out of the comparison. The best that can be hoped for is a qualitative idea of how well P O I S S O N agrees with the true fields (i.e. whether there are systematic discrepancies). Figure D.5 shows the comparison of the P O I S S O N calculation, the R A Y T R A C E expected field and the actual field maps for the mid excited dipole without a clamp (map 1). T h e R A Y T R A C E field is not valid to the left of the peak. Like the P O I S S O N field, it is calculated along the perpendicular to the pole edge and is projected onto the field map track. As mentioned above, this projection is only valid for the external field and distorts the internal R A Y T R A C E field to the left of the peak. The internal fields predicted by P O I S S O N agree well with the observed fields except for tracks 5 and 6 which are out near the edge of the pole where the field is expected to drop. Seeing the effects of the dipole sides on the internal field supports the premise that the bulge observed in the contour plots is caused by the dipole side. The P O I S S O N fringe fields agree approximately with the actual fringe fields having about the same extent and slope. The peaks agree well which indicates the original assumption that the fringe field starts to dominate at the start of the bevel is reasonable. This indicates that the actual curvature of the bevel has to be carefully controlled to produce the correct V F B curvature. The fact that the actual fringe field is not consistantly above the P O I S S O N fringe field indicates that the there is little aperture filling (see section 4.1) by the flux. That is, the different parts of the aperture do not interact with each other. These calculations were repeated with the high excitation field maps (2) with APPENDIX D. THE LEPS 198 1.0-0.8-0.6 H 0.4-0.2-0.0--.2-100 — i — 120 i i LEPS11 50000.0 Bpmrj.-VS606.8rN ol »44.0O0CIT ttTmo.-l7064.473 ol W2.000cn--pott «dg. 0.000 vfbp.--2.S10 top erigm-PE di.t 37.455 Fit t»] i«rOidi#KorL(rinoe)epil(ni -field rap Bmar-1124* 149.000 t « * - t S 2 0 140 "l60 180~ z (cm) 200 220 240 1.0 O.B-I 0.6 0.4 0.2 0.0 - .2 _1_ _l_ _1_ 100 LEPS21 50000.0 Bpma*-U2eiJ13 ol m.OOOcrr RTu>o»-l65I5J24 at O4.000crr-pot. edg. 0.000 rfbp.—2.465 kp origin-Pi dnrl 36.680 FM: [2] i»rt>^diricon.lrinoe)ep»Ufr -ffeu mop Bmt»-I299C lepedge-149.000 ocale-1.440 1 120 140 —I 1 — 160 180 z (cm) 200 220 240 1.0-0.8-0.6 I 0.4 0.2 0.0--.2 100 J I I LEPS31 50000.0 T27B0.OO0 ol R4 OOOcrr RTma.-C700.IOe ol HO.OOOcm-pofc edg. 0.000 v fbp . -2 .6 lS kp origin-PC dot 4LO60 Flit [3] u»rO{dt*icon.fringe)epsl'n- -field map Bmax-T26H • 149J)00 « c d e - U 4 0 1 120 140 160 180 z (cm) 200 220 240 1.0 0.8 0.6-1 X D I 0.4 CD 0.2 0.0 -.2 100 LEPS41 50000.0 Bpmar.T2387.667 ol VM OOOcrr HTrrB.-e076.3S3 at 142.000cm-04)00 rtbpe—2.753 kjp origin-PC dst 41.990 FM: [4) usr0{dmcan.tringe)epsl.fn Held mop Brno»-T226 149.000 ocok-1280 120 140 160 z (cm) 180 200 220 1.0 0.8-0.6 X o I 0.4 CQ 0.2 0.0 100 j LEPS51 50000.0 Bpmos.t2046.444 at H4 OOOcrr RTmoi-144o7.002 at KO-OOOcrr -polo edg. 0X100 vtbpe—2.622 lop orioin-PE artl 43.370 Fet [5j uorO{dtfican.tring<)epsUni -field map Bma»-ieB4 kJpodge-M94>00 ocott-UOQ 120 140 160 WO 2 (cm) 200 220 1.0 0.B 0.6-X o I 0.4 0.2 0.0 -.2 100 120 LEPS61 50000.0 Bpma»-ir73e5Se ol » 4 .OOOcrr RTmo«-H023.S11 ot HOOOOem-potf edg. 0.000 vfbpe—2.732 lep crigin-PF. di<t 43.600 FM. [6] uorO{duncan.tringc)epsUfT -few mop Brno r-11300 trpodgo-149.000 tcole- lKO 140 160 Z (cm) 180 200 220 Figure D.5: The P O I S S O N predicted fringe fields (points) compared with the R A Y -T R A C E expected field (dashed line) and the measured fringe field (solid line) for the mid excited dipole without a clamp (map 1). Because of transformations done to the P O I S S O N and R A Y T R A C E data, the R A Y T R A C E field is not valid to the left of the peak. APPENDIX D. THE LEPS 199 . , .1 l . . . •••• .1 . I • I I II II I I I I 1  II I l l l l I 111 I I l l l l I III II I 111. I '• Ml | | ||j 11111111111111111111111111 r 11111 175 20 0 22S 2S0 CYCLb = 620 -0 25 SO 75 100 12S PROB. NW1E * LEP0LL23 6400D Figure D.6: Cross Section of the L E P S modelled with P O I S S O N ; used to study the saturation effects on the magnet. almost the same results as expected from the contour plots. D.3 Modelling the LEPS Interior Fields The study of the L E P S ' s internal field was prompted by the saturation effects predicted by P O I S S O N in section 3.3. Figure D.6 shows a cross section through the L E P S modelled with P O I S S O N (the P O I S S O N problem file is listed in appendix E ) . The top plate and return yoke are much larger than the actual ones on the L E P S . This is because the 2D P O I S S O N model would predict much more leakage flux from the iron than is the case on the real L E P S with its extensive return yoke (figure D . l ) . The leakage flux lowers the field in the dipole air gap. This was remedied by increasing the size of the return yoke and top plate until the field at the central gap of the dipole agreed roughly with that actually observed in the dipole. T h e dimensions shown in figure D . l are the result. Figure D.7 shows an excitation spectrum of the L E P S internal field predicted by P O I S S O N similar to the one produced for the S A S P (figure 3.9). Unlike the S A S P , the L E P S does not suffer centre field sagging at low excitation. Th is is because the APPENDIX D. THE LEPS 200 Figure D.7: T h e profile of the L E P S internal field for various dipole excitations. Starting from the upper left, the excitations are 14000NA, 34000NA, 60000NA, and 64000NA. APPENDIX D. THE LEPS 201 20 H 1 1 1 1 1 h excitation (NA) Figure D.8: T h e measured L E P S excitation curve (solid) and several different attempts to model it with P O I S S O N (points). L E P S is a C magnet and the flux tends to spread more uniformly into the center of the pole piece. A t low excitations, the field is uniformly 0.362% high over most of the pole. Which means that the fringe field region extends slightly past the central radius gap of 8.5cm. A t maximum excitation, the L E P S has a saturation distortion of (subtracting the base level), 1.737% - 0.362% = 1.375% which is approximately a third larger than the S A S P ' s distortion at maximum excita-tion. One would expect the L E P S dipole to be worse than the S A S P dipole because it only has one return yoke making it easier for flux to leak out. However this could be off set by the L E P S ' s smaller size and the greater saturation of the L E P S is probably not significant. Figure D.8 shows the measured excitation curve (at the L E P S ' s field probe posi-tion) and various attempts to model it with P O I S S O N . The curves are normalized so that the linear portions at low excitation agree with each other. T h e upper P O I S S O N curves use the geometry shown in figure D.6 based on two different types of steel exci-tation curves. Again , the rational behind this geometry was to match the field in the P O I S S O N model with that observed in the real system. Thus it is not surprizing that APPENDIX D. THE LEPS 202 the P O I S S O N curves are close to the observed curve. Th is is, of course, a cheat since by changing the geometry around, the L E P S curve could be matched exactly. T h e lower curve was thought to be a more honest approach. Here, the ratio of the widths of the pole piece and return yoke in the P O I S S O N model were set equal to the ratio of the areas of the pole piece and return yoke for the real magnet, ,„POISSON Areal WPP _  APP ,.,POISSON Areal Wry A r y while the top plate in the model was made the same thickness as that on the real magnet. The idea is that since P O I S S O N models the cross section through an infinitely long dipole, setting the ratio of width's in the model to the ratio of areas in the real magnet should produce the same reluctance. The top plate was kept the same, since it projects onto the two dimensional plane as it would on the real magnet. O n reflection, this was a mistake. If