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Optical properties of 50 CM Browne-Buechner spectrograph Lee, Kai-Chung 1975

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OPTICAL PROPERTIES OF 50 CM BROWNE - BUECHNER SPECTROGRAPH by LEE KAI-CHUNG B . S c , U n i v e r s i t y of Wisconsin, 1973 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENT FOR THE5DEGREE OF MASTER OF SCIENCE i n the Department of P h y s i c s We accept t h i s t h e s i s as conforming to the re q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA December,1975 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced deg ree at the 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 , I a g r ee t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r ag ree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y pu rpo se s may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f t - * - " ^ The U n i v e r s i t y o f B r i t i s h Co l umb i a 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5 Date if XC/76 ABSTRACT A 50-cm Browne-Buechner broad-range spectrograph i s being used a t TKIOH.F to measure the production c r o s s s e c t i o n of low energy pions when H and other n u c l e i are bombarded by e n e r g e t i c protons. These pions are detected by a s c i n t i l l a t i o n counter hodoscope l o c a t e d on the f o c a l plane of the spectrograph. T h i s t h e s i s d e s c r i b e s the i n v e s t i g a t i o n of the o p t i c a l p r o p e r t i e s of the spectrograph,such as m a g n i f i c a t i o n , d i s p e r s i o n and a b e r r a t i o n . In order to measure these p r o p e r t i e s t o the accuracy d e s i r e d , i t was necessary to o b t a i n the complete f i e l d maps of the magnet of the spectrograph. In knowing the o p t i c a l p r o p e r t i e s , o n e can then o b t a i n accurate e v a l u a t i o n of the acceptance which can then be d i r e c t l y used to convert event r a t e s i n t o c r o s s s e c t i o n s . The f i e l d i n the magnet was measured at s e v e r a l l e v e l s of e x c i t a t i o n , u s ing both a NMR probe and a H a l l probe. F i r s t , the f i e l d was measured by a NMR probe at 2.5, 5 and 10 kG, and second, the f i e l d was re-measured using a H a l l probe at 5.0 and 10.0 kG. The f i e l d maps were used as data f o r the ray t r a c i n g programme to o b t a i n the complete determination of the magnet o p t i c s of the spectrograph. The f i t s showed t h a t f o r the c e n t r a l ray,the d i s p e r s i o n was 229. 4 cm/(Ap/p) (3%),the m a g n i f i c a t i o n was 2.38 (6%), and the t o t a l r e s o l u t i o n was 0.157 Ap/p (11.?). Also f o r the c e n t r a l r ay, the a b e r r a t i o n (the s i z e f o r a p o i n t source of AAA, monoenergetic p i o n s a t t h e f o c u s ) was 1.436 cm (15%). The v a l u e s i n t h e p a r e n t h e s i s are percentage d e v i a t i o n from t h e v a l u e s f o r a u n i f o r m f i e l d Browne-Buechner s p e c t r o g r a p h o f 50 cm e f f e c t i v e r a d i u s . For t h e n o n - c e n t r a l r a y o p t i c s , the 50 cm u n i f o r m Browne-Buechner model a l s o compared as f a v o r a b l y . Table of contents ^ Page Abstract JUL Table of contents -CV Acknowledgement \j Introduction 7 Chapter 1 Magnetic Field Measurements And Field Maps Of Spectrograph A) General Method 8 B) Field Measurements By NMR Probe 9 C) Field Measurements By Hall Probe 10 D) Comparision of Field Maps 75 Chapter 2 Theory Of Ray Tracing Programme And Physical Properties Of Spectrograph A) • •7G-b?ord'i-n&te• Sy•s^'em1s^^)f^Mag,h•ets:ri.ctio .^s 28 B) Ray"-"T-racihgpPr-ogSamme2.And Restriction - 28 C) Ma'gfiSM&^-ieMs^gSnsioia^.'^st ,. 36 D) Focal Plane Of Effective Radius = 52 cm Uniform Field 39 Magnet E) Focal Plane Of Measured Field Magnet 42 F) Magnet Centre And Effective Radius Of Spectrograph ' 45 Chapter 3 <DB'temi;ha-ti.drP O^Opm^a^'Wop'eVtie's- OfyrspecWbgraph A) Polynomial Fitting For Magnet Optics 55 B) Optical Properties Of Spectrograph 58 Chapter 4 Conclusions 75 • " Appendix,A.^ajEharov's'vRay Tracing-Programnie ^ Appendix B Optical-Prop.er^ 88 ; Bibliograhy 97 V ACKNOWLEDGEMENT I would l i k e to thank my thesis supervisors Professor Garth Jones f o r suggesting the experiment, and valuable c r i t i c i s m and advice on early part of research work i n measuring the magnetic f i e l d s of the spectrograph; and Dr. Patrick L. Walden for providing the Hall probe f i e l d data, valuable suggestions i n extending the magnetic f i e l d maps, obtaining the multiple regression f i t s and o p t i c a l properties of spectrograph, providing sources of documents related to the experiment, and countless hours involved in discussing the organization of t h i s thesis. I would l i k e to thank Dr. Richard R. Johnson f o r updating the CERN ray tracing programme; Miss Marj K i l i a n f o r several computing technigues; Mr. Al Stevenson for electronics work in maintaining the power supply to the spectrograph and help in operating several e l e c t r o n i c s instruments. I would l i k e to thank Dr. Tom Masterson for providing co-ordinates of the s c i n t i l l a t i o n counters i n the hodoscope box. I am most grateful to Mr. David Mieng Pai for providing the control cards of the UBC-FMT programme to print out t h i s t h e s i s . The f i n a n c i a l assistances of the National Research Council, TRIDMF and the University of B r i t i s h Columbia are very much appreciated. F i n a l l y , I am indebted to my parents and my wife for t h e i r t i r e l e s s encouragement and patience. INTRODUCTION A 50 cm pole face magnetic spectrograph has been assembled at TRIDMF on beam-line 1 between target position T1 and T2. This spectrograph, together with a f o c a l plane hodoscope consisting of 24 s c i n t i l l a t i o n counters and 2 transmission s c i n t i l l a t i o n counters w i l l be used to measure the production cross section of (P,TT) reactions on d i f f e r e n t targets. However, the derivation of the production cross section requires knowing the acceptance of the spectrograph, which necessitates the precise measurement of i t s magnetic f i e l d . From t h i s measured f i e l d used as input to the ray tracing programme, we can not only determine the acceptance of the spectrograph from the successful rays which pass through the magnet and are detected by the hodoscope elements, but we can also determine the o p t i c a l properties of the spectrograph as well. The spectrograph i s a Browne-Buechner type manufactured by Spectromagnetic Industries cf C a l i f o r n i a . A detailed discussion i s given i n references 1 & 2. This spectrograph, designed for precise measurement of momenta and in t e n s i t y of charged p a r t i c l e s from nuclear reactions, has pole faces with an e f f e c t i v e radius of 49.83 cm and gap width of 1.905 cm (See chapter 2 section F). It can be set at any magnetic f i e l d strength between 0 and 12 kG by varying the output of the power supply. Ihe iron of the magnet can be e f f e c t i v e l y 2 saturated to a f i e l d value of about 12 kG, as shown i n Fig 1. The magnet i s mounted on a rotating c a r r i e r . I t can be rotated from 35° to 145°, and from 215° to 325° with respect to the beamline di r e c t i o n . The l i m i t a t i o n i s set by the physical layout of the spectrograph and beamline. The physical structure of the magnet i s shown in Figs 2 and 3. The hodoscope box, containing 24 s c i n t i l l a t i o n counters, i s mounted on the top of the e x i t flange. The box and other structures surrounding the magnet are made from non-magnetic materials such as aluminum and s t a i n l e s s s t e e l . The position of the s c i n t i l l a t i o n counters along the f o c a l plane was determined for the i d e a l case of a Browne-Buechner spectrograph with an uniform f i e l d and an e f f e c t i v e pole face radius of 52 cmv I t was estimated by Browne-Buechner1 that the e f f e c t i v e radius of the magnet was approximately egual to the pole face radius plus one gap width. In our case t h i s i s 52 cm. In addition to the f o c a l plane counters, there were two more s c i n t i l l a t i o n counters, CO and C1, used for coincidence counting mounted above the 24 counters (see F i g . 3) . Evacuation of the magnet chamber, which i s continuous with the hodoscope counter box was required to reduce multiple scattering, energy loss, and nuclear absorption between the target and the hodoscope array at the f o c a l plane. A pressure of less than 100M was reguired i n order to enable c a l i b r a t i o n of the spectrograph with alpha p a r t i c l e s . 3 Magnetic f i e l d s t r e n g t h of spectrograph vs, current of power supply 20 40 60 80 Power supply current 100 120 (amps) 140 E i g . 