the objective is to make the reluctance per unit length of each part of the model the same as the total reluctance of the real magnet, then, +POISSON Areal hp _ A t p ...POISSON ~ Areal WPP  APP where ATT*AL is the cross sectional area of the top plate — which is the same relation as for the return yoke. What should be defined as the cross section of the top plate is not obvious since the return yoke wraps around it (figure D. l ) . What could be done is to make the top plate wedge shaped, with the thickness of the wedge corresponding to the cross section along a circle centered on the pole piece. Th is would make the top plate small over the pole piece, and large over the return yoke. It turned out that the pole piece and return yoke lengths used in figure D.6 were very close to the proper ratio. Decreasing the thickness of the top plate to the real thickness (which is not a valid model) produced the lower curve in figure D.8. While this model is not the best guess, it does illustrate the sensitivity of the excitation curve to changes in the yoke and top plate. What it also says is that try-ing to model the excitation spectrum of the L E P S or the S A S P with P O I S S O N is a risky business. Both magnets are extreme examples of a three dimensional parallel path problem. That is, flux from the pole piece can travel by many different routes through the yoke and this can not be simulated in two dimensions. B y fiddling with the geometry, the P O I S S O N excitation curve could be matched to the observed curve but once done, the same procedure could not be applied to the S A S P since it has a APPENDIX D. THE LEPS 203 much different geometry. This problem requires the application of a three dimensional magnetostatic code such as T O S C A . D.4 Summary of the LEPS Studies T h e nature of the L E P S data is such that qualitative agreement could be made with the P O I S S O N predictions but not quantitative. It is a successfully passed null test, meaning that while the L E P S data do not disagree with the P O I S S O N calculations, they do not necessarily agree either. The main points to be learned from this experience, are: 1. T h e undamped L E P S fringe field bulges — more likely due to the proximity of the side of the pole piece than any effects of the aperture itself. 2. The side plateless field clamp behaves like a snake as expected. 3. The importance of a large enough return yoke on a field clamp is illustrated by the clamp with one yoke. 4. Predicting B - H curves from a P O I S S O N model is difficult. 5. T h e P O I S S O N model produced an internal field profile of the L E P S that agreed with the observed internal field along the probe tracks. Appendix E POISSON Input Files This appendix contains sample problem files for the four types of P O I S S O N simulation done for the S A S P and L E P S dipoles. A P O I S S O N problem file contains the description of the regions to be simulated. The problem file is converted into a logical mesh file from which the grid P O I S S O N preforms the over-relaxation on is generated. The last section has the P O I S S O N input file that describes the B - H curves used for the S A S P dipole pole piece. A P O I S S O N input file is used to set various options in the program and define new B - H curves. The options include the tolerance P O I S S O N is to carry the calculation to. E . l SASP Field Clamp This is a sample problem file for the S A S P field clamp. The cross section is through the x = 0 position on the V F B with the end of the clamp 14.1546cm from the pole edge (located at 153.0cm). The dipole is set for an excitation of 67000NA. This file produces figure 4.16. There are five regions specified in the file. T h e first is an air box that outlines the entire problem region. The second region defined is the dipole itself which is assumed to be made from 1020 steel. The third and fourth regions are the coils and the fifth region is the field clamp. CLAMP2080 SF=14.1546#67000.0 $REG NREG=5 MAT=1 DX=2.0 DY=1.5 XMTN=0.0 XMAX=250.0 YMAX=150.0 NP0INT=5$ $P0 X=0.0 Y=0.0 $ $P0 X=250.0 Y=0.0 $ $P0 X=250.0 Y=150.0 $ $P0 X=0.0 Y=150.0 $ 204 APPENDIX E. POISSON INPUT FILES 205 $PO X=0.0 Y=0.0 $ $REG MAT=4 NPDINT=13 $ $P0 X=0.0 Y=0.0 $ $P0 X=65.0 Y=0.0 $ $PO X=65.0 Y=26.0 $ $P0 X=88.0 Y=26.0 $ $P0 X=88.0 Y=6.8333 $ $P0 X=142.5159 Y=5.2845 $ $P0 X=148.0000 Y=8.4508 $ $P0 X=153.0000 Y=27.1110 $ $PO X=153.0 Y=30.0 $ $P0 X=173.0 Y=30.0 $ $P0 X=173.0 Y=90.0 $ $PO X=0.0 Y=90.0 $ $P0 X=0.0 Y=0.0 $ $REG MAT=1 CUR=67000.0 NP0INT=5 $ $PO X=70.0 Y=9.0 $ $P0 X=86.0 Y=9.0 $ $PO X=86.0 Y=25.0 $ $PO X=70.0 Y=25.0 $ $P0 X=70.0 Y=9.0 $ $REG MAT=1 CUR=-67000.0 NP0INT=7 $ $P0 X=157.0000 Y=9.8577 $ $P0 X=167.2114 Y=9.5676 $ $PQ X=171.4942 Y=9.4459 $ $P0 X=171.5793 Y=12.4447 $ $P0 X=172.0195 Y=27.9384 $ $PO X=157.5254 Y=28.3502 $ $P0 X=157.0000 Y=9.8577 $ $REG MAT=4 CUR=0.0 NP0INT=13 $ $PO X=240.0000 Y=0.0000 $ $P0 X=240.0000 Y=16.2067 $ $P0 X=230.2098 Y=16.2067 $ $P0 X=187.3264 Y=6.9953 $ $P0 X=167.1546 Y=7.5684 $ APPENDIX E. POISSON INPUT FILES 206 $ P O X = 1 6 7 . 2 1 1 4 Y = 9 . 5 6 7 6 $ $ P O X = 1 7 1 . 4 9 4 2 Y = 9 . 4 4 5 9 $ $ P O X = 1 7 1 . 5 7 9 3 Y = 1 2 . 4 4 4 7 $ $ P O X = 1 8 6 . 8 6 5 7 Y = 1 2 . 0 1 0 4 $ $ P O X = 2 3 0 . 3 5 6 0 Y = 2 1 . 3 5 2 1 $ $ P O X = 2 4 5 . 0 0 0 0 Y = 2 1 . 3 5 2 1 $ $ P O X = 2 4 5 . 0 0 0 0 Y = 0 . 0 0 0 0 $ $ P O X = 2 4 0 . 0 0 0 0 Y = 0 . 0 0 0 0 $ E.2 SASP Dipole Interior Field This is the cross section through the centre of the S A S P dipole. T h e pole piece is made out of a 1020 steel (the default P O I S S O N magnetic material — type 2), while the pole piece in made from 1010 steel (material type 4 defined in section E.5). This file produces figure 3.6. D I P 0 L E 3 6 D = 8 4 . 6 1 5 7 7 0 0 0 . 0 $ R E G N R E G = 5 M A T = 1 D X = 2 . 5 0 X M I N = - 1 3 . 0 X M A X = 2 7 5 . 0 Y M A X = 9 5 . 0 N P 0 I N T = 5 $ $ P 0 X = - 1 3 . 0 Y = 0 . 0 $ $ P 0 X = - 1 3 . 0 Y = 9 5 . 0 $ $ P 0 X = 2 7 5 . 0 Y = 9 5 . 0 $ $ P 0 X = 2 7 5 . 0 Y = 0 . 0 $ $ P 0 X = - 1 3 . 0 Y = 0 . 0 $ $ R E G M A T = 4 N P 0 I N T = 1 1 $ $ P 0 X = - 1 3 . 0 Y = 0 . 0 $ $ P 0 X = 4 3 . 0 Y = 0 . 0 $ $ P 0 X = 4 3 . 0 Y = 3 2 . 5 $ $ P 0 X = 7 3 . 0 Y = 3 2 . 5 $ $ P 0 X = 1 8 4 . 5 Y = 3 2 . 5 $ $ P 0 X = 2 0 7 . 5 Y = 3 2 . 5 $ $ P 0 X = 2 0 7 . 5 Y = 0 . 0 $ $ P 0 X = 2 6 3 . 0 Y = 0 . 0 $ $ P 0 X = 2 6 3 . 0 Y = 8 9 . 0 $ $ P 0 X = - 1 3 . 0 Y = 8 9 . 0 $ $ P 0 X = - 1 3 . 0 Y = 0 . 0 $ APPENDIX E. POISSON INPUT FILES 207 $REG MAT=3 NP0INT=9 $ $P0 X=73.0 Y=32.5 $ $PO X=76.5 Y=6.7 $ $PO X=79.5 Y=4.7 $ $PO X=89.0 Y=5.00 $ $PO X=173.615 Y=7.50 $ $PO X=183.0 Y=7.75 $ $PD X=184.5 Y=9.277 $ $P0 X=184.5 Y=32.5 $ $PO X=73.0 Y=32.5 $ $REG MAT=1 CUR=77000.0 NP0INT=5 $ $PO X=47.0 Y=26.774 $ $P0 X=47.5 Y=8.774 $ $P0 X=62.0 Y=9.202 $ $P0 X=61.5 Y=27.202 $ $PO X=47.0 Y=26.774 $ $REG MAT=1 CUR=-77000.0 NP0INT=5 $ $PO X=188.5 Y=30.955 $ $PO X=189.0 Y=12.955 $ $PD X=203.5 Y=13.382 $ $P0 X=203.0 Y=31.382 $ $P0 X=188.5 Y=30.955 $ E.3 LEPS Dipole Fringe Field The P O I S S O N problem file used to model the L E P S fringe field along track #3 (see figure D.2). The pole face slope, points (88.0,7.3802) - (149.0488,4.5192) is set at the angle seen by the probe while the bevel, (149.0488,4.5192) - (153.0,11.3629) is taken from a plane perpendicular to the pole edge. This file produces a geometry similar to D.6. LEPS33 50000.0 $REG NREG=5 MAT=1 DX=2.0 DY=1.5 XMIN=0.0 XMAX=250.0 YMAX=150.0 NP0INT=5 $ $P0 X=0.0 Y=0.0 $ APPENDIX E. POISSON INPUT FILES 208 $P0 X=250.0 Y=0.0 $ $P0 X=250.0 Y=150.0 $ $P0 X=0.0 Y=150.0 $ $P0 X=0.0 Y=0.0 $ $REG MAT=2 »P0INT=13 $ $PD X=0.0 Y=0.0 $ $P0 X=65.0 Y=0.0 $ IPO X=65.0 Y=26.0 $ $P0 X=88.0 Y=26.0 $ $P0 X=88.0 Y=7.3802 $ $P0 X=149.0488 Y=4.5192 $ $P0 X=151.0000 Y=7.8988 $ $P0 X=153.0000 Y=11.3629 $ $P0 X=153.0 Y=20.0 $ $P0 X=173.0 Y=20.0 $ $P0 X=173.0 Y=90.0 $ $P0 X=0.0 Y=90.0 $ $P0 X=0.0 Y=0.0 $ $REG MAT=1 CUR=50000.0 NP0INT=5 $ $P0 X=70.0 Y=9.0 $ $P0 X=86.0 Y=9.0 $ $PQ X=86.0 Y=25.0 $ $P0 X=70.0 Y=25.0 $ $P0 X=70.0 Y=9.0 $ $REG MAT=1 CUR=-50000.0 NP0INT=6 $ $P0 X=158.0 Y=8.0 $ $P0 X=170.0 Y=8.0 $ $P0 X=170.0 Y=11.0 $ $P0 X=161.0 Y=20.0 $ $P0 X=158.0 Y=20.0 $ $P0 X=158.0 Y=8.0 $ $REG MAT=2 CUR=0.0 NP0INT=5 $ $P0 X=163.0 Y=4.0 $ $P0 X=163.0 Y=7.0 $ $P0 X=171.0 Y=7.0 $ APPENDIX E. POISSON INPUT FILES 209 $PO X=171.0 Y=4.0 $ $PO X=163.0 Y=4.0 $ E.4 LEPS Dipole Interior Field T h e P O I S S O N problem file used to model the internal field of the L E P S dipole. This file produces the geometry shown in figure D.6. LEP0LE230 94000.0 $REG NREG=4 MAT=1 DX=2.0 XMIN=0.0 XMAX=250.0 YMAX=150.0 NP0INT=5 $ $P0 X=0.0 Y=0.0 $ $P0 X=250.0 Y=0.0 $ $P0 X=250.0 Y=150.0 $ $P0 X=0.0 Y=150.0 $ $P0 X=0.0 Y=0.0 $ $REG MAT=3 NP0INT=13 $ $P0 X=0.0 Y=0.0 $ $P0 X=75.0 Y=0.0 $ $P0 X=75.0 Y=20.0 $ $P0 X=98.0 Y=20.0 $ $P0 X=98.0 Y=6.65 $ $P0 X=101.0 Y=6.5 $ $P0 X=146.0 Y=4.25 $ $P0 X=152.5 Y=3.925 $ $P0 X=163.0 Y=20.0 $ $P0 X=168.0 Y=20.0 $ $P0 X=168.0 Y=44.0 $ $P0 X=0.0 Y=44.0 $ $P0 X=0.0 Y=0.0 $ $REG MAT=1 CUR=94000.0 NP0INT=6 $ $P0 X=78.0 Y=8.0 $ $P0 X=90.0 Y=8.0 $ $P0 X=90.0 Y=20.0 $ $P0 X=87.0 Y=20.0 $ APPENDIX E. POISSON INPUT FILES 210 $P0 X=78.0 Y=11.0 $ $PO X=78.0 Y=8.0 $ $REG MAT=1 CUR=-94000.0 NP0INT=6 $ $PO X=168.0 Y=8.0 $ $P0 X=180.0 Y=8.0 $ $PO X=180.0 Y=11.0 $ $PO X=171.0 Y=20.0 $ $PO X=168.0 Y=20.0 $ $PO X=168.0 Y=8.0 $ E.5 POISSON Input File T h e P O I S S O N input file that defines the B - H curves of two new steel types, #3 a n d #4. #4 is a good 1010 steel that is used for the S A S P dipole pole piece. 0 DUMP *6 0 *18 2 *85 1.0E-3 1.0E-3 S CON 3 1.0 1 600 .0 0 .001667 1900 .0 0 .001053 5700 .0 0 .0007017 7200 .0 0 .0006944 9200 .0 0, .0007608 11100 .0 0, .000909 14000 .0 0, .0014285 16500 .0 0 .0030303 18200 .0 0 .0054945 20000 .0 0, .0100000 20900 .0 0 .019139 21041 .0 0 .023763 21772 .0 0 .04593 22890 .0 0, .08737 25961 .0 0 .1925965 30985 .0 0. .322737 APPENDIX E. POISSON INPUT FILES 211 40997.0 71004.0 121006.0 4 1.0 1 0.1 1.0E2 8.5E2 1.997E3 3.3E3 6.094E3 9.600E3 1.190E4 1.320E4 1.470E4 1.610E4 1.710E4 1.770E4 1.860E4 1.920E4 1.990E4 2.025E4 2.099E4 2.212E4 2.159E4 3.022E4 4.023E4 7.024E4 -1 DUMP 0.48784 0.704186 0.82641 S 0.001 0.0025 0.00235 0.0015 0.001212 0.00098 0.00104 0.00126 0.00152 0.00204 0.00310 0.00438 0.00565 0.00806 0.0104 0.0150 0.0247 0.0476 0.0904 0.1985 0.3309 0.4970 0.7118 S Appendix F Aperture Specification This appendix contains specifications of the pole edge, field clamp, and vacuum box geometry for the S A S P dipole apertures. It is a first order design and will be modified before a final engineering design is achieved. For example, the exact widths of the dipole apertures and the relative positions of the return yokes are not yet known. Specified here are: • Pole edge bevel shape. • Dipole entrance and exit aperture curvatures. • F ie ld clamp design. • Dipole entrance and exit vacuum box designs. T h e basic dipole geometry and the coordinate systems used are shown in figure 3.3. Positions along a V F B (XVFB) is referred to by the x coordinate of the appropriate coordinate system, B or C . Note that z is positive out for both. Some assumptions made about the dipole geometry are: • The coil cross section is 14.5 cm horizontal by 18.5 cm vertical • The coils lie in planes parallel to the dipole pole faces. • Coi l — pole edge separation is 2.186559in (5.5538599cm) for the entrance aperture and 2.317238in (5.8857845cm) for the exit aperture. • T h e lower extent of the coil is 5.0 cm above the plane of the pole face. • T h e vacuum box is made from material 2.0 cm thick. • Both the dipole and field clamp are made of 1020 hot rolled steel as specified by P O I S S O N material #4 (the B - H curve for which is shown in appendix C) . 212 APPENDIX F. APERTURE SPECIFICATION 213 F . l Pole Edge Bevel A Rogowski bevel is approximated for the S A S P dipole's apertures by taking 75° and 30° cuts perpendicular to the pole edge as shown in figure F . l . If the pole pieces were flat, a cross section through the bevel would follow the three points A , C , and D shown in figure F . l b . C would be 5cm and A 11cm from the pole edge (which due to the S A S P dipole's geometry is perpendicular to the median plane). However, the dipole's pole pieces are sloped making the 30° cut intersect the pole face at B rather than A . Because the apparent slope of the pole face varies depending on where the perpendicular to the P E B is taken, the distance from the pole edge to B would vary over the aperture. The resulting curves made by points B and C are shown in figure F . l a . T h e effect of the sloped pole face on the distance from B to the pole edge is exaggerated. One possible method for cutting the bevel is shown in figure F.2. Point E , the corner of the uncut pole piece is taken as a reference point. Note that because the dipole gap varies over the extent of the aperture, the distance of E from the median plane varies as well. First a 75° cut (with respect to the median plane) is made such that the distance from E to D is 22.1244cm (figure F.2b). Then a 30° cut is made such that the height of C above E is 3.4641cm (figure F.2c). Th is method makes it unnecessary to calculate the distance from B to the pole edge. F.2 Vacuum Box Design To simplify the construction of the vacuum boxes, it is desirable to construct them from flat plates. As shown in figure 4.15, each box consists of a flat plate that sits under the coil and field clamp parallel to the dipole pole face. The inner side of this plate becomes a flange that seats against the side plates of the dipole. The end of the plate that protrudes from the clamp butts against a sloped plate that flares to accomadate the spreading ray bundle (figure F.3). F.2.1 Entrance T h e entrance vacuum box must connect the dipole to the exit of the second quadrapole. T h e distance between the quadrupole and the dipole along the optical axis is ap-proximately 120cm normally. T h e exact distance will depend on the final design of the quadrupole. However, the S A S P will be run in some configurations where the quadrupole is 30cm closer to the dipole. The beam pipe leaving the quadrupole is the APPENDIX F. APERTURE SPECIFICATION 214 line of constant gap b) 18.6603cm 3.4641cm 22.1244cm median plane pole edge Figure F . l : S A S P dipole bevel, a) The cuts are made perpedicular to each point on the P E B . b) T h e bevel is comprised of a 75° and a 30° cut. The intesection of the 30° cut and the pole face varies over the aperture. APPENDIX F. APERTURE SPECIFICATION 215 Figure F.2: Cutt ing the bevel, a) The uncut pole piece, b) First the 75° cut is made, c) Followed by the 30° cut. Figure F.3: Simplified drawing of an aperture vacuum box. Does not show the changing gap width or curvature of the aperture. The field clamp is a tube symmetric about the median plane that wraps tightly about the vacuum box. APPENDIX F. APERTURE SPECIFICATION 216 28cm wide cruciform shaped pipe discussed in section 5.2.2 The rays fill most of the region between the quad and the dipole. In the nonbend plane, the quad has a 28 cm wide aperture compared to the dipole's aproximately 10 cm aperture. T h e vacuum box will have to flared beyond the coils to connect to the quad. This flare must accomodate both the 120cm and 90cm separations between the quadrupole and dipole. In the bend plane the rays leave the quadrupole over its entire 28cm aperture and enter the dipole aperture over a range of approximately -47.5 cm to 44.0 cm in co-ordinate system C (figure 3.3). Thus while the vacuum box must increase in width from the dipole to quad in the nonbend plane, it must decrease in the bend plane. T h e vacuum box has to fit under both the coil and the field clamp (figure 4.15). The coil - pole face separation is 5.0 cm; the tongue of the field clamp that projects under the coil is 2.0 cm thick; the vacuum box is 2.0 cm thick. Th is leaves a gap of 1.0 cm between the vacuum box and the pole face. Without this gap, the solid angle of the spectrometer would be reduced by approximately 10%. Figure F.4 shows a side