1 4 50.56 Entrance port E x i t port C i r c u l a r pole face edge Pole face A l l u n i t s i n cm MO— centre of c i r c u l a r pole faces 0-— o r i g i n of H a l l probe map Fig.2 P h y s i c a l s t r u c t u r e of spectrograph and u s e f u l dimensions 5 H o r i z o n t a l F i g . 3:; Spectrograph and hodoscope box w i t h 24 s c i n t i l a t i o n counters and C-0 & C l coincidence counters 6 An o i l d i f f u s i o n pump coupled to a mechanical pump was used to obtain good pumping e f f i c i e n c y at these pressures. The spectrograph can accept pions from energies of 10 HeV to 160 MeV. The lower l i m i t i s imposed by the thickness of the s c i n t i l l a t i o n counters, and the upper l i m i t i s imposed by the maximun attainable magnetic f i e l d of the spectrograph. The v e r t i c a l angular acceptance of the spectrograph i s 15.6° which i s imposed by the size of the entrance flange and the horizontal angular acceptance i s 0.8° which i s imposed by the gap width between the pole pieces. For p r a c t i c a l reasons, the pole faces are shaped so that a magnet f i e l d exists only over the region where the p a r t i c l e t r a j e c t o r i e s w i l l e x i s t . The c o i l s of the magnet are cooled by water. Hhen the c o i l s were absorbing 3kW from the power supply, the temperature difference of the i n l e t and outlet water was 5.2°C. This i s within the s p e c i f i c a t i o n of the blue-print supplied by the manufacturer. There were two power supplies used for maintaining constant D.C. Current passing through the magnet. One was b u i l t at 0.B.C. and was used for the NMR probe measurement. The maximum power output was 7.5 kw" for 100 amps and 75 vo l t s . Its s t a b i l i t y was about 0.4%, and was water-cooled. The other one was b u i l t by Canadian Dynamics (Control D i v i s i o n ) L t d , Model No VCR50/100. The maximum power output was 5 kW for 100 amps and 50 v o l t s . It had a much better s t a b i l i t y (0.02%) than the U.B.C. made power supply. 8 Chapter 1 Magnetic Fi e l d Measurements And Fie l d Maps Of Spectrograph 11 General Method Because of the way the spectrograph was constructed, the only access to the pole face was either through the entrance port or the exit port. The pole faces were designed to he the walls of the vacuum box so that the vacuum box i s i n t e g r a l with the spectrograph, and cannot be removed to gain access to the pole faces. The purpose of t h i s feature was for the sake of increasing the horizontal acceptance of the spectrograph as the gap width of the magnet i s only 1.905 cm. Therefore, we had to measure the f i e l d i n two regions, one through the entrance port and one through the exit port. The resu l t i n g measurements had to be matched up i n the overlapping region to obtain the complete f i e l d map. Because the gap width of the spectrograph i s small (1.905 cm), i t was only possible to measure the f i e l d i n the median plane owing to the size of the probes. However, this i s s u f f i c i e n t since by design, the main entrance and exit rays are normal to the pole face edges. Therefore, the component of the fringe f i e l d which i s normal to the edge cannot contribute a s i g n i f i c a n t focusing or dafocusing e f f e c t at the edge of the magnet. Hence i t i s not necessary to measure, considering the accuracy desired, the off-median 9 plane fringe f i e l d . 11 Ii§ld Measurements Ej H I Probe The NMR system used was a model 3093 manufactured by the Alpha Division of Systron Dcnner. The system consists of 3 components, the resonance probe, the remote freguency converter unit and the d i g i t a l display. The resonance probe was mounted on an arm of length 82 cm. The probe i t s e l f measured 0.42 cm i n radius and 0.94 cm i n length; The remote unit transmitted the induced NMR s i g n a l to the d i g i t a l display. The display 1converted the resonance freguency of the probe into magnetic f i e l d i n kilogauss d i r e c t l y . The setting of the resonant freguency was e a s i l y done by tuning the o s c i l l o s c o p e display to be symmetrical on both sides of the central l i n e . In order to obtain the f i e l d map, the remote unit plus the probe arm was mounted on a wooden board on which was l a i d out a co-ordinate grid «ith a point to point seperation of 2 cm horizontal and 2 cm v e r t i c a l l y . The system i s shown diagrammatically i n Fig. 4. The probe arm and remote unit assembly was moved frcm grid point to grid point manually. In t h i s manner a complete f i e l d map could be obtained over the course of several hours. In a moderate inhomogenous f i e l d (most central portion-of the pole face; about 10 gauss/cm at a median f i e l d of 10 kG), the signal-to-noise r a t i o at the TO oscilloscope display was s u f f i c i e n t l y large that a resonance could be e a s i l y located. However, the signal could s t i l l be observed for an inhomcgeneity up to 25 gauss/cm. Ihe s t a b i l i t y of the power supply used for the spectrograph was inadequate (0.4SI fluctuation over several hours) for the precision desired i n the measurements. Thus, a secondary NME probe was inserted into the magnet (as shown i n Fig 4) for monitoring the power supply output. In t h i s way, a complete f i e l d map cculd be normalized to t h i s secondary NHB reading. £ f i e l d Measurement Ey. Hall Probe The NHE probe has'the advantage of providing absolute magnetic f i e l d measurements. It . i s incapable, however, of being used to measure the fringe f i e l d area because of the large f i e l d inhomogeneity there. Therefore, to obtain the complete f i e l d map of the spectrograph, we had to resort to a H a l l probe measurement; The Hall probe has the advantage both of being able to measure the f i e l d strength i n regions where the f i e l d i s non-homogenous and also of providing a d i r e c t measure of the f i e l d unlike BUS probes which require tuning of the resonance freguency for each reading. The l a t t e r property i s convenient for computer control. Their disadvantages, of course, involve the need for careful c a l i b r a t i o n and due concern for thermal and other ef f e c t NMR probe p o s i t i o n e d on g r i d p o i n t s d r i l l e d on board j power s t a b i l i t y c o n t r o l F i g . 4 Measuring boards w i t h NMR probes i n mounting p o s i t i o n 72 a f f e c t i n g r e p r o d u c i b i l i t y . The Hall probe used was a model BH-703 3D Hall-Pak manufactured by F.W. B e l l Inc. Columbus, Ohio. Its size was 0.41 cm by 0,41 cm. The maximum temperature dependence of the H a l l voltage was about -0.03SS/°C, the temperature dependence of resistance was approximately +0.15%/°C and the maximum temperature dependence for n u l l voltage was 0.5 V/°C. The operating temperature range was from -40 °C to 100 °C. The above information was obtained from the s p e c i f i c a t i o n sheet. The probe was mounted inside a Hall probe holder which measured 1.9 cm by 1,8 cm by 2.3 cm. The holder was i n turn fixed to a supporting arm of length 76.2 cm. Referring to Fig 5, for the measurement of the entrance port region, the supporting arm was d i r e c t l y mounted to a m i l l i n g table and moved in steps into the entrance'port. The probe can be inverted so that a bigger region can be measured near the top and the bottom of the entrance port ( see probe orientation 1 and 2 i n Fig 5). For the measurement of the e x i t port region, a brace was designed to s i t on the m i l l i n g table and to hold the supporting arm i n a v e r t i c a l l y downward di r e c t i o n into the e x i t port< see probe orientation 3 i n Fig 5). The- probe can also be inverted for covering more region as i n the entrance pert measurement. Mounting the probe i n t h i s way enabled most of the f i e l d region to be covered. However, there s t i l l remained an 13 important f i e l d - r e g i o n that could not be reached by either of the above mounting methods. In crder to reach t h i s region, the supporting arm was mounted into a 60 degree holder attached to the brace>(see probe orientation 4 i n Fig 5). Thus, in addition to the fringe f i e l d regions, the H a l l probe measurements provided readings f c r regions which the NMR probe was unable to reach. The measurements were made using the IBIUMF magnet measuring system 3 . This system consisted of a Hewlett-Packard computer(•type- 2116B ) , a magnetic tape deck and the m i l l i n g table mentioned above. This milling table which was under complete computer control , could be moved precisely i n 0.25 inch steps between each measurement. The Hall voltage plus the Hall current were read d i r e c t l y by the computer and were written onto a magnetic tape. This tape was compatible with the U.B.C IBM 370/168 computer system. After the-Hall voltages were obtained, they could be converted into f i e l d readings by the c a l i b r a t i o n of the Hall probe voltage. To do t h i s c a l i b r a t i o n ; the Hall 1 probe was placed adjacent-to a NMR probe i n a highly homogenous f i e l d , which i n t h i s case was the combination magnet on beamline 1 at TRIUMF. The magnetic f i e l d was then varied from 2 tc 10 kG, and the readings were taken from both probes at each f i e l d setting; The Hall voltage was f i t t e d to the NMR measurements using a least sguare f i t programme* involving a 125.5cm _ _._ „ .. _ End of brace supported by hydraulic table Milling table / / / / / / / i I I 78.7cm / / / / / 60 77/ • j Supporting ' '^f arm / < i .4 £-... ... ~E i Hall probe holder £\ 3 "1 V Position l is positive polarity Position 2,3& 4 are negative polarity (Refer to Table 1 for polarity f i t ) Fig. 5 TRIUMF magnet measuring system on computer controlled milling table i t 15 t h i r d order polynomial of the Hall voltage( see Table 1 ) . Since the probe had been inverted during measurement, c a l i b r a t i o n f o r both p o l a r i t i e s were needed.