and bottom view of the entrance vacuum box. As shown, it consists of one section (probably cast as part of the main dipole vacuum box) extending back from the dipole to the quad where it couples with the quad's vacuum pipe. This does not consider the need to move Q2 back towards the dipole. A n improved design would have the flange closer to the dipole with an intermediate box between the dipole box and quadrupole. F.2.2 Exit Figure 6.3 shows the cross sections of ray bundles after the S A S P for 6= -10%, 0%, and + 15%. Also shown are the maximum widths of the ray bundles at each cross section (in the nonbend plane). These bundles are for the largest target spot that would be used with the spectrometer, ± 5 c m (x) by ± 2 c m (y). The vacuum box will have to flare slightly in the nonbend plane to clear the ray bundle. The exit window for the vacuum box should lie on the focal surface of the spectrometer. However, the studies of the S A S P dipole indicate that the exact location of the focal surface will not be known until the spectrometer is built and its field mapped. Thus the vacuum box will be built in two sections. The first will be attached to the dipole vacuum box with a flange located after the field clamp. The second section will be added later to extend the vacuum to the focal surface. Figure F.5 shows a side view of the exit vacuum box. APPENDLXF. APERTURE SPECIFICATION 217 coil BOTTOM VIEW Q2 , Figure F.4: Entrance Vacuum box design, side and bottom view. Figure F.5: Exit Vacuum box, side view. Also shown are the extreme rays for -10% and +15% as well as points on the focal surface. APPENDIX F. APERTURE SPECIFICATION 219 F.3 Field Clamp Design As shown in figure F.3, the clamps consist of tubes fitted tightly around the vacuum boxes and butting against the coils with'tongues projecting under the coils. T h e thick-ness of the tongue is 2.0 cm while the balance of the tube is 5.0 cm thick. T h e clamp is symmetric about the median plane. The tubes must protrude beyond the end of the fringe field which extends aproximately 30 cm out from the pole edge (figure 4.16b). A reasonable value for the extent of the tube is 50 cm from the pole edge. P O I S S O N studies indicate that 5cm thick return yokes is adequate to carry the flux from both apertures. However, because the P O I S S O N calculations are uncertain, it is important to leave room to add more steel to the return yokes if necessary. T h e clamps are shown in figures F.6 and F.7. The 2 cm tongue under the coil carries most of the flux in the clamp. Making the clamp fit tightly to the coil relieves this bottle neck and helps prevent saturation problems. This means that there are two complex curves to be fashioned for each clamp: the fit to the coil, and the leading edge of the tongue. It would be desirable to fashion the clamp such that the tongue detaches to make modifications to it if necessary after construction of the dipole. As stated in the design procedure, polynomial descriptions of the pole edge ( P E B ) and field clamp edge ( F C B ) were generated. These descriptions are in the local coordinate systems shown in figure 3.3 (the same as the original V F B descriptions). where z and x are in cm. T h i s p o l y n o m i a l is g o o d for the r a n g e x = ±50.0c?n for the e n t r a n c e a n d x = —80.0cm t o i = 100.0cm for the ex i t . Outside this range the polynomials give unreliable results. The coefficients are listed in table F . l and points on the P E B and F C B are listed in tables F.2 and F.3. Also listed are points on the inner and outer boundaries of the coils. The curves produced by the coefficients in table F . l are shown in figures F.8 and F.9. Also shown in the figures, are the coil positions. F.4 PEB and FCB Specification 8 t=0 APPENDIX F. APERTURE SPECIFICATION 220 SIDE VIEW return Figure F.6: Entrance field clamp, side view. APPENDIX F. APERTURE SPECIFICATION 221 SIDE VIEW vacuum return yoke Figure F.7: Exit field clamp, side view. APPENDIX F. APERTURE SPECIFICATION 223 E n t r a n c e E x i t P E B F C B P E B F C B 0.46932 14.624 0.50705 14.341 Ol 1.34820 x 10~ 2 -3.24589 x 10" 2 -1.73076 x l O - 2 3.09870 x 10" 2 a 2 2.99889 x 10"3 3.07946 x 10~ 3 3.10143 x l O " 3 3.25823 x 10~ 3 a.3 1.09609 x 10" 5 -3.08958 x 10" 6 1.18375 x 1 0 - 5 2.49444 x 10~5 a,4 4.47183 x 10" 7 1.53804 x 10" 7 -1.63113 x l O " 7 -3.29191 x 10~ 7 a 5 1.22918 x 10"8 1.65351 x 10" 8 3.75418 x l O - 9 1.33917 x l O " 1 0 o 6 -3.43090 x l O " 1 1 -9.39666 x l O " 1 3 0.7 a8 Table F . l : P E B and F C B coefficients APPENDIX F. APERTURE SPECIFICATION 224 Dipole Entrance C o i l XVFB P o l e E d g e F i e l d C l a m p inner outer in. in. in. in. in. -20.000000 1.931569 7.895737 4.095641 9.713022 -19.000000 1.881544 7.958900 4.067156 9.756620 -18.000000 1.800169 7.956359 3.999649 9.734820 -17.000000 1.695130 7.901020 3.902583 9.663625 -16.000000 1.573181 7.804357 3.784232 9.556796 -15.000000 1.440210 7.676508 3.651757 9.426028 -14.000000 1.301298 7.526349 3.511288 9.281106 -13.000000 1.160781 7.361573 3.368006 9.130070 -12.000000 1.022311 7.188780 3.226219 8.979378 -11.000000 0.888918 7.013558 3.089454 8.834068 -10.000000 0.763072 6.840564 2.960522 8.697924 -9.000000 0.646742 6.673606 2.841611 8.573633 -8.000000 0.541460 6.515725 2.734365 8.462953 -7.000000 0.448380 6.369284 2.639957 8.366874 -6.000000 0.368342 6.236040 2.559180 8.285784 -5.000000 0.301933 6.117236 2.492520 8.219623 -4.000000 0.249545 6.013681 2.440241 8.168056 -3.000000 0.211440 5.925828 2.402466 8.130634 -2.000000 0.187812 5.853863 2.379253 8.106951 -1.000000 0.178843 5.797783 2.370681 8.096811 Table F.2: Entrance P E B , F C B , and coil curves. APPENDIX F. APERTURE SPECIFICATION 225 Dipole Entrance cont'd C o i l XVFB P o l e E d g e F i e l d C l a m p inner outer in. in. in. in. in. 0.000000 0.184772 5.757480 2.376929 8.100394 1.000000 0.205949 5.732827 2.398356 8.118411 2.000000 0.242904 5.723752 2.435582 8.152274 3.000000 0.296400 5.730333 2.489570 8.204255 4.000000 0.367500 5.752868 2.561704 8.277655 5.000000 0.457629 5.791965 2.653875 8.376954 6.000000 0.568632 5.848626 2.768553 8.507990 7.000000 0.702836 5.924320 2.908879 8.678108 8.000000 0.863114 6.021077 3.078736 8.896334 9.000000 1.052943 6.141564 3.282834 9.173526 10.000000 1.276467 6.289169 3.526791 9.522552 11.000000 1.538562 6.468087 3.817214 9.958439 12.000000 1.844888 6.683393 4.161777 10.498542 13.000000 2.201960 6.941136 4.569305 11.162710 14.000000 2.617207 7.248416 5.049851 11.973441 15.000000 3.099026 7.613464 5.614783 12.956050 16.000000 3.656856 8.045732 6.276858 14.138835 17.000000 4.301231 8.555969 7.050307 15.553234 18.000000 5.043839 9.156307 7.950913 17.233988 19.000000 5.897592 9.860340 8.996093 19.219301 20.000000 6.876683 10.683211 10.204982 21.551022 APPENDIX F. APERTURE SPECIFICATION 226 Dipole Exit C o i l XVFB P o l e E d g e F i e l d C l a m p inner outer in. in. in. in. in. -31.000000 -4.129920 2.614647 -0.513139 6.835637 -30.000000 -2.859769 3.242228 0.461686 7.431477 -29.000000 -1.789150 3.793691 1.288558 7.948064 -28.000000 -0.896370 4.274421 1.982541 8.391273 -27.000000 -0.161433 4.689624 2.557595 8.766718 -26.000000 0.434033 5.044322 3.026611 9.079751 -25.000000 0.906882 5.343372 3.401496 9.335490 -24.000000 1.272533 5.591455 3.693212 9.538823 -23.000000 1.545056 5.793086 3.911839 9.694418 -22.000000 1.737251 5.952611 4.066628 9.806731 -21.000000 1.860735 6.074216 4.166049 9.880023 -20.000000 1.926015 6.161923 4.217845 9.918366 -19.000000 1.942563 6.219594 4.229077 9.925645 -18.000000 1.918890 6.250937 4.206171 9.905575 -17.000000 1.862613 6.259501 4.154961 9.861705 -16.000000 1.780526 6.248684 4.080738 9.797425 -15.000000 1.678661 6.221734 3.988282 9.715978 -14.000000 1.562351 6.181747 3.881909 9.620453 -13.000000 1.436290 6.131673 3.765504 9.513809 -12.000000 1.304588 6.074316 3.642557 9.398862 -11.000000 1.170827 6.012337 3.516202 9.278307 -10.000000 1.038112 5.948250 3.389241 9.154709 -9.000000 0.909116 5.884434 3.264181 9.030514 -8.000000 0.786134 5.823124 3.143262 