^ The difference i n the positive and negative voltage c a l i b r a t i o n was about 0.7%. Using these f i t t e d polynomials, the Hall voltages were converted into f i e l d values. For the Ha l l probe measurement, we used the TBIUMF commercial power supply mentioned i n the introduction. I t s s t a b i l i t y was about 1/5000, and as such i t was far better than the one used in the NMfi measurement. The power supply was continuously monitored by the secondary 8MB probe described i n the previous section. 21 Comparison Of F i e l d Haps The regions measured by the HMB.-a.nd Hall probes are shown in Fig 6 6 7 respectively. In order to compare the f i e l d maps at d i f f e r e n t f i e l d strengths, the contour maps were plotted with a same scale and same contour ' values. The contours were drawn in;steps of 0.25% of the f i e l d value at a reference point, and the f i e l d values are scaled (where appropriate ) so "that thei'samei f i e l d - v a l u e s ' a r e ^ shown at the -reference ^po'int ( see F i g s . 9, 11 & 1 2 ; ) . . For both the NBB and Hall probe f i e l d maps (Figs 8, 9, 11, 12, and 13), there were two reproducible bumps occurring in regions A S B (as shown i n Figs 6 S7). As a conseguence, 16 Table 1 Polynomial f i t f o r magnetic f i e l d from H a l l probe v o l t a g e of p o s i t i v e and negative p o l a r i t y ( 2 kG « B * 10 kG) C o e f f i c i e n t P o s i t i v e p o l a r i t y 3 f i t ( p j x 0,0149987 -0.0846634 0.576877E-4 -0.173283E-6 Negat i v e R S j a r i t y f i t (Nj) -0.668944E-2 -0.0853910 -0.604243E-4 -0.184892E-6 N./P. -0.446001 1.0085942 -1.0474389 1.0669956 where, 2 3 B = A + A, v + A. v + A „ v , o 1 2 3 where B i s the magnetic f i e l d i n kG & v i s the H a l l probe v o l t a g e (mV) 77 E x i t port Measuring grid separation = 2 cm F i g . 6 F i e l d measurement by NMR probe Exit port Measuring grid seperation =0.635 cm L0.0914 cm .0.5359 cm J { Entrance port Center of magnet — .1.43 cm below ' o ' o Spacers F i g . 7 F i e l d measurement by H a l l probe 0 — Measured region Extended region H a l l probe -measuring o r i g i n F i g . 8 H a l l probe f i e l d map measured at 5 kG F^g. 9 H a l l probe f i e l d map measured at 10 kG 20 F i g . 8 Contour . sep a r a t i o n = F i e l d values i n kG 12.5 gauss 27 F i g . 9 G r i d contour separation = 25 gauss F i e l d values i n kG. — j — Reference point f o r f i e l d maps comparsion 22 Extended | f i e l d F i g . 1.0 Extended H a l l probe map at 10 kG. (Nature of f i e l d extension described i n Chapter 2 s e c t i o n B) 23 F i g . 11 Normalized NMR f i e l d map at 10 kG. F i g . 12 Normalized NMR f i e l d map at 5 kG. F i g . 13 Normalized NMR f i e l d map at 2.5 kG. 24 25 F i g . 12 Contour separation 12.5 gauss F i e l d values i n kG 26 F i g . 13 Contour s e p a r a t i o n 6.25 gauss F i e l d values i n kG 27 the higher energy p a r t i c l e s which traverse these regions w i l l he bent more than in the uniform f i e l d case. This w i l l increase the energy range acceptance of the spectrograph, but at the same time t h i s produce aberrations i n the focus, which i s not a very desirable e f f e c t . From the 5 and 10 kG Hall probe maps (Fig 8 & 9), note that the magnet tends to be less uniform in the central exit f i e l d region at lower f i e l d strength. This very same c h a r a c t e r i s t i c i s also observed i n the NHE maps. The e f f e c t i s e s p e c i a l l y pronounced in the f i e l d map at 2.5 kG (see Fig 13) . 28 Chapter 2 Theory Cf Hay Tracing Programme find Physical Properties Of Spectrograph A £2zO£dinate systems of magnet There w i l l be 2 co-ordinate systems referred to in t h i s thesis (see Fig 14):- one i s the 45 degree t i l t e d system (Z,X) described i n section B \, the or i g i n of t h i s system (0) was taken a r b i t a r i l y at the grid point shown i n Fig 14. The second system was the magnet center system(X ,Y ),which had an o r i g i n (MO) at the centre of the magnet defined by the c i r c u l a r pole pieces. This system had the same axis orientation as the magnet f i e l d maps. The or i g i n of the magnet centre system was at (36.0890,-2.3034)cm with respect to the 45 degree t i l t e d co-ordinate system. The conversions between the 2 systems were:-2 ^  ( K -4- y') - I-3 of • X - cy - * ) / t-"f Oh" - "*><>• oZq • *' = ( f- - x, - 13-j^r? ) / |.*»* • . ; ; : , , (2.2) y' - (~z t * ' + i 8-11 21 Say Tracing, Programme And Restrictions The programme orginated from the CERN momentum f i t t i n g programme i n analysing spark chamber data. I t was o r i g i n a l l l y written by Zacharov 5, 6 and then up-dated by Johnson 7. The Detectors 30 basic idea of the programme st a r t s from the Lorentz eguation 5 = e A'F =he% * 8 (2.3) where F i s the force on charged p a r t i c l e with charge e t r a v e l l i n g with velocity v i n the magnetic f i e l d B i n the (z,x) plane (see Fig. 14). If we consider the x and z components of (II. 3), we have ,v L- I*. a ( v, 6 - 8 v ) and Since we are only interested i n the position dependence and not the time development of the p a r t i c l e trajectory, the time dependence i s eliminated by equating time derivatives i n terms of motion along the z-axis, which i s assumed to be the main input d i r e c t i o n of the p a r t i c l e s . He then have and likewise, from (2.7), we obtain 37 H - * X V/ (2.8) and substituting i n (2.4) and (2.5), we get Introducing the t o t a l v e l o c i t y , v„ = (v* + vy* + v* ) > l and curvature parameter k=eB„ /(pc), where B0 i s the constant f i e l d i n the region and p i s p a r t i c l e momentum, we obtain which i s S i m i l a r l y , f o r di r e c t i o n p a r a l l e l to the pole face edge, we (N.B'.S.Trkel;a'b.oye d e r i v a t i o n persentes i s mainly based on rion-relativistic dynamics. However, the reader can e a s i l y show that t h i s i s also true (^e§atfv^s©iWlily-?)*iv&" m j-eseaLed ±C b ' S ^ w f wsv •<••<. - ,, dynamos. How?.'--, cba reader cau tr. a l l y show . • c'.» L i v i s t i c a l l y . ) He now : have a second order d i f f e r e n t i a l eguation with the time parameter removed. The ray tracing programme 32 i n t e g r a t e s (2.7) by choosing the s e t of s i x i n i t i a l parameters, namely x ,y and z i n order t o d e f i n e the f i e l d and x ,y and k. Then Bunge-Kutta i n t e g r a t i o n i s used to n u m e r i c a l l y e v a l u a t e these second order d i f f e r e n t i a l e q uations. From the i n p u t mesh of data,(see next s e c t i o n ) , the h o r i z o n t a l c o o r d i n a t e (z) i s t r e a t e d as independent v a r i a b l e f o r the numerical i n t e g r a t i o n . A l l the remaining f i v e v a r i a b l e s are e v a l u a t e d at each value of z. The above case i s the g e n e r a l three dimensional approach. However, f o r our case, only the planar motion of t r a j e c t o r i e s of p a r t i c l e s are d e s i r e d s i n c e we only measured the By component as d e s c r i b e d i n Chapter 1 s e c t i o n A. He can then s i m p l i f y the above equations t o a 2 dimensional case with the x-z plane as the p r i n c i p a l bending plane, and B as the p r i n c i p a l component of the magnetic f i e l d a c r o s s the pole f a c e s . Thus we have 6, » 5e y >" = y = o (2.10) which then reduces (2.9) to (2.11) (N.B. However the programme uses 2.9 but with 2.10 imposed) 33 However, the use of t h i s programme was not e n t i r e l y free from d i f f i c u l t i e s . The programme had 2 r e s t r i c t i o n s : - (1) The gradient ( x ) of the t r a j e c t o r i e s must be f i n i t e at any point ( that i s , the trajectory must not make an angle of 90 degree with respect to the z-axis }. (2) The r a t i o of the s p a t i a l separation of the v e r t i c a l grid points to the s p a t i a l separation of the horizontal grid points had to be greater than any p a r t i c l e trajectory gradient (x ). That i s . G r i d r a t i o V e r t i c a l s p a t i a l g r i d s e p e r a t i o n H o r i z o n t a l s p a t i a l g r i d s e p e r a t i o n ^ T r a j e c t o r y gradient C r i t e r i o n (1) was s a t i s f i e d by simply r o t a t i n g the magnetic f i e l d by 45 degrees. Then the input ray could have a slope of -1 and the output ray of slope +1 (that i s , a 90 degree v e r t i c a l bend without having the slope going to i n f i n i t y , see Fig 14). But, since the system was to be rotated by 45 degrees, f i e l d data had to be juggled to f i t the new t i l t e d g r i d system. This was done as follows, r e f e r r i n g to Fig 15, the dotted l i n e s represent the o r i g i n a l measured g r i d which has 0.25 i n between each point.- Note that these points f i t exactly onto every other point of a rotated g r i d represented by the s o l i d l i n e s . If we choose t h i s 45 degree rotated system, we have an unmeasured point, eg B5 which i s . ^ adjacent to 4 measured points, eg B1, B2, B3 and B4. Therefore, B5 can be assigned a f i e l d value by simply X Old g r i d (Dotted l i n e ) New g r i d ( r otated by 4 5°)) ( S o l i d l i n e ) -\ ( y{ S y S / t S. V \ N r } i s S \ \ N V . V \ / y y / r \ V V •' X V \ \ / / • • s f ^ \ \ \ / y y y y / V \ \ \ V y / y * y / N \ V \ V X 7 (-<r \ V \ v ? • / / / / / k \ \ V \ \ V \ ) s • • / \ \ X \ \ \ • / / * * / r ( ^ B2 k \ \ V V N B5 > y y r / C B \ \ V V N • / BB 1 V V \ \ • y • y y y y > \ pig 15 Jugg l i ng of f i e l d data i n to 45° t i l t e d co -o rd i na te system 35 averaging the f i e l d values of the measured points around them. Note that by rotating the grid system i n t h i s manner, we have doubled the number of the f i e l d grid points. Note that the grid r a t i o of t h i s 45 degree t i l t e d system equals 1. This means from c r i t e r i o n (2), we can only successfully integrate a p a r t i c l e t rajectory whose- f i n a l or i n i t i a l slope i s less than one. From inspecting the t r a j e c t o r i e s i n the t i l t e d system, i t was r e a l i z e d that approximately half of the t r a j e c t o r i e s that successfully passed through the spectrograph had output gradients greater than 1. Thus the grid r a t i o had to be increased i n order to determine the t r a j e c t o r i e s of these' p a r t i c l e s . This was simply done by eliminating every alternate horizontal row of the 45 degree rotated system so that the grid r a t i o became 2. In t h i s way, we could obtain any trajectory whose absolute value of gradient did not exceed 2. However, there were s t i l l a few t r a j e c t o r i e s with output gradients s l i g h t l y greater than 2 occurring near the low energy focus. Since these t r a j e c t o r i e s were only a small f r a c t i o n of the t r a j e c t o r i e s i n t h i s region, they were - not important; There was s t i l l one more t r i v i a l requirement to be met before Zacharov's programme could be implemented. As mentioned previously, the programme integrated t r a j e c t o r i e s in 3 dimensional space, therefore a 3 dimensional magnetic f i e l d grid had to be presented to the programme. This was 36 simply done by duplicating the existing 2 dimensional grid system and placing i t on both sides of the o r i g i n a l g r i d . Note that the l a p l a c i a n eguation f o r the magnetic f i e l d i s not used as we assume an uniform f i e l d d i s t r i b u t i o n between the small gap of the magnet. C]_ Magnetic F i e l d Extension' The f i n a l problem i n determining t r a j e c t o r i e s f o r d i f f e r e n t energies was that some t r a j e c t o r i e s passed through regions i n which neither NMS cr Hall probe f i e l d measurements existed. This was due to the access problem mentioned i n Chapter I section A wherein i t was not physically possible to place a probe i n these regions. In such cases, the f i e l d was extended by using polynomial f i t s to the measured data. The areas where the f i e l d had to be extended are shown i n Fig 10. For areas I and II, cubic f i t s were made to the f i e l d data of the form:-6 - A,M 4 y + d,(*> yl + /V«;J ( 2. 1 2 ) where B i s the f i e l d value i n kG of a v e r t i c a l l y arranged column of grid points, y i s the v e r t i c a l position i n the f i e l d map, and A. (x) are the f i t c o e f f i c i e n t s at a given h o r i z o n t a l ' f i e l d map position, x. Using these f.its> the f i e l d could be extended into the un-measured v e r t i c a l y regions at 37 each given position,x; The order of the polynomial f i t was chosen so that a good f i t to the measured data was obtained. In order to be reasonably sure that the f i e l d extension was correct, i t was demanded that the resultant extended magnetic f i e l d had to look physically reasonable (see Pig 13) . Simil a r l y for area I I I , the same procedure was used except that the f i t s were guadratic, and the f i t d i r e c t i o n was along the horizontal instead of the v e r t i c a l d i r e c t i o n , that i s , the the polynomial f i t for a given horizontal arranged row of data becomes 6 = A, <y> + A,*/) * + Vy> *' (2.13) where B i s the f i e l d value in kG of horizontally arranged row of grid points;x i s the horizontal position i n the f i e l d map, and A^ (y) are the f i t c o e f f i c i e n t s at a given v e r t i c a l f i e l d map position y. In order to be reasonably sure that the f i e l d extension was correct i n t h i s region, the resultant extended magnetic f i e l d was compared with the NMR maps, which i n t h i s region covered a larger area. No contradiction between the 2 contour maps was found, and both maps looked reasonably close (see Figs. 10 5 11). A check on the accuracy of these f i t s could also be found by inspecting the measured f i e l d s with the extrapolated f i e l d values(see Fig 16). Also, we can use the chi-square © NMR reading -4-Hall probe reading G r i d p o i n t s F i g . 16 • 39 method to t e s t f o r the adequacy of the a n a l y t i c f i t i n the region where the f i e l d value i s measured' . . This* t e s t showed- t h a t we are i n a l e v e l o f conf i d e n c e b e t t e r than 99%. T h i s extended f i e l d data was then duely j u g g l e d i n t o the 45 degree t i l t e d c o - o r d i n a t e system. This map now covered a l l a v a i l a b l e r e g i o n s between the pole f a c e s . During i n t e g r a t i o n i n the Zacharov's programme, a tra c k which entered an unmeasured r e g i o n was now co n s i d e r e d to have h i t an o b s t a c l e and the ray t r a c i n g i n t e g r a t i o n was terminated immediately. 2 , ZfiSSl Plane At E f f e c t i v e Radius = 52 cm Uniform F i e l d Magnet The ray t r a c i n g programme was t e s t e d by- producing t r a j e c t o r y f i t s of p a r t i c l e s i n an uniform f i e l d of a browne-Buechner spectrograph and using them t o d e f i n e o p t i c a l p r o p e r t i e s which could be compared to the f i r s t order a n a l y t i c a l s o l u t i o n given by Browne-Buechner 1. The f i r s t t e s t c o n s i s t e d of o b t a i n i n g the c e n t r a l pion ray f o r the 10 kG f i e l d ( i . e . the r a y which e n t e r s h o r i z o n t a l l y at 0-degrees and comes out v e r t i c a l l y , making an angle of 90 degrees with the h o r i z o n t a l a x i s ) * T h i s was ob t a i n e d by t r i a l and e r r o r of observing s u i t a b l e t r a j e c t o r i e s . The energy of the c e n t r a l pion obtained i n t h i s manner was 69.74 Me? corresponding t o a momentum o f 156.0 MeV/C. T h i s agrees with t h a t expected from the r a d i u s of 40 c u r v a t u r e f o r the r e l a t i o n P ( X v/c) = 01 * 8(k<*) ' R^<«») ( 2 . 1 4 ) where E e ^ was chosen to be 52 cm( chosen because p o l e face r a d i u s p l u s 1 gap width i s 52 cm, see i n t r o d u c t i o n ) . The second t e s t c o n s i s t e d of comparing the f o c a l plane -obtained from rays produced by the ray t r a c i n g programme to the f o c a l plane obtained from the a n a l y t i c a l eguation given i n Browne-Buechner's paper 1. The q u a l i t y of the f o c a l plane obtained i s shown i n F i g 17 where the s o l i d l i n e i s e x p r e s s i o n (2.15) and the c r o s s e s are the i n t e r c e p t i o n s of r a y s e n t e r i n g the spectrograph at d i f f e r e n t angles. The r a y s u s e d f o r t h i s t e s t o r i g i n a t e d from a p o i n t source l o c a t e d at 2 e f f e c t i v e r a d i i (104 cm) from the magnet c e n t r e as r e g u i r e d f o r a Browne-Buechner s p e c t r o g r a p h 1 . They were generated at angles±0.04, .