8.908047 Table F.3: Exit P E B , F C B , and coil curves. APPENDIX F. APERTURE SPECIFICATION 227 Dipole Exit cont'd C o i l XVFB P o l e E d g e F i e l d C l a m p inner outer in. in. in. in. in. -7.000000 0.671117 5.766416 3.028479 8.789526 -6.000000 0.565722 5.716274 2.921615 8.677049 -5.000000 0.471343 5.674519 2.824260 8.572607 -4.000000 0.389152 5.642843 2.737830 8.478086 -3.000000 0.320128 5.622801 2.663595 8.395255 -2.000000 0.265093 5.615819 2.602689 8.325788 -1.000000 0.224732 5.623186 2.556132 8.271241 0.000000 0.199626 5.646063 2.524843 8.233071 1.000000 0.190270 5.685482 2.509650 8.212623 2.000000 0,197094 5.742342 2.511309 8.211131 3.000000 0.220483 5.817417 2.530506 8.229721 4.000000 0.260786 5.911349 2.567867 8.269403 5.000000 0.318336 6.024656 2.623966 8.331072 6.000000 0.393453 6.157726 2.699327 8.415504 7.000000 0.486454 6.310821 2.794426 8.523346 8.000000 0.597657 6.484074 2.909697 8.655123 9.000000 0.727382 6.677496 3.045522 8.811229 10.000000 0.875953 6.890968 3.202238 8.991912 11.000000 1.043691 7.124245 3.380125 9.197287 12.000000 1.230908 7.376956 3.579401 9.427310 13.000000 1.437902 7.648605 3.800221 9.681789 14.000000 1.664943 7.938570 4.042658 9.960366 15.000000 1.912257 8.246096 4.306695 10.262508 16.000000 2.180015 8.570306 4.592215 10.587511 APPENDIX F. APERTURE SPECIFICATION 228 Dipole Exit cont'd C o i l XVFB P o l e E d g e F i e l d C l a m p inner outer in. in. in. in. in. 17.000000 2.468311 8.910198 4.898980 10.934479 18.000000 2.777137 9.264637 5.226620 11.302320 19.000000 3.106363 9.632360 5.574611 11.689736 20.000000 3.455713 10.011978 5.942254 12.095219 21.000000 3.824726 10.401971 6.328658 12.517026 22.000000 4.212730 10.800686 6.732709 12.953183 23.000000 4.618808 11.206342 7.153054 13.401463 24.000000 5.041760 11.617028 7.588060 13.859385 25.000000 5.480059 12.030693 8.035803 14.324195 26.000000 5.931811 12.445158 8.494020 14.792842 27.000000 6.394713 12.858104 8.960086 15.261992 28.000000 6.866001 13.267085 9.430978 15.727989 29.000000 7.342399 13.669504 9.903238 16.186853 30.000000 7.820072 14.062634 10.372941 16.634260 31.000000 8.294560 14.443609 10.835642 17.065538 32.000000 8.760732 14.809418 11.286351 17.475630 33.000000 9.212715 15.156904 11.719476 17.859097 34.000000 9.643832 15.482774 12.128787 18.210098 35.000000 10.046546 15.783578 12.507369 18.522360 36.000000 10.412376 16.055727 12.847569 18.789183 37.000000 10.731833 16.295469 13.140946 19.003397 38.000000 10.994357 16.498924 13.378232 19.157360 39.000000 11.188216 16.662041 13.549256 19.242939 40.000000 11.300447 16.780605 13.642915 19.251476 Appendix G Random Numbers E A S Y , being a Monte Carlo simulator, makes extensive use of r a n d o m n u m b e r s , p a r t i c -ularly uniformly and normally distributed ones. Algorithms for uniform r a n d o m n u m -ber generators are common [18]; E A S Y generates uniform random numbers b e t A v e e n 0 and 1 using the standard routines built into the C programing language. Th is r a n d o m number generator is initialized at the start of the program, meaning if a simulation is run twice exactly the same results will be produced each time. Th is is advantageous when debugging a simulation, each run will be identical making it easier to t r a c k d o w n problems. O n the other hand one often wants to duplicate a Monte Carlo u s i n g dif-ferent numbers to see if an observed effect is due to chance statistics or if i t is a r e a l phenomena. The only way to do this as E A S Y is currently written is to increa se the number of events simulated. Ideally, the seed number of the random number g e n e r a t o r should be set in the program to either duplicate a random number sequence o r to gen-erate a new one. The usual random number generator produces a sequence o f n u m b e r s that appear random but in fact will eventually repeat after a large number of i t e r a t i o n s — this is independent of the seed number used to start the generator. B u t a M o n t e Carlo uses a large number of iterations and there is a (albeit small) chance t h a t the simulation will start to repeat itself. The user thinks he is getting good s t a t i s t i c s f r o m a run but in fact they are almost useless. One method of avoiding this is for subsequent runs of the Monte Carlo to use different algorithms for the random number genera tor s . There are many algorithms available for calculating random numbers with a G a u s -sian distribution (see [16]). Most of these methods are conserned with using a p p r o x -imations to make the calculation as fast as possible. In E A S Y calculating a r a n d o m number is only a small part of the total simulation so speed is not a factor. T h u s a slow but exact method of calculating Gaussian random numbers is used. Starting with two uniform random numbers between 0 and 1, u and v, the t r ans -229 APPENDIX G. RANDOM NUMBERS 230 formation x = V— 2 In u cos(27rt>) y = y—21nwsin(27ri») is used. Th is produces two normally distributed random numbers with mean 0 a n d standard deviation 1. To convert x and y to numbers with mean m and s t a n d a r d deviation cr, x' = m + (a * x) y' = m + (cr * y) A nice feature of this method is that it takes only a little more time to calculate t w o numbers than one. Appendix H 2 Body Kinematics H . l Calculating C M Energies and Momentums Suppose that one of the incident particles (1TI2) is stationary (as in a target) Lab Frame and the other (mi) is moving with an kinetic energy T i in a beam. E2 = m2 The total C M energy available to the reaction W is given by, The magnitude of the momentums of the reaction products P 3 = P4* is calculated by, W2 = 2Txm2 + ( m i +m2)2 then calculating 7 * and /3*, W Tx + mi + m2 W (W2 - (m3 + m4Y) [W* - (m3 - mrf) AW2 231 APPENDIX H. 2 BODY KINEMATICS 232 H.2 Aiming at a Detector At what angle 9* does one point a scattered particle with momentum P* in the centre of mass frame, to hit a detector lying at an angle of 9 in the lab frame? Start with the relation, E* = 7 ( E - pP cos 9) E* — = E - /3P cos 9 7 and defining A = (~r) and B = P cos 0, A = (E - BP)2 = E2 - 2BEP + B2P2 substituting E2 = P2 + m2 A = P2 + m2- 2B\/P2 + m2P + B2P2 rearranging and squaring both sides, one gets a quadratic in P 2 , [(1 + B 2 ) 2 - AB2] P4 - [2(1 + B2)(A - m2) + Am2B2\ P2 + (A - m2)2 = 0 for which there are two solutions, _ (l+B2)(A-m2) + 2m2B2 2By/m2(l + B2)(A - m2) + m<B2 + (A- m2~)2 ( 1 - B 2 ) 2 (l-B2)2 What do the two solutions for P mean? For a given E" and 0, there are two possible values of P. From E* = i*(E — P*P cos 9) there are two cases, 0 < f E* < E 9 > \ E* > E So when choosing P from the two possible solutions, 0 <9 < § 1<9<TT choose larger P choose smaller P APPENDIX H. 2 BODY KINEMATICS 233 Note that if P* is less that 7*/?*m, there is a maximum scattering angle, a n d then using P ' s i n f T = Psinf? one can solve for 8* 8* = sin-1 ( - ^ s i n f ? Appendix I EASY Input Files This appendix lists the files used to generate the various Monte Carlo studies of the S A S P using E A S Y . To describe a problem to E A S Y , the user must create three files: the device file; the data acquisition file and the command file. Each of these files has the same name as the problem (in this case it will called sol id) and a distinctive suffix; .dev, .dac, and .cmd respectively. As well, the problem must be identified to E A S Y in the problem file called E A S Y . P R B . This is done with the line: Qproblem 1 s o l i d where solid is the name of our problem. A l l these examples are for a ray bundle of +10% (accept for the focal plane mapping) assuming the central momentum of the S A S P is 660.0 M e V / c . To change to a different momentum, the variable ray momentum in the @ m a p s o l i d command is changed to the appropriate value. 