±,0,12 ,and 0 r a d from the h o r i z o n t a l d i r e c t i o n . Note t h a t i n F i g 17, the r a y s do not i n t e r c e p t at a p o i n t . T h i s i s mainly due to the f a c t t h a t the sudden i n c r e a s e or decrease i n magnetic f i e l d at the edge of the uniform f i e l d i s not adeguately t r e a t e d by the programme (2.15) ( With o r i g i n at magnet centre ) Focal plane of uniform f i e l d w i t h R f ~ =52 cm 4.0 50 6 0 70 80 90 100 (cm) 45 deg t i l t e d . co-ordinate system (- b e e e 9 t n - ( 2 . 1 ) ) F i g - 17 42 due to the inherent coarseness of any magnetic f i e l d map grid . The point to notice in Fig 17 i s that the scatte r of f o c i follows the f o c a l plane calculated from (2.15). This observation was taken as proof that the programme was working. I Focal Plage Of Measured F i e l d Magnet After being s a t i s f i e d that the programme was working properly, we went to the case of the physically measured f i e l d and tested that the programme handled the f i e l d map co r r e c t l y . For the f i r s t t r i a l , rays were again generated from the point source located at 104.0 cm as before, assuming we had a 52 cm spectrograph. However, the f o c i of the intercepting output rays did not l i e close to the B a f ( =52 cm a n a l y t i c a l f o c a l planei I t lay closer to that of a B c f ( = 50 cm spectrograph instead. Therefore, the point source was moved to 100.0 cm from the magnet centre, and rays were generated as before. The resultant f o c a l plane using the interception of these rays are shown in Fig 18* The s o l i d l i n e in that figure i s the B ^ = 50 cm Browne-Buechner a n a l y t i c a l focal plane. Note that because the f o c i are too close together, only the interceptions of the ±0.12 rad rays and the horizontal rays are shown i n Fig 18. The *x' points on t h i s figure^are the interceptions of the horizontal and -0.12 rad rays. Likewise, the »*» points are that of the ± Focal plane of measured H a l l probe f i e l d Browne-Buechner a n a l y t i c a l f o c a l plane R r-r =50 cm 20 30 40 50 60 70 80 90 100 110 120 130 140 150 (cm) 45 deg t i l t e d '"' " ~" ' 7-co-ordinate system ( S e e e 9 t n - (2.1)) F i g . 18 44 Oi12 rad rays and the points are that of the horizontal and +0.12 rad rays. Due to inherent aberration e f f e c t s of the spectrograph, these ray interceptions do not f a l l on the f o c a l plane. An i n t e r e s t i n g pattern was noticed for these interception points. At the low energy region, the pattern was - * which reversed i t s e l f at the central ray f o c i to + become * at the high energy region ; To make sure such a pattern was r e a l and not an a r t i f i c i a l r e s u l t of the ray tracing programme, P.Halden 8 solved a n a l y t i c a l l y the t r a j e c t o r i e s for rays 1, 2 S 3 i n an uniform f i e l d magnet and obtained the same pattern along the f o c a l plane; Notice that the aberration or spread of f o c i about the f o c a l plane i n Fig 18 are much more closely packed together than those formed in the uniform f i e l d case (Fig 17). This was due to the fact that the sharp edges of the uniform f i e l d were replaced by the smooth edges of the'physically measured f i e l d , and these edges were adequately treated by the s p a t i a l resolution of the f i e l d map which i s not the case of the uniform f i e l d . From the nicely packed nature of the f o c i , i t was assumed that the programme handled the r e a l measured f i e l d map cor r e c t l y . Since the point source was just an i d e a l case, we ran the case of a source with the approximate dimension of a beam spot at the target. Thus, the object was treated as a 2 dimensional square of length 1 cm and s i t e d at 100.0 cm from 45 the magnet centre. Bays of d i f f e r e n t gradients (+0.12 rad and -0.12 rad to horizontal) were generated at the 1 corners and centre of the square. The r e s u l t i n g images formed around the f o c a l plane of the spectrograph are shown in Figs 19 S 20. From Fig 20, one can see from the images that the magnification e f f e c t i s l e s s pronounced i n the c e n t r a l energy region than i t i s in the higher energy region. Note that the image of our 1 cm square pion source has a span of about 2 cm in horizontal and v e r t i c a l d i r e c t i o n s . Knowing t h i s , we can see that i f the hodoscope counters are placed anywhere within this s p a t i a l l i m i t , the resolution of the spectrograph w i l l not be compromised. F Magnet Centre And E f f e c t i v e Radius Of Spectrograph The magnet centre was f i r s t assumed to l i e on the horizontal axis of the Hall probe co-ordinate system ( see Figs. 21 6 2). Since the f o c i of the p a r t i c l e s ( as discussed i n section E ) corresponded to a R€^ = 50 cm magnet instead of the expected Re^ = 52 cm magnet, i t was conjectured that there was probably some misalignment of the pole pieces with respect to the centre of the entrance flange. In order to determine the- misalignment of the pole pieces, a f i x i n g of the extremities of the pole pieces(points A •& B of Fig 21) with respect to the f i e l d map co-ordinate system had to be done. This was obtained by the following F i g . 19 Image formation of 1 cm object on Browne-Buechner spectrograph 47 (cm) 37 35 33 Image formation of 1 cm by 1cm object at 10 kG along a n a l y t i c a l f o c a l plane Pion energy=64.75Mev A n a l y t i c a l f o c a l plane 64 • 45 t i l t e d co-ordinate system 66 68 70 (cm) 72 F i g 20 E x i t f l a nge 48 r 10± (0') O r i g i n df H a l l probe co-ordinate (0,0)1 Centre of Beamline I(BLI) B ;..JCX0 ,48.58) A C10.-16.Y.) Assumed magnet centre * ; (CM) J X i ^ Real magnet centre 49.18 C X o , Y 0 ) 0.81 Entrance flange (1ALL UNITS IN CM ) From survey 0' i s 0. 62 above BLI CM i s . i . 4 3 . below 0' Thus CM i s 0.81 below BLI 'ig . 21 E x t r e m i t i e s at entrance and e x i t flanges and magnet centres of c i r c u l a r pole faces 49 p r o c e d u r e . From the b l u e - p r i n t of the spectrograph and independent measurements, i t was known that the pole face began 10.16 cm i n s i d e the i n n e r edge of the entrance f l a n g e and a l s o 10.16 cm i n s i d e the i n n e r edge of the e x i t f l a n g e . Thus, we are given x = 1 0 . 1 6 cm f o r point A and y =48.58 cm f o r p c i n t B(see f i g 21)."In order to determine x of p o i n t B, the f i e l d value B was p l o t t e d a g a i n s t x at y = 48.58 cm. T h i s i s shown i n F i g 22-b. The maximum f i e l d value i n d i c a t e s the ex t r e m i t y of the pole f a c e at the e x i t f l a n g e . S i m i l a r l y , y o f p o i n t A was determined-by p l o t t i n g B vs. y at x = 10.16 cm (see F i g . 22-a). Thus, we obt a i n the p o s i t i o n of p o i n t A ( 10.16, -1.43 cm ) , and p o s i t i o n o f point B ('60* 18 48.58 cm ) . From t h i s , one can see t h a t the magnet c e n t r e i s • mi s a l i g n e d by 1.43 cm below the h o r i z o n t a l a x i s of the H a l l probe map. He can a l s o confirm t h a t the r a d i u s of the magnet poles i s indeed 50.0 cm. as Radius of pole face at entrance = x(A) - x(B) = 60.18- 10.16 = 50.02 cm Radius of pole face at exit =y(B) - y(A) = 48.58- (-1.43).= 50.01 cm . Since we know q u a l i t a t i v e l y the the e f f e c t i v e r a d i u s was about 50 cm,and now knowing the exact p o s i t i o n of the magnet c e n t r e ; a d i r e c t measure of the e f f e c t i v e r a d i u s ; B ^ co u l d be done. T h i s was achieved by e v a l u a t i n g the i n t e g r a l (kG) 8.5 ^ F i g 22-a B vs Y. atox =10.16 cm (Entrance f l a n g e ) 1 1 ^ ^~—i 1 1 1-, 56 58 60 62 64 66 68 X ( c m ) F i g 22-b B vs.. X, at_ Y .= 48.58 cm •.••(Exit flange) / bd •J? where B i s the magnetic f i e l d strength along d 1 and B' i s the e s s e n t i a l l y constant" f i e l d s t r e n g t h w i t h i n the magnet pole faces , _ and the l i m i t s are taken from the point p • . of strength E' to an arbitrary point f where the f i e l d i s zero. This integral was done for both the entrance and e x i t port. However, we only had f i e l d data up tc the entrance and ex i t flanges, where B>0, then we extrapolated the f i e l d data beyond the measured region by an exponential f i t . Sith the f i e l d map and the f i t , the integral was computed by Simpson's Sule to obtain the ef f e c t i v e r a d i i . For the entrance port, B' =10. 098 kG and R E F F =51,51 cm; for the exit port, B ' =9.978 kG and R e W =51.26 cm. (see Fig 23-a) . To account for the position of the f o c i formed around the f o c a l plane (see section E, Be^ ~> 50 cm), we note that the dominating e f f e c t i v e radius i s that along the exit aperture (see Fig 23b )• As stated before, the position of the magnet centre i s 1.43 cm. below the centre of the entrance aperture,therefore, the resultant e f f e c t i v e radius of the magnet R < W = (51. 26-1. 43) =49. 83 cm which accounts for the observation- i n section E.)p thus, we had a spectrograph of pole face radius 50.0 cm, gap width 1.905 cm, and e f f e c t i v e radius 49.83 cm. The e f f e c t i v e radius was not larger than the Magnet f i e l d i c (kG) 52 Magnet i c f i e l d (kG) E f f e c t i v e radius of magnet at e x i t port ^ , _ j — . : 1 — i (i 1—'. i— ; f 48 ~" 50 : - 52 7 , : 54 7 "56 58 60 (cm) Distance from magnet centre F i g . 23 a 53 Magnet pole pieces - - - - - 6 W ~.:f ^ S i g n i f i c a n t change i n t r a j e c t o r y d i r e c t i o n as change i n v e r t i c a l \ i E f f e c t i v e \ ^ ' r a d i u s of magnet \ Assumed magnet centre Displaced magnet centre \ . \ * \( v t / . i / i i I 'Real r a d i u s / F i g . 23rb E f f e c t of v e r t i c a l displacement of magnet centre on i n f e r r e d e f f e c t i v e r a d i u s . p o l e f a c e r a d i u s a s t h e magnet c e n t r e i s 1.43 cm t o o 55 Chapter 3 Determination Of Optical Properties Of Spectrograph II PollSomial F i t t i n g for Macjnet Optics I f t e r f i n i s h i n g the f o c a l plane analysis, we went on to determine the o p t i c a l properties of the spectrograph. A set of 1500 random rays in the x-z plane were traced through the spectrograph using the ray tracing programme. For each ray generated, the following quantities were stored on a computer disc f i l e : Z,9 the o r i q i n of the ray at the target and i t s i n i t i a l slope C the position of the ray at the entrance flange D the position of the ray at the f o c a l plane Q the momentum parameter of the ray which i s defined by where P i s momentum, and P„ = 0.3*B*fi { Refer to equation 2.14 ) and P0 i s 149.9178 fteV/C for the 10 kG measured f i e l d . The quantities Z, 0, C, D are shown i n F i g . 24. To get a best evaluation of the o p t i c a l properties, polynomial f i t s r e l a t i n g the above various quantities were obtained. S p e c i f i c a l l y , i t was desirable to have P = P. c i + a ) (3.1) D D ( H , e, cn (3.2) 56 F i g . 24 Parameters f o r m u l t i p l e r e g r e s s i o n programme 57 Table II Polynomial coefficients of parameter f i t s by Multiple Regression Programme D = D (z,e,Q) Q = Q (Z,C,D) Pr oducts Term:i Z ez Q ee QQ zzz eee QQQ QQe zz zeq z'e ZQ 6Q ZZ8 ZZQ eez eecj Qoz Coefficient(C i) 2.37725 0.35370 222.93933 0.10082 304.55273 0.09076 361.93604 321.76587 158.69020 -0.147,26:; 40.20866 4.87232 10.24690 54.58368 3.11941 1.30355 18.83777 4144624:27/7 16.34491 Producte.TiefmX T . ) X X-t. z c D DD zzz ccc DDD DDC zz ZCD ZC ZD CD ZZC ZZD CCZ CCD DDZ Coefficient(C ± ) x i o ' -108938.31250 -2737.81641 45160.77344 714.00610 -268.00903 8744.19141 -147.99379 1.16815 1.01421 3734.31104 -44.52469 -2814.69287 -640.35132 -176.45595 -2828.36426 92.58025 676.92529 -49.30330 -5.92782 constant -0.82828 constant 34097.22656 * 2 Product Term = Product of listed pararmeters eg. ZZQ== Z Q Example: Q = i[ T cbristar.t• )\ D ='E T. C. + gonstanti) i 1 1 ' ' " 5S A N D Q -- & (2, Cj o) (3.3) expressed as a polynomial f i t of the variables shown. The multiple regression programme (BMD:02R) i n the OBC computer centre was used f o r t h i s purpose. This programme computes a seguence of multiple regression equations in a stepwise manner, at each step, one variable ( example Z, C, D as i n equation 3.3 or any combination of them ) i s added to the regression eguation. This variable added i s the one which makes the greatest reduction i n the error sum of squares. Eventually, tftei'dependent variable(example-a Q i n equation 3.3 ) i s expressed i n terms of the independent variables or combinations of them. The r e s u l t s of the f i t s are shown i n table I I . SL Optical Properties Of Spectrograph From the polynomial f i t s obtained in the abive section, we derived the following o p t i c a l properties: I)Magnification Magnification occurs due to difference i n object distance and image distance from the spectrograph. This phenomenon i s defined by the r a t i o of the size of the image 59 on the f o c a l plane to the size of the object, which leads to S i z e of image M a g n i f i c a t i o n Size of object which i n our case as having D = D( Z, 6, Q), we o b t a i n M a g n i f i c a t i o n = -^ 5-o-Z Z=6=0 and evaluating t h i s polynomial f i t D = D(Z,9,Q), we obtain M a g n i f i c a t i o n -r^„.^0 2-3 ? +• to-2f U-i*Gr (3.4) This expression (3.4) i s plotted with crosses i n Fig 25 with the uniform f i e l d case shown as a s o l i d l i n e . Note i n the figure that magnification i s s l i g t h l y higher than the uniform f i e l d case around the central energy region, but i s s l i g h t l y lower than the uniform f i e l d case at the extremes of the f o c a l plane. II)Dispersion Dispersion i s defined as a change in position of rays at the focal plane due tc a change i n the momentum of the p a r t i c l e s . This can be expressed as D i s p e r s i o n = change i n di s t a n c e along f o c a l plane momentum of p a r t i c l e x * = AD_x P A P = P X b3 ^P change i n momentum (fo r small AD & AP) Z=0=O 60 Hodoscope position , 12 18 24 H ' > i '— t— (— I 1 1 t—i—i—t—t—t—i 6 0 7 0 ^ 90 100 (MeV) Pion energy at 10 kG Fig. 25 67 and from P = P0 ( 1 + Q ) i.e. dP = P. dQ , we get Dispersion = ( 1+Q ) z=e=o Evaluating t h i s expression using the polynomial f i t Q = Q (Z,C,D) , we obtain This expression (3.5) i s plotted with crosses in Fig 26 with the uniform f i e l d case shown as the s o l i d l i n e . Note i n the figure that the measured f i e l d case matches up with the uniform f i e l d case i n the low energy region, and at about 80 MeV pion energy the measured dispersion drops below that of the uniform f i e l d . Ill)Q-values of hodoscope counter elements Since there are 24 s c i n t i l l a t i o n counters located at d i f f e r e n t positions along the f o c a l plane, each element w i l l be at a ce r t a i n momentum focus which can then be assigned a Q-value from the equation Dispersion (3.5) a = p - I (3.6) which comes from equation (3.1), Thus,from the polynomial f i t Hodoscope p o s i t i o n v_ . 6, 12 18 24 •* 1 1 1 1 ( 1 + 1 • 1 ., f :-> i 1 h J » — H 1 "t h. 1 3 • 40 50 60 70 80 90 1 0 0 (MeV) Pion energy at 10 kG F i g 26 63 Q=Q(Z,C,D), with Z=0, S c=0, and knowing the D value f o r each counter from the hodoscope box construction drawings provided by T.Masterson9, the Q for each of the 24 counters can be obtained. These 24 Q values are plotted with crosses i n Fig 27 with the unifom f i e l d case plotted as a s o l i d l i n e . Note that the Q-values in the measured f i e l d case matches up c l o s e l y with that of the uniform f i e l d case. The positions of the hodoscope elements so-obtained have also been shown in Figs. 25 & 26. IV)Aberration Since the position of a ray along the f o c a l plane of the spectrograph i s not completely independent of the incident ray angle , we have the aberration e f f e c t that a point source of mono-energetic p a r t i c l e s w i l l produce a f i n i t e spot size at the f o c a l plane. This necessitates the determination of the upper and lower maximum incident ray angles B, & 0Z . In order to obtain 6, S 6* approximately, we use the assumption of an uniform magnetic f i e l d of 10 kG with an e f f e c t i v e radius of 50 cm. This was then overlapped on the drawing of the physical structure of the spectrograph. Then, rays were produced from a point source (100 cm from the centre of the magnet) and c i r c u l a r t r a j e c t o r i e s were drawn through the magnet. Bays h i t t i n g an obstacle were then rejected. By t r i a l and error, 0, 6 0Z of successful rays of 64 • 'i F i g 27 , ' Q-values of 24 hodoscope elements < . (MeV/C) (MeV) Hodoscope' . Momentum (p)' Energy element no. Q-values =149.9178* (1+Q) -,.:=(p2+139.62)-139.6 1 -frj-i-bVirt— 124,4916 "" 47,"t4&5 . 2 -0,147«8 127,7476 49,6290 3 -0 , 12758 130 .7912 ' 51 ,6968 — 4 -0 . 1 0708 - J 133,8646 53.8112 5 -0,03543 137,1100 S6.0714 fa -0,06524 140,1371 58,2043 7 -f>v(H4-9-8— 143, 1741 60,3679 ' 8 .-0,02395 ^ 1.46,3274 62,6371 9 -0.00036 149,8640 65,2106 I 0 0, 0 174 4— 152,5405 67,1771 I I 0.03535 155,2169 69,1593 ' 12 0,05291 157,8504 71,1247 ' —H> O-rfrT-^ ^H— 160,4785 73, 1005' j 14 0,08647 • 162.88J0 74,9189 ; 15 0, 10335 165,4123 76,8472 ' —1 t> .- - 0 ,1 1927- 167,7984 78,6761 17 0,13870 170,7112 80,9231 18 0,15476 173,1188 82,7922 — ^ ^ J ^ X J J } ^ 175,3983 84 ,571 1 , 20 . 0,18492 177,6409 86,3302? 21 0,20016 179,9256 88,1310 — . 2 2 0 . f 2 l 5 6 8 _ 182,2620 89,9972. 23 0,23133 184,5967- 91,8408 , 24 0,24715 . 186,9703 9:3*7368 65 66 t h i s c e r t a i n energy were obtained. From t h i s range of incident angles, d i f f e r e n t angles were generated. These incident angles were fed into the f i t t i n g polynomial D=D(Z,0,Q) with the condition Z-0 imposed, to produce D, the distance along the f o c a l plane. The values of D were plotted vs. the incident angles. The plot for Q=-0.170 i s shown i n Fig 28. The aberration i s obtained as the maximum displacement of the rays along the f o c a l plane. This same procedure was repeated for the 24 counter Q-values, and the aberration obtained for each of the 24 counter energies were plotted i n Fig 29. Note that the aberration i s a minimum around the 6 th counter (Q=-0. 06524) of about .9 cm up to a maximum of about 6.5 cm at the l a s t (24 th ) counter. V)Total resolution of hodoscope counters The momentum resolution (Ap/p) i s the measure of how accurately the detection system w i l l be able to measure the momentum of a p a r t i c l e . In our case, t h i s i s a combination of 3 factors: the target s i z e , the counter size and the aberration e f f e c t . The momentum resolution due to target size (Az) can be calculated using the dispersion= p^- and the magnification = ^-£-. This can be done by Az Distance along f o c a l plane D (cm) s t A b e r r a t i o n of 1 counter element Q = -0.170 33.0 •• 32.5 7 32.0 1 31.5 Incident ray angle (rad) -0.130 -0.0705 -0.011 0.0485 0.108 F i g 28 69 R - / ( P § > ( 3 . 7 ) _ A M a g n i f i c a t i o n D i s p e r s i o n The target si z e was taken to be 1 cm and the expression (3.7) was evaluated and plotted i n Fig 30 with the cross symbols. The target size resolution for the uniform f i e l d case (see Appendix II) i s plotted as the s o l i d curve. Note i n the figure that the momentum resolution due to target si z e i s s l i g h t l y lower than the uniform f i e l d case i n the low energy region ( <50 BeV) ,but i s s l i g h t l y higher than the uniform f i e l d case at higher energies and reached a maximum around 80 MeV pion energy. The same treatment was done for a target of height 1 mm to observe ' f o r - the effect on the resolution of a change i n beam siz e ( see F i g . 32 ). The momentum resolution due to counter size can be derived as follows. The hodoscope elements are of 3 cm i n width. However, the counters were mounted with 2 cm separation between their centres. This placement r e s u l t s i n dividing each counter into 3 regions: 2 overlapping regions of 1 cm on the ends and one non-overlapping region i n the centre. To account for the momentum span (Ap) of each counter, the position of the both ends of the central region are determined by the co-ordinate provided 9 and the corresponding momenta were obtained from Q=Q(Z,C,D) as i n section I I I . Also, the Q-value of the centre of the central 70 Momentum r e s o l u t i o n due to t a r g e t s i z e at 10 kG ,0115 Momentum Res o l u t i o n 0' p/p) .0110 4 .0105 1 H a l l probe + measured? f i e l d case + 4-+ + ++ + t +• -h .0100 T ,0095 •+ + + + Browne-Buechner uniform f i e l d case ,0090 4 + ,0085 Hodoscope p o s i t i o n 1 6 12 18 24 H 1 1 1 1 1 > — • 4 *—i 1 1 r^—f h 1 r — i 1 r ( — , H •4-(Mev) -+ 40 50 60 55.0 80 90 100 Pi o n energy a t 10 kG F i g . 30 71 r e g i o n i s taken as the Q-value of the counter element. By d e f i n i t i o n of P = P (1 + Q), we r e a d i l y o b t a i n Momentum r e s o l u t i o n due to counter s i z e R c AP P 1 + Q AQ (3.8) T h i s e x p r e s s i o n i s shewn i n F i g s 31 B 32. Note from these f i g u r e s t h a t the momentum r e s o l u t i o n due to counter s i z e decreases from the lower energy counters to higher energy c o u n t e r s , and reaches a minimum around the 20 th counter of the hodoscope a r r a y . The momentum r e s o l u t i o n due to a b e r r a t i o n can be obtained from the a b e r r a t i o n of a p o i n t source f o r the counter elements d e r i v e d i n s e c t i o n IV. The momentum r e s o l u t i o n i n t h i s case can be obtained by where AB i s the image s i z e along the f o c a l plane, and expr e s s i o n (3.9) i s eval u a t e d and p l o t t e d i n F i g 31 & 32.. Note t h a t the momentum r e s o l u t i o n due to a b e r r a t i o n decreases from the lower energy counters to a minimum around the 6 t h counter and r i s e s up i n higher energy counters as expected from the a b e r r a t i o n and d i s p e r s i o n p r o p e r t i e s . Taking the above three r e s o l u t i o n e f f e c t s , we caii o b t a i n the t o t a l r e s o l u t i o n of the spectrograph, R T . Assuming t h a t ^Momentum r e s o l u t i o n due to a b e r r a t i o n t, [111 f- O* ; Ir \ i AP A D x AP _ A b e r r a t i o n (3.9) ab P PAD D i s p e r s i o n .021 ,019 .017 I .015 J Momentum esolutionfap/p) 72 .013 f _|_ .011 .009 ,007 17 Momentum r e s o l u t i o n o f counting system vs. Q-values of 24 hodoscope elements (targe t s i z e = 1 cm) Tot a l r e s o l u t i o n R, Res o l u t i o n by t a r g e t s i z e d^lcm) R t + + R e s o l u t i o n by ab e r r a t i o n R , Re s o l u t i o n by counter s i z e R: Q-values of counters - i — .10 ,03 -.04 -.11 F i g . 31 .18 .25 H \ 1 1 1 1 1 1 1 1 1 1 ( 1 1 1 * 6 12 18 Hodoscope p o s i t i o n - t— I h—( 1 H 24 F i g . 31 ,0205 .018 ,0155 ,0130 .0105 .008 .0055 .003 Momentum Res o l u t i o n 73 -.17 Momentum r e s o l u t i o n of counting system vs. Q-values of 24 hodoscope elements (Target s i z e = 0*1 cm) Res o l u t i o n by, t a r g e t s i z e (1 mm) R^ -.10 -.03 ,04 .11 .18 ,25 Hodoscope p o s i t i o n - * 1 1 >- _( ) | -4 1 1 1 1 1 1 1 1 1 I 1 12 18 24 F i g . 32 74 the e f f e c t s discussed above are independent of each other, E T can be obtained by This i s shown in Fig 31 for target s i z e of 1 cm, and also shown in Fig 32 for target size of 1 mm. Note that from both f i g u r e s , the change i n target size only improves the minimum t o t a l resolution from 0.0156 Ap/p to 0.01125Ap/p (28%),and the maximun t o t a l resolution from 0.0202Ap/p to 0.0180Ap/p (11%). (3.10) - 75 Chapter 4 Conclusion The results obtained i n t h i s t h e s i s can be summarised as f o l l o w s : -P h y s i c a l s t r u c t u r e Gap width Maximum magnetic f i e l d Radius of pole face E f f e c t i v e r a d i u s ( e x i t ) (entrance) P o s i t i o n of magnet centrebeiow (HalibT*probe map origin)o' ) (Beamline I) O v e r a l l e f f e c t i v e r a d i u s : ( t a r g e t at 0' ) (ta r g e t along BLI) I O p t i c a l p r o p e r t i e s a a t 10 kG Measured C e n t r a l energy (MeV), E„ 65. C e n t r a l momentum (MeV/c), P. 149. Maximum Q-value of counters -0. Minimum Q-valuesof counters 0. Maximum accepted momentum (MeV/c)1186. Minimum accepted momentum ' ' 124. Maximum accepted energy (MeV) 93. Minimum accepted energy ,, 47. Momentum accepted (AP/P. ) 0. Energy accepted (AE/E„ ) 0. At c e n t r a l energy, M a g n i f i c a t i o n D i s p e r s i o n (cm/(£.P/P)) A b e r r a t i o n (cm) Re s o l u t i o n due to : (ta r g e t s i z e = 1 cm) (ta r g e t s i z e = 1 mm) (counter s i z e ) I ( a b e r r a t i o n ) T o t a l momentum r e s o l u t i o n (AP/P) (t a r g e t s i z e = 1 cm) (ta r g e t s i z e = 1 mm) 2. 222. 1. 0. 0. 0, 0, 1.905 cm 12 kG 50.0 cm 51.51 cm 51.26 cm 1.43 cm 0.808 cm 49.83 cm 50.45 cm I I I&II f i e l d Uniform f i e l d D i f f e r e n c e 22 65.31 .1% 92/3 150.00!; .1% 1696 -0.170 .2% 2472 0.243 2% 9703 '184.77 1% 491b 122.73- 1% 7468 91.98 2% 4§63 46.28 2% 42;.8 0.41 . 8% 7«1>98 0.70: 1% 38 2.t 24 6% .94 223.61 .3% .436 1.692 15% .0107 0.01002 6% ,00107 0.001002 6%» ,00957 0.00985 29% ,00645 0.00756 17% .0157 0.0160 2%. .. .0116 0.0125 8% 76 CO o CL >-to 2 CO c> o o . Q. ro • ' CO . 1/1 I w to . o a. 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W 3 I- O rj) rr •a L U rt <r fj; ar LU -^ L.J J 5 — r\i ro a- in -T r-~. o- cr cr cr cr-ch o-ru ru rvi ru ru ru ru 2 9 9 c x z IS THE- BENDING- PL ANE-»-AND-A-,-B-ARE-OX/DZ> - D Y/DZ- RESPECTIVELY-30 0 C 301 DIMENSION 01 (4),02(4),03(4),04(4),F(3) > 302-303 COMMUM 8FLU.N£NT(4),GRID(h00j,FIELD(2S«0i6) 30a C O M M O N /VTRAJl/EN,PPlL3,ZPAK,DXDZ,ErEST;0TEST 30b •— — 1/TAPENO/ ITA=E,LTAPE,NTAPE - --307 2/FLOPAR/ XC,rC,iC,21,H2f^LfHCENT,^S,'JS2,Nl,N2,N3,NP,NPl,NP3f 3 0 K 3J.SYM(S),iCENT,JZC6),INTC3J -3 0-9- * / - T - R A L-<-/--T* r*-H-^ 310 5wY(10),SSO(2) 311 6/THAPAH/ KEVENT,V(6),ER(5) 312 7/I1EBU3/ I I , I T ER AT, P ATH (-50 0 Z£DFC'»X^AX ;YMAX -313 C 314 C WilNGE-KUTTA INTEGRATION _3..i.5 ^_ . _.. 31b MM=0 316,25 MTAPE=9 317 . - ZED.= ZL — 310 00 50 JJ=1,3,ICENT 319 IC=0 - 3 20 I-P-(-J-J--£-eT.l-)-5x>_^ r4J_^ 1_ 321 IF(JJ.EQ,2)G0 TO 332 322 IF(JJ,ECi.3)G0 TO 333 32 3 - 331 IN = I\-T ( 1 ) -32a GO T3 334 325 332 IN=I\iT(25 -3 2 6 —-G 0—1-D-.3-3-» 3 2 7 3 3 3 I * = I N T < 3 ) 3 2 6 3 3 4 C O N T I N U E 3 2 9 C I N = I M T ( J J ) -— - - - CERN 3 3 0 EsFL0AT(l-IAB3(IN))*HCENT 3 3 1 J J J = 2 * J J - 1 _ 3 - j ? t t t = 0 3 3 3 L = J Z ( J J J ) - I N 33« 51 L = L +1 M 335 - - • • LL = L + I N - -•-3 3 6 • LLL=LLL+1 3 3 7 H s A 0 S ( G R I O ( L L ) - G R I D ( L ) J + E -3-3 ft H H = 0-r5-« H 3 3 9 • S=H 3 4 0 MsO 3 « 1 " 2 2 0 0 1 0 1 = 1 , a- - - --- - — -3 4 2 G O T O ( 1 , 2 , 2 , 3 ) , I 3«3 1 X T = X A - 3 4 4 Y-T-5 Y-A-3 4 5 A T = A 3 0 6 8T=8 34 7 • - Z A = Z E D - - - - - - -• - -3 4 8 0 2 = 0 , 0 3 4 9 G O T O 4 - 3 5 t r - 2—J s-h - t -351 XT = XA + o,5*D! ( J ) 352 YT=YA+0.5*02(J) 353 AT = A+0i5*O3(J) •-35« . 