1.1 Solid Angle The S A S P ' s "high resolution" solid angle was calculated by generating rays smeared out over a rectangle 2cm by 1cm in the target plane. The system consists only of the sasp_opt ics . The function m a r k e r tells the command @ m a p s o l i d that the ray made it successfully past this point (in this case the end of the S A S P optics). E A S Y automatically calculates the solid angle of the device and prints it out to the problem's output file, solid.out. T h e command @setsasp sets the central momentum of the S A S P and the command @ e n d tells E A S Y that the simulation is over. D e v i c e F i l e (so l id .dev ) d e f i n e main{ s a s p _ o p t i c s 234 APPENDIX I. EASY INPUT FILES 235 } Data Acquisition File (solid.dac) #address saspend=[sasp_optics] # funct ion mymark=marker(flsaspend/x;) Command File (solid.cmd) flsetsasp 660.0 \ s e t Po of SASP to 660.0 MeV/c (Dmapsolid s \smeared source -1 .0 1.0 2.0 \xmin, xmax, xstep -0.5 0.5 2.0 \ymin, ymax, ys tep 726.0 70.0 \ r a y momentum, d i s t to aper ture 0.100 \ s c a t t e r i n g cone s i z e ( i n s t r ) 100000 \number of rays (Send 1.2 Limiting Aperture This problem is almost the same as for the high resolution solid angle except that a larger spot size is used and the command @saspcuts prints the S A S P aperture cuts scatter plots to the output file solid.out. Three sets of scatter plots are generated for each aperture for a total of 27. The P A S S E D plots show all the rays that pass the aperture cut. T h e M I S S E D plots show all the rays that hit the aperture. The T H R U plots show the rays that made it through the entire system as they appear at each aperture. Device File (solid.dev) d e f i n e main{ sasp_opt ics } Data Acquisition File (solid.dac) #address saspend=[sasp_optics] • f u n c t i o n mymark=marker(Qsaspend/x;) Command File (solid.cmd) APPENDIX I EASY INPUT FILES 236 fisetsasp 660.0 \set Po of SASP to 660.0 MeV/c Qmapsolid s \smeared source -5.0 5.0 2.0 \xmin, xmax, xstep -2.0 2.0 2.0 \ymin, ymax, ystep 726.0 70.0 \ray momentum, dist to aperture 0.100 \scattering cone size (in str) 100000 \number of rays (Osaspcuts \print out SASP aperture cuts (Send 1.3 Target Plane Acceptance While the device and data acquisition files are the same as for the above studies, the command © m a p s o l i d is changed to produce 2000 rays at each point on a grid of size 40cm by 20cm with a 2cm spacing. The solid angle for each point on the grid is printed to a file called (in this case) solid.sol along with the coordinates of the grid point. This data can then be changed into a matrix with 21 rows and 11 columns by the program V E C T O M A T . C for plotting by a program such as P L O T D A T A (a standard T R I U M F plotting package). D e v i c e F i l e (so l id .dev ) define main{ sasp_optics } D a t a A c q u i s i t i o n F i l e (so l id .dac) #address saspend=[sasp_optics] #f unction mymark=marker(<Bsaspend/x;) C o m m a n d F i l e ( s o l i d . c m d ) Gsetsasp 660.0 \set Po of SASP to 660.0 MeV/c (Dmapsolid p \smeared source -20.0 20.0 2.0 \xmin, xmax, xstep -10.0 10.0 2.0 \ymin, ymax, ystep 726.0 70.0 \ray momentum, dist to aperture 0.100 \scattering cone size (in str) APPENDIX I. EASY INPUT FILES 237 100000 \number of rays (Send 1.4 Downstream Ray Profiles The downstream ray profile calculation uses the same target spot size as the limiting aperture calculation but in addition to transporting the rays through the S A S P , they are drifted downstream in 50cm steps. The data acquisition file sets up scatter plots of the ray bundle at each step and the command @ a n a l i z e prints them out. D e v i c e F i l e (so l id .dev ) define drifting{ drift aperture entrance none length 50.0 radlength -1.0 aperture exit none \50.0 cm \vacuum } define main{ sasp_optics drifting drifting drifting drifting drifting drifting drifting drifting \ 50cm downstream \100cm  \150cm  \200cm  \250cm  \300cm  \350cm  \400cm } D a t a A c q u i s i t i o n F i l e (so l id .dac) #address saspend=[sasp_optics] #address #address #address dl=[drifting,1] d2=[drifting,2] d3=[drifting,3] APPENDIX I. EASY INPUT FILES 238 #address d4=[drifting,4] #address d5=[drifting,5] #address d6=[drifting,6] #address d7=[drifting,7] #address d8=[drifting,8] #f unction mymark=marker(<8saspend/x;) scat(sasp exit) saspend/y(y cm) min=-100.0 max=100.0: saspend/x(x cm) min=-100.0 max=100.0; scat( 50cm downstream) <Bdl/y(y cm) min=-100 0 max=100 0 <Ddl/x(x cm) min=-100 0 max=100 0 scat(100cm downstream) <Bd2/y(y cm) min=-100 0 max=100 0 <8d2/x(x cm) min=-100 0 max=100 0 scat(150cm downstream) ®d3/y(y cm) min=-100 0 max=100 0 <0d3/x(x cm) min=-100 0 max=100 0 scat(200cm downstream) <0d4/y(y cm) min=-100 0 max=100 0 <Bd4/x(x cm) min=-100 0 max=100 0 scat(250cm downstream) <Dd5/y(y cm) min=-100 0 max=100 0 <Dd5/x(x cm) min=-100 0 max=100 0 scat(300cm downstream) <8d6/y(y cm) min=-100 0 max=100 0 (Dd6/x(x cm) min=-100 0 max=100 0 scat(350cm downstream) <Sd7/y(y cm) min=-100 0 max=100 0 Qd7/x(x cm) min=-100 0 max=100 0 scat(400cm downstream) <8d8/y(y cm) min=-100 0 max=100 0 <Dd8/x(x cm) min=-100 0 max=100 0 C o m m a n d F i l e ( s o l i d . c m d ) (Ssetsasp 660.0 \set Po of SASP to 660.0 MeV/c (Dmapsolid s \smeared source -1.0 1.0 2.0 \xmin, xmax, xstep -0.5 0.5 2.0 \ymin, ymax, ystep 726.0 70.0 \ray momentum, dist to aperture 0.100 \scattering cone size (in str) 100000 \number of rays (Danalize \print out downstream cross sections APPENDIX I. EASY INPUT FILES 239 (Send 1.5 Focal Surface Device File (solid.dev) d e f i n e main{ sasp_opt ics } Data Acquisition File (solid.dac) #address saspend=[sasp_optics] # funct ion mymark=marker(Qsaspend/x;) Command File (solid.cmd) (Dsetsasp 660.0 \ s e t Po of SASP to 660.0 MeV/c (Bfocus -1 .0 1.0 -4.70 \ x : min max D\M -85.0 85.0 \ t h e t a : min max 10.0 24.0 0.5 \ d e l t a : min max step -0.05 0.05 0.005 \ y : min max step -50.0 50.0 5.0 \ p h i : min max step 500 \number (Bend Appendix J Track Reconstruction Using VDCs The E A S Y generic function v d c 2 reconstructs the trajectory of a ray passing through two vertical drift chambers at some arbitrary plane (called the canonical plane) using the position outputs of the V D C s . This plane is not necessarily parallel to the V D C s and is usually the focal plane. The geometry using two identical chambers is shown in figure J . l . Each chamber has an x plane and a u plane readout where the u axis is at some angle 6U to the x axis. The chambers are parallel and set at an angle By DC to the optical axis. T h e lower chamber is called V D C l and the upper, V D C 2 . T h e x origin of each chamber lies on the optical axis while the u = 0 position lies on the perpendicular running through x = 0. The spacing between x planes of the two chambers is D and the separation between the X i plane and the cannonical plane (which is set at some angle 6rp to the optical axis) is drp. 1. Convert the individual chamber readouts to the ( x l 5 U i ) coordinate system. x1 = X i u[ = Ui x'2 = X2 + 0 2 1 U2 = U2 + 0 2 1 cos 8U where 0 2 i is the distance between the origins of x i and x 2 . 2. Calculate the ratios vx (^2 ^ l ) Vz ~ D _ (u'2 — u[) vz~ D 240 APPENDIX J. TRACK RECONSTRUCTION USING VDCS 241 Figure J . l : Reconstructing a track using two V D C ' s APPENDIX J. TRACK RECONSTRUCTION USING VDCS 242 where u x , vu, vz are the x, u, and z velocities of the particle. 3. The «i plane is above (or downstream) of the xx plane. Calculate the u position of the ray in the x plane. / uT = u. — a— vz 4. Calculate the y position and velocity in the x a plane. uT x r r sin 9U tan #u vz sin #u tan 6U 5. Rotate the velocity to a plane parallel to the canonical plane. vXc _ Sf c o s 9r + sin 9r vzc — ^  sin 9r + cos 0 r vzc — ^  sin #r + cos 9T where 9r = 8Tp — 9VDC is the angle between the canonical plane and t h e V D C plane. 6. Calculate the distance to the canonical plane and the displacement of the c a n o n -ical plane's origin. drpp = dr + xr sin 9r if drp < 0 (the canonical plane is below the V D C l ) 0 P = 0n + xr cos 9T else 0P = xT cos 9r — 0r\ and the position on the canonical plane, APPENDIX J. TRACK RECONSTRUCTION USING VDCS 243 yc = aTpp (- yr Vzc 7. Calculate the angles the canonical ray makes with the zc axis, ^ t a n " 1 ( ^ ) \vzcJ ^ = tan" 1 f ^ ) \v2cJ Appendix K EASY's SASP Model The complete E A S Y definition of the S A S P spectrometer is kept in the two files S A S P J 3 E F . D E V and S A S P J D E F . D A C . K . l Device File \*************************************************************** \ \ SASP D e f i n i t i o n F i l e \ \ 26 J u l y 1988 \ \ The geometry used f o r t h i s model i s approximate only. \ \ To use t h i s SASP d e f i n i t i o n , d e f i n e the f o l l o w i n g device i n \ your DEV f i l e , \ \ d e f i n e sasp{ \ sasp_fec \ sasp_optics \ sasp_stack \ } \ \ To run the SASP without a f r o n t end chamber, d e l e t e the SASP.FEC \ from the d e f i n i t i o n . \ 244 APPENDIX K. EASY'S SASP MODEL 245 d e f i n e sasp_cathode{ d r i f t \mylar aperture entrance none length 0.00254 radlength 28.7 aperture e x i t none > d e f i n e sasp_vdc_gas{ d r i f t \gas aperture entrance none length 1.5875 radlength 34 aperture e x i t none } \ Estimated rad length f o r 50'/, Ar and 50'/, isobutane at STP de f i n e sasp_vdc_gas_bulged{ d r i f t \gas aperture entrance none length 2.8575 radlength 34 \estimate aperture e x i t none } d e f i n e sasp_fec_gas{ d r i f t \gas aperture entrance none length 1.27 radlength 16930 \isobutane <B STP (should to be 1/3 atm) aperture e x i t none APPENDIX K. EASY'S SASP MODEL 246 d e f i n e sasp.fec_gas_bulged{ d r i f t \gas aperture entrance none length 2.54 radlength 16930 \isobutane 0 STP (supposed to be 1/3 atm) aperture e x i t none def i n e sasp_fec_window{ d r i f t \mylar window aperture entrance r e c t -10.0 10.0 -10.0 10.0 length 0.00254 radlength 28.7 aperture e x i t none \ The SASP's FEC c o n s i s t s of f o u r wire planes seperated by \ gas and cathodes. The c e l l i s p r e s s u r i z e d at 1/3 atm and \ the windows bulge out 0.5". While the FEC i s r e a l l y a \ p r o p o r t i o n a l counter, the VDC pos sensor i s used because the \ PC option i s not yet implimented j u s t ingore the angle \ i n f o r m a t i o n , d e f i n e s a s p _ f e c _ c e l l { sasp.fec_window sasp_fec_gas_bulged sasp.cathode sasp_fec_gas pos.sensor -10.0 10.0 0.80 0.0 \x0 plane vdc 0.05 0.5 0.0 sasp.fec_gas sasp_cathode APPENDIX K. EASTS SASP MODEL 247 sasp_fec_gas pos.sensor -10.0 10.0 0.80 0.0 \x0> plane vdc 0.05 0.5 0.0 sasp_fec_gas sasp.cathode sasp_fec_gas pos.sensor -10.0 10.0 0.80 90.0 \y0 plane vdc 0.05 0.5 0.0 sasp_fec.gas sasp_cathode sasp_fec_gas pos.sensor -10.0 10.0 0.80 90.0 \y0' plane vdc 0.05 0.5 0.0 sasp_fec_gas sasp.cathode sasp_fec_gas_bulged sasp_fec_window \ The SASP's FEC c o n s i s t s of a d r i f t from the t a r g e t \ plane to the FEC, t r a n s p o r t through the FEC, and then \ a bactrack to the t a r g e t plane. This p o s i t i o n s the \ ray ready f o r t r a n s p o r t a t i o n through the SASP_0PTICS \ device, d e f i n e sasp_fec{ d r i f t \vacuum aperture entrance none length 50.0 radlength -1 aperture e x i t none s a s p _ f e c . c e l l backtrack 65.25778 } APPENDIX K. EASTS SASP MODEL 248 define sasp_vdc_vindow{ d r i f t \mylar window aperture entrance rect -100.0 100.0 -20.0 20.0 length 0.00254 radlength 28.7 aperture exit none } \ The SASP's VDCs consist of a window followed by g a s , a cathode \ plane, gas, wire plane (x), gas, cathode, wire p l a n e ( u ) , ga s , \ cathode, gas, window. The gas mixture i s 50*/, Ar and 50% \ isobutane. define sasp_vdc{ sasp_vdc_window sasp_vdc_gas_bulged sasp_cathode sasp_vdc_gas pos.sensor -90.0 90.0 0.80 0.0 \x plane vdc 0.015 0.5 28.0 sasp_vdc_gas sasp_cathode sasp_vdc_gas pos_sensor -100.0 100.0 0.80 30.0 \u plane vdc 0.015 0.5 28.0 sasp_vdc_gas sasp_cathode sasp_vdc_gas_bulged sasp_vdc_window } \ The SASP.STACK transports the ray starting at the exit of the \ dipole through the vacuum to the exit window of the SASP vacuum, \ to and through the 3 VDCs APPENDIX K. EASY'S SASP MODEL 249 \ the t r i g g e r paddles, and the 2 downstream s c i n t i l l a t o r s . \ The ray i s r o t a t e d to a frame where the VDCs appear h o r i z o n t a l \ are d r i f t e d through them, and then r o t a t e d back to the l a b frame \ (z p o i n t s v e r t i c a l l y up) to be d r i f t e d through the downstream \ s c i n t i l l a t o r s . d e f i n e sasp_stack{ d r i f t \vacuum aperture entrance none length 158.0 radlength -1 aperture e x i t none r o t a t e y -46.0 d r i f t \mylar vacuum window aperture entrance none \ change t h i s l e ngth 0.00254 radlength 28.7 aperture e x i t none \ change t h i s d r i f t \ a i r aperture entrance none length 1.3867767 radlength 30420.0 aperture e x i t none t r a n s l a t e 6.0468723 0.0 0.0 sasp.vdc d r i f t \ a i r aperture entrance none length 5.7023 radlength 30420.0 aperture e x i t none t r a n s l a t e 18.411728 0.0 0.0 sasp.vdc d r i f t \ a i r aperture entrance none length 5.7023 APPENDIX K. EASY'S SASP MODEL 250 radlength 30420.0 aperture e x i t none t r a n s l a t e 18.411728 0.0 0.0 sasp_vdc d r i f t \ a i r aperture entrance none length 25.78 radlength 30420.0 aperture e x i t none t r a n s l a t e 22.087778 0.0 0.0 s c i n t aperture entrance r e c t -105.0 105.0 -21.0 21.0 length 0.5 radlength 42.4 aperture e x i t r e c t -105.0 105.0 -21.0 21.0 t r a n s l a t e 0.51776515 0.0 0.0 r o t a t e y 46.0 d r i f t \ a i r aperture entrance none length 110.97782 radlength 30420 aperture e x i t none s c i n t aperture entrance r e c t -90.0 90.0 -22.0 22.0 length 0.5 radlength 42.4 aperture e x i t r e c t -90.0 90.0 -22.0 22.0 d r i f t \ a i r aperture entrance none length 59.5 radlength 30420 aperture e x i t none s c i n t APPENDIX K. EASY'S SASP MODEL 251 aperture entrance r e c t -100.0 100.0 -25.0 25.0 len g t h 0.5 radlength 42.4 aperture e x i t r e c t -100.0 100.0 -25.0 25.0 K.2 Data Acquisition File \ \ Standard EDAC d e f i n i t i o n s f o r SASP.DEF.DEV \ \ 25 Jan 1988 \ \ #address sasp. _xl» [sasp; sasp_vdc, 1; pos, .sensor,1] #address sasp. _ul= [sasp;sasp_vdc,1;pos .sensor,2] #address sasp. .x2= [sasp;sasp.vdc,2;pos .sensor,1] #address sasp. _u2= [sasp;sasp.vdc,2;pos .sensor,2] #address sasp. .x3= [sasp;sasp.vdc,3;pos .sensor,1] #address sasp. _u3= [sasp;sasp_vdc,3;pos .sensor,2] #address sasp. -tp= [sasp;scint,1] #address sasp. _sl= [sasp;scint,2] #address sasp. _s2= [sasp;scint,3] #address sasp. _x0= [sasp_fec;pos_sensor ,1] #address sasp_x0p=[sasp.fec;pos.sensor,2] #address sasp_y0=[sasp.fec;pos.sensor,3] #address sasp_y0p=[sasp.fec;pos_sensor,4] APPENDIX K. EASY'S SASP MODEL 252 •boolean s a s p _ v d c l _ f i r e d = ( s a s p _ x l && sasp_ul) •boolean sasp_vdc2_fired=(sasp_x2 && sasp_u2) •boolean sasp_vdc3_fired=(sasp_x3 && sasp_u3) •boolean sasp.fec_fired=((sasp_xO I I sasp_xOp)&&(sasp_yO I I sasp_yOp)) •boolean sasp_tp_fired=(sasp_tp) •boolean s a s p _ s c i n t s _ f i r e d = ( s a s p _ s l && sasp_s2) •boolean s a s p _ v d c s _ f i r e d = ( s a s p _ v d c l _ f i r e d && sa s p _ v d c 2 _ f i r e d && sasp_vdc3_fired) •boolean s a s p _ t r i g g e r = ( s a s p _ v d c s _ f i r e d && s a s p _ t p _ f i r e d && s a s p _ s c i n t s _ f i r e d && s a s p _ f e c _ f i r e d ) • f u n c t i o n sasp_vdcl2=vdc2(sasp_xl,sasp_ul,sasp_x2,sasp_u2; 17.78,3.17754,30,46,160,0) • f u n c t i o n sasp_vdcl3=vdc2(sasp_xl,sasp.ul,sasp_x3,sasp_u3; 35.56,3.17754,30,46,160,0) Appendix L EASY's MRS Model The complete E A S Y definition of the M R S spectrometer is kept in the two files: M R S - D E F . D E V M R S - D E F . D A C L . l Device File \**************************************************************** \ \ MRS Device Definition F i l e \ \ 26 July 1988 \ \ \ The geometry used i n this f i l e i s approximate only. \ \ \ To use this MRS def i n i t i o n , define the following device in \ your DEV f i l e : \ \ define mrs{ \ mrs_fec \ mrs.optics \ mrs.stack \ > 253 APPENDIX L. EASY'S MRS MODEL 254 \***************************************** d e f i n e mrs_gas_bulged{ d r i f t aperture entrance none length 2.54 radlength 34 aperture e x i t none } \estimate of 50'/,Ar and 50'/, isobutane mix at STP r a d l e n g t h d e f i n e mrs_gas{ d r i f t aperture entrance none length 1.27 radlength 34 \estimate aperture e x i t none } d e f i n e mrs_fec_gas{ d r i f t aperture entrance none length 1.27 radlength 16930 aperture e x i t none } \isobutane at STP radlength (NOTE that c o r r e c t pressure \ i s 1/3 atm). de f i n e mrs_fec_gas_bulged{ d r i f t aperture entrance none length 2.54 APPENDIX L. EASY'S MRS MODEL 255 r a d l e n g t h 16930 a p e r t u r e e x i t none } d e f i n e mrs cathode{ d r i f t a p e r t u r e entrance none l e n g t h 0.00254 r a d l e n g t h 28.7 ape r t u r e e x i t none } d e f i n e mrs ,vdc_window{ d r i f t a p e r t u r e entrance r e c t -52.5 52.5 -17.5 17.5 l e n g t h .00254 r a d l e n g t h 28.7 ape r t u r e e x i t none \ MRS VDC c o n s i s t s of 2 w i r e planes sandwiched between gas and \ cathode l a y e r s . d e f i n e mrs_vdc{ mrs_vdc_window mrs_gas_bulged mrs_cathode mrs_gas pos.sensor -50.0 50.0 0.80 0 \x plane vdc 0.015 0.5 28.0 mrs.gas mrs.cathode mrs_gas pos.sensor -50.0 50.0 0.80 30 \u plane vdc 0.015 0.5 28.0 } APPENDIX L. EASY'S MRS MODEL mrs_gas mrs_cathode mrs_gas_bulged mrs_vdc_window de f i n e mrs_fec_window{ d r i f t aperture entrance r e c t -4.0 4.0 -4.0 4.0 length 0.00254 radlength 28.7 aperture e x i t none \ The MRS FEC c o n s i s t s of 4 wire planes (2 x and 2 y) \ with gas at 1/3 atm. Using VDC pos sensor type i n s t e a d \ of PC because PC not implimented yet. Just ignore the \ angle information, d e f i n e m r s _ f e c _ c e l l { mrs_fec_window mrs_fec_gas_bulged mrs_cathode mrs_fec_gas pos.sensor -4.0 4.0 0.80 0.0 \x0 plane vdc 0.015 0.5 28 mrs_fec_gas mrs_cathode mrs_fec.gas pos_sensor -4.0 4.0 0.80 0.0 \x0' plane vdc 0.015 0.5 28 mrs_fec_gas mrs.cathode mrs_fec_gas pos.sensor -4.0 4.0 0.80 90.0 \y0 plane vdc 0.015 0.5 28 APPENDLX L. EASY'S MRS MODEL mrs_fec_gas mrs.cathode mrs_fec_gas pos.sensor -4.0 4.0 0.80 90.0 \y0' plane vdc 0.015 0.5 28 mrs_fec_gas mrs_cathode mrs_fec_gas_bulged mrs_fec_window } \ The complete MRS FEC device. D r i f t s the ray through vacuum \ to the FEC, through the FEC and then back to the t a r g e t plane \ so i t i s ready f o r t r a n s p o r t though the MRS_0PTICS. d e f i n e mrs_fec{ d r i f t \vacuum aperture entrance none length 56.0 radlength -1 aperture e x i t none m r s _ f e c _ c e l l backtrack 71.25778 \ The complete MRS detector stack. D r i f t s the ray from the \ e x i t of the MRS d i p o l e through vacuum, out the vacuum window, \ through the 2 VDCs, the t r i g g e r paddles (approximated as a \ s i n g l e s c i n t i l l a t o r ) and thourgh the downstream s c i n t i l l a t o r s . \ Note the Fo c a l Plane Polarimeter i s not implimented. d e f i n e mrs_stack{ d r i f t \vacuum aperture entrance none length 419.6 radlength -1 APPENDIX L. EASY'S MRS MODEL 258 aperture e x i t none r o t a t e y -45.0 d r i f t \vacuum window aperture entrance none length 0.00254 radlength 28.7 \mylar aperture e x i t none d r i f t \ a i r aperture entrance none length 1.4124175 radlength 30420 \ a i r aperture e x i t none t r a n s l a t e 5.2292135 0.0 0.0 mrs_vdc \VDC1 d r i f t \ a i r aperture entrance none length 23.952272 radlength 30420 \ a i r aperture e x i t none t r a n s l a t e 34.124973 0.0 0.0 mrs_vdc \VDC2 d r i f t \ a i r aperture entrance none length 10.0 radlength 30420 \ a i r aperture e x i t none t r a n s l a t e 16.3577 0.0 0.0 s c i n t \TP aperture entrance r e c t -60.0 60.0 -30.0 30.0 \garbo length 0.5 radlength 42.4 \polystyrene s c i n t i l l a t o r aperture e x i t none t r a n s l a t e 0.5 0.0 0.0 r o t a t e y 45.0 d r i f t \ a i r APPENDIX L. EASY'S MRS MODEL 259 aperture entrance none length 36.074388 radlength 30420 aperture e x i t none s c i n t \ S 1 aperture entrance r e c t -60.0 60.0 -30.0 30.0 \garbo length 0.5 radlength 42.4 \polystyrene aperture e x i t none d r i f t \ a i r aperture entrance none length 56.65 radlength 30420 aperture e x i t none s c i n t \ S 2 aperture entrance r e c t -60.0 60.0 -30.0 30.0 \garbo length 0.5 radlength 42.4 \polystyrene aperture e x i t none L.2 Data Acquisition File \ \ Standard EDAC d e f i n i t i o n s f o r MRS_DEF.DEV \ \ 25 Jan 1988 \ \ •address mrs_xl=[mrs;mrs_vdc,l;pos_sensor,1] •address mrs_ul=[mrs;mrs_vdc,1;pos_sensor,2] APPENDIX L. EASY'S MRS MODEL #address mrs_x2=[mrs;mrs_vdc,2;pos_sensor,1] •address mrs_u2=[mrs;mrs_vdc,2;pos_sensor,2] •address mrs_tp=[mrs;mrs_stack;scint,1] •address mrs_sl=[mrs;mrs_stack;scint,2] •address mrs_s2=[mrs;mrs_stack;scint,3] •address mrs_xO=[mrs_fec;pos_sensor,l] •address mrs_xOp=[mrs_fec;pos_sensor,2] •address mrs_yO=[mrs_fec;pos_sensor,3] •address mrs_yOp=[mrs_fec;pos_sensor,4] •boolean mrs_vdcl_fired=(mrs_xl && mrs_ul) •boolean mrs_vdc2_fired=(mrs_x2 && mrs_u2) •boolean mrs.fec_fired=((mrs_xO I I mrs_xOp)&&(mrs_yO II mrs_yOp)) •boolean mrs_tp_fired=(mrs_tp) •boolean m r s _ s c i n t s _ f i r e d = ( m r s _ s l && mrs_s2) •boolean mrs_vdcs_fired=(mrs_vdcl_fired && mrs_vdc2_fired) •boolean mrs_trigger=(mrs_vdcs_fired && m r s _ t p _ f i r e d ftft m r s _ s c i n t s _ f i r e d && m r s _ f e c _ f i r e d ) • f u n c t i o n mrs_vdcl2=vdc2(mrs.xl,mrs_ul,mrs_x2,mrs_u2; 34.124973,2.54254,30,45,424.9942859,0) Bibliography [1] P.L. Walden, M . J . Iqbal Proceedings of the DASS/SASP (Dual Arm Spectrometer System/ Second Arm Spectrometer) Workshop TRI-86-1 December 1986 [2; [3: K [5. [6 [7; [s; [9 [10 [11 [12: [13 [14 Stanley Yen SASP as a Neutron Detector T R I U M F Design Note to be published Stanley Yen Optics of the Q-Q-Clamshell Second Arm Spectrometer T R I U M F 1987 Alan Otter Proposed DASS/SASP Clamshell Dipole TRI-DN-87-1 H.A . Enge in A . Septier Focussing of Charged Particles Vol II, Academic Press, 1967 K a r l L. Brown, Sam K. Howry TRANSPORT/860 A Computer Program for De-signing Charged Particle Beam Transport Systems S L A C Report No.91 July 1970 S. Kowalski, H .A.Enge. RAYTRACE 1986, M I T . See for example Richard L. Burden, J . Douglas Faires Numerical Analysis T h i r d Edi t ion, P W S Publishers 1985 H .A . Enge, Rev. Sci. Instr. 35, 278 (1964) Wi l l iam H. Hayt, Jr . Engineering Electromagnetics McGraw-Hi l l , 1981 C M . Braams Nuc. Instr. Meth. 26, 83-89 (1964) W . Rogowski, Arch. f. Elektrotechnik, 12 1 (1923) A . G . Drentje, et. al. Nuc. Instr. Meth. 133, 209 (1976) R . F . Herzog, Z. Physik 89, 447 (1934) 261 BIBLIOGRAPHY 262 [15] H. Hubner, H. Wollnik, Nuc. Instr. Metk. 86, 141 (1970) [16] F . James Rep. Prog. Phys., Vol 43, 1980 [17] J . M . Hammersley, D . C . Handscomb Monte Carlo Methods John Wiley & sons 1964 [18] See for example Wi l l iam H. Press et. al. Numerical Recipes Cambridge University Press 1986 [19] Fraser Duncan EASY, A Monte Carlo Simulator T R I U M F Design Note to be published [20] C . Kost , P. Reeve REVMOC A Monte Carlo Beam Transport Program T R I - D N -82-28 Feb. 20, 1984 [21] SASP Detector Array T R I U M F Design Note to be published [22] W . Bertozzi, et. al. Nuc. Instr. Meth. 141, 457 (1977) [23] TRIUMF Medium Resolution Spectrometer (MRS) Manual T R I U M F 18 June 1986, Revision 1.0 [24] K . L . Blomqvist Consideration for Application of the Energy Loss Principle to Single Arm Scattering Experiments Bates Linear Accelerator, Internal Report 82-3, March 1982 [25] A . G . A . M . Armstrong, C . J . Collie, N .J . Diserens, M . J . Newman, J . Simkin, C . W . Trowbridge GFUNSD User Guide July 1979 Rutherford Laboratory [26] J . Simkin, C . W . Trowbride Three-dimensional nonlinear electromagnetic field com-putations, using scalar potentials I E E Proc. Vol .127, P t . B , No.6, November 1980 [27] A lan M . Winslow Journal of Computational Physics 1, 149 (1967) [28] James B. Scarborough Numerical Mathematical Analysis P g . 410 T h e Johns H o p -kins Press 6th E d . 1966 [29] R . F . Holsinger POISSON Group Programs User's Guide F ield Effects Inc., 1981 

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