9T = H + ci ,5*04 ( J ) 355 'ZASZEO + HH 3 5 „ q i - ( : . , - . w / s  357 GO TO 4 00 INS S3 1 —4 > tt t o -<t =J .c O K to _ I - s » u. — in * -o •- 2 » is. to >-I • tn t to - s - ro to I O tO 3 • *• + ci a c- to <Z + +• lit V >- < 30 tvj I It II 11 11 >- < E N Q s: i - o rt >- X *-< <I to ll :r II x tn -s I  — _ w J - . r-j u. w S l _ «t-I o ru : o »-l - s • ar ^ « <J> ru w i s-2 -s X 1— I " > > *~* w to It II w w r u L t " I X w « U - i in t o t o >. to to ~ i — — < in LL. OJ •K CO i— ^ s - to Cl CO to ^ o • cr —• u> - — • 7z — -a. ed r u N' — It to 'a. fx I I w - o K LL " >- " LL 7. <t t n * * -< o to • 11 ~< to II rst o rr- ro cr. ^ -x r n i t cp. ac tn ct 3£ rt 2 O o 2 to OJ rt o o ~ tj 3 to » 2 -O <t t - CU 2 y o -tt O r s ! t— o a. >- i— Ct TC t-z. U J . _ ! _ l 2 -tf/ O (J t- -L|J Jfc J. ti> a. li-2 2 LD CJJ i tBto to tL. II II fx tft i i . ru rs o o 0 o o o o — V . » |» 1 r~* -C £) « fs -S \ -f. 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U . * C | * ^ , II II II II II II T • O <I r ^ s J l l l J O N a j ^ r . i-{-2 x £ if r T : x <i• r-, w x J - T r r y K (\i e c i w ^ 0- w w w < w a J : D x i x x x r II u. -NJ < i— i— •— >— v— <r--'ww < < < < < < 7 li_ ii u. ci_ CL ci. a_a CL <!«—•.-<»—« i £ i l 5 c c r- ^  ^ r- r- f- r- r- :x; x x> rxj ^  ^  3 LT. .£ r-d a " I 9 C - * : U T E < 6 , 5 2 ) X A , Y A , Z E O , A , 9 , P U ) , F C 2 ) , F ( 3 ) , A N A , E N , D X D Z 4 2 0 5 2 F O R M A T ( 1 0 F 1 0 , 5 , 2 F 9 , 4 ) 4 2 1 C -U22 — v-P-I £-a (-Z-eO-rX-M-36r3-7-0 0 ) » , 7 3-74-« 2 J X f I E = - ( X A - Z E O + 3 3 , 7 8 5 1 ) * , 7 0 7 1 4 2 " Y P I E = Y P I E / 2 . 5 U + 2 , 1 2 5 " 2 5 - - X P I t = ( X P I E + <jl , 9 9 ) / 2 , 5 u - , 7 S -4 2 b C C H A N G E TU j v ! , i . P A T ' S C F N T R E « 2 7 I F ( X ? I E , L T , 0 , ) G O TO 9 9 I-F-C fc-N , Ee-rE-T-E-ST-T^OvO-Xr^^^T^T-E-Sf-)^0—T-9-^99-« 28-- ....... . . , 4 2 9 I F C Y P l E , L T t 2 " f 0 , A N D , F ( 2 ) , L T , l , E - 2 0 ) G O TO 9 8 4 3 0 I F C Y P I E . L T , 2 4 , 1 , O R . Y P I E . G T , 2 7 * 0 ) G O TO 9 9 -• " 3 1 - I F ( Y P l f , L T . 2 4 . 0 . A N D , F ( 2 ) , L T y l t F . - 2 0 ) G O " T O 9 8 - - • " 3 1 , 2 5 0PAr< = P P i L B / l i i 9 . 2 3 3 9 - 1 ,0 4 3 2 - < R I T E ( 6 , 5 2 ) x A , Y A , Z E D , A , r ) , F U ) , F ( 2 ) , F ( 3 ) , A N A , E N , D X D Z 4 3 3 G- wH-I-T-E C ^ A P E - r S - r ^ A - r X - ^ — 4 3 u « R I T E ( 7 , 5 2 ) A , X A , Z E D , E N , P P I L 3 , Q P A R , D X D Z , Z P A R , Y P I E , X P I E 4 3 5 E T E S T S E N -- 4 3 h O T E S T s O x D Z ----- - --- - - - - • -• • 4 3 7 GO TO 9 9 " 3 9 9 8 W R I T E ( b , 5 9 6 ) E N , 0 X 0 Z , Y P I E , X P I E 4-39 C — • « S-I-T- ir <»••? &-P &-,-5-9*-H^-7-^*B-Z-7^-P-K-^-P-l-c 4 0 0 , F. T fc S T ; F N 4 " l ' O T E S T s D x O Z 4 4 2 5 9 8 F O R M A T ( 2 X » 1 * * * * * * * * * * * * * , F 1 0 , 3 > 1 M E V - P I IN A T » , F 1 3 , 2 , ' - - HITS A T ' , 2 F 1 3 , 2 , ' * * * * * * * * * ' 3 4 4 3 9 9 C O N T I N U E 4 4 " C N T A N i A S A N A + 4 5 , 0 4-45 £ *R 1 T E-( b , 5-2-)—Y-P-I-E , X P I-E-rA-rC N T A N A , X A - r-V-A-^-fl^rR -g-) — 4 4 6 K 1 = K 1 + I N 4 4 7 K 2 = K 2 t I N • — 4 4 8 - •• K 3 = K 3 + I N -•— -- - -UU9 KUSK4+ I N 4 5 0 I F ( L L L - J Z ( J J J + 1 ) ) 5 1 , 5 0 , 5 0 5^_ Gn.vT - l.NUE- . . 4 5 2 RETUK'N 4 5 3 C • 4 5 4 3 1 1 1 = 1 — - - - - — • " 5 5 R E T U R N 4 5 6 E N D J U 5--, s y tt H u ,j F ^ E — f r v T P O L (lyrfriTW-T-H-T-l-r-J-rK-Tb-l-4 5 8 C I N T P O L R O U T I N E FOR L I N E A R I N T E R P O L A T I O N I N 3 - 0 F I E L D M E S H 4 5 9 C 4 6 0 D I M E N S I O N D C 3 ) , L L ( 2 ) - -4 6 1 C 4 6 2 COMMON B F L O , N E N T ( U ) , G R I D ( 6 0 0 ) , H ( 2 5 4 0 l 6 ) -a 6-3 fr+Ap^Q-/—h^.?.£rrt^p^i^jp_£r y o 4 2 / F L O P A R / X C , Y C , Z C , Z l , Z 2 , Z E D , H C E N T , N S , N S 2 , N 1 , N 2 , N 3 , N P , N P 1 , N P 3 , 4 6 5 3 I N D I C E C 1 3 ) 4 b 6 - C - - -- - - - -4 6 7 I F ( ( 5 ( 3 ) . - Z E O ) l , l , 2 4 6 8 C S P E C I A L S E T T I N G - U P S E C T I O N F O R F I R S T E N T R Y I N A G I V E N S A Y - T R A C E -U 6 9 C T-H -1-5-1-8—S K A-tt—*d-r^£-mfE-N-f--tN4-R4 E-^ : 4 7 0 4 7 1 1 K = 2 ' ex. • 4 7 2 - L = 0 -4 7 3 0 0 7 M = l , 2 4 7 u - L = L + N E N T ( M ) -4 7 5 O-Q-i^-^X - f-L--4 7 b I F ( D ( M ) - G W I O ( J ) ) 1 0 , 9 , 9 4 7 6 - - L L C M ) = J - 1 • - " - - ••—~ - - - • -4 7 9 . GO T U 11 4 8 0 9 C O N T I N U E > 4 &1 1 i — K = K-r-rvj : • . 0 6 2 7 C O N T I N U E 4 8 3 I X = L L f 1 ) 4 6 a - I Y = L u (?.) ' - -- -a«5 I = N P 1 * C l Y - N l - 1 J + N 3 * ( I X - 1 ) • ! I - N S 2 4 8 6 J = I + N P 1 i) fl -7 K- S-J-+-H 3 — : • 4 8 8 L = I + N 3 4 6 9 0 X = I) C 1 ) 4 9 0 - D Y = D ( 2 ) - — - - - -- -4 9 1 GO TO 1 2 4 9 2 C 4 9-3 C -TE 8 4 — * H E T H E R T H E P A R T I C L E H A S C-R-6S-SED A GR I D - t I*S"SIN C-E—T-H EH.A-8T • • — 4 9 4 C E N T R Y I N T O 1 N T P 0 L 4 9 5 C . ' — • 4 9 b - 2 I F ( O C 1 J - G R I D C I X + 10) 1 3 , 13> 14 • •-- — - - - - - . 4 9 7 1 4 I X = I. X 11 4 9 b I = I+iM3 • — . q 9 9 j s - J +TV 3 — —-5 0 0 K = K + N 3 5 0 1 L=L+N3 •- 5 0 2 - G O TO 1 5 — - - • 5 0 3 1 3 I F ( D ( 1 ) - G R I 0 ( I X ) ) 1 6 , 1 5 , 1 S 5 0 4 l b I X = I X - i 505 1 = ;-•••< 3 — •—• • -. 5 0 b J = J - ^ 3 5 0 7 K = K - N 3 5 0 9 1 5 D X = ( D ( . l ) - G R I D ( I X ) ) / ( G R I O ( I X + n - G R I D ( I X ) ) 5 1 0 I F ( 0 ( 2 ) - G R I D ( I Y + l ) H 7 , 1 7 , 1 8 -5 1-1 •—1-8-I-Y-S-I-Y-+-1 • — — — : • :  5 1 2 I = I + N ? 1 5 1 3 J = J+N.?1 -- 5 1 4 - - • K = K + W P l - - - - - --• -5 1 5 L=L+MP1 5 1 6 GO TO 1 9 51 7 • 1-7—I F ( 0 ( - t - h & H t i l t r f t - t f e , 1 9 , 1 - 9 = 5 1 b • 2 0 I Y = I Y - 1 5 1 9 I = I - N ? 1 • - 5 2 0 J = J - N ? I - - - - - - ----- - - --- --5 2 1 K S K - N P I 5 2 2 L=L-NPI 5 2 3 1-9—P-Y °-fO(-2-)--^GRI 0 (-I-Y-H / ( G R H> ( I Y11 ) - f r f r - f r L H - H - B > 5 2 4 C 5 2 5 C I N T E R P O L A T I O N F O R A L L 3 F I E L O C O M P O N E N T S - 5 2 b - C - - - - • . - - • • • 5 2 7 1 2 A = 1 , 0 - 0 X 5 2 6 8 = 1 ,0-O.Y 5 2 9 - — — H -= I-H-** T ^ • 5 3 0 J 1 = J + I N T .' ex, 5 3 1 K 1 = K H N T ' <Jt \ - • 5 3 2 . - • L 1 = L H N T - - - - -5 3 3 0 0 6 MM=I , 3 5 3 a R E 1 = H ( I ) ' ^ 5 3 5 R£-2=H-(-f~B 5 3 b R E 3 = H f J ) -• 5 3 6 - - R E 5 = H ( K ) - - -5 3 9 R E 6 = H ( K 1 ) 5 4 0 R E 7 = H f L ) , 5 M j s b 6 = H . ( . 5 4 2 N Z E R 0 = 8 5 « 3 I f - - ( H ( l ) , G T , l , E - 3 ) e S T D s H ( n 5UU I F ( H ( J ) • G T . 1 > F . - 3 ) « S T D = H ( J-) - . - - - - - - -5 " 5 I F ( H ( K ) , G T , 1 . E - 3 ) 8 S T D = H ( X ) 5 4 b I F ( r i ( L ) , G l , . 1 , E - 3 ) B S T 0 = H ( L ) 5 4-7 I F C-M-( 1 M + . + i W W - 3 - H ) - e f « — 5 4 8 2 0 1 1 I F C H ( J ) . L T , 1 . E - 2 0 ) G 0 T O 2 0 0 2 5 4 9 2 0 1 2 I F C H ( K ) f L T , l , E - 2 0 ) G 0 T O 2 0 0 3 - - 5 5 0 • 2 0 1 3 I F ( H ( L ) . L r, 1 , E - 2 0 ) G 0 TO 2 0 0 4 - - — - - - — - - --5 5 1 GO T O 2 0 0 7 5 5 2 2 0 0 1 N Z E R 0 = N Z E R 0 - 1 — 5 5 3- --H-C-H = 8 - S H 5 :  5 5 " G O T O 2 n ) 1 5 5 5 2 0 0 2 N Z E R O = N Z E R O - l •- 5 5 6 H ( J ) = BSro -• — - - .• - - — - - - - - -5 5 7 GO T O 2 0 1 2 5 5 9 2 0 0 3 N Z E R 0 = N 7 E R 0 - 1 5 £>c» H - (-*-)=-» S^T{? : :  5 6 0 GO T O 2 0 1 3 5 6 1 2 0 0 4 N Z E R ' J = N Z F R 0 - 1 ••• 5 6 2 . H ( L ) = B S T D -- - '• - - - - -- — - -~ -- -- - -- - ------5 6 3 2 0 0 7 I F ( M ( I 1 ) , U T , ) , E - 2 1 ) N Z E R O = N Z E R O - 1 5 6 " I F ( H ( J i ) ,LT , 1 , E - 2 1 ) N Z E R O = N Z E R O - l 5 6 5 • ——I - F - ( - W . ( - ^ - i - ) - R L - 4 - r 4 . E - 5 1 ) N Z e N Z E R 0 - 1 :  5 6 6 I F C H ( u l ) . L r . l , t - 2 1 ) N Z E R Q = N Z E r < 0 - l 5 o 7 I F ( M Z E R 0 . L T , 4 ) G O T O 2 0 0 8 5 6 8 - - - I F ( M M , N E . 2 ) G Q T 0 7 9 2 - - - - - -----5 6 9 C H ( I 1 ) . N E , H U ) 5 7 0 I F ( M ( I l ) , L T , 1 , E - 2 1 ) H ( I 1 ) = 8 S T D . 5-7..! IF-(-B-fiH-H b - t - r t - . - E — 2 ^ - ) H C J 1 )-aa-S4-9 5 7 2 I F ( H C M ) , L T . l . E - 2 3 ) H ( K l ) = t ) S T D 5 7 3 I F C H ( U l ) , L T , 1 , E - 2 3 ) H ( L l ) = 8 S T 0 5 7 5 C 5 7 b C 5 7-7- I F C H ( I -HrH-C-J-3 « H ( L ) « H C-K ) « H C I I ) * H C - c ^ i H ^ - V H - ^ 5 7 8 w R I T E C 6 , 5 " ) 5 7 9 5 4 F O R M A T ( 5 0 X , 4 0 H T R A J E C T O R Y MET F I E L D E D G E F O R N E X T E N T R Y ) -- 5 6 0 • - 7 9 2 C O N T I N U E - - - - - - - - - - — -5 B 1 D ( M M ) = C 1 , - D Z ) * ( C H C I ) * 8 + H C J ) * D Y ) * A + D X * C H C L ) * 8 + H C K ) * D Y ) ) + 5 8 2 1 D Z * C C H C I l ) * 8 + H ( J l j * D Y ) * A + O X * ( H C L 1 ) * 3 + H C K 1 ) * D Y ) ) 5 8-3 G-O-T-tJ—2-0-0-9- — — — — 5 8 " 2 0 0 8 D ( M M ) = 0 , 5 8 5 2 0 0 9 C O N T I N U E - 5 6 6 - H ( I ) = R E 1 - - ~ " ' - — — - - - - -5 8 7 H ( I 1 ) = R E 2 .  5 8 8 H ( J ) = R E 3 5 6 9 K-H-l-)-=-Rcna : : — • — — 5 9 0 H C K ) = R E 5 ,' ' 5 9 1 H C K 1 ) = K E b . ' . 5 9 2 -- - H ( L ) = 9 E 7 - - - - ; - - - -• O N 5 9 3 H ( H ) = K E 8 5 9 " I = I • f i ? v 5 9 5- • r i a J - t - M ^ 5 9 6 K=K+NP 87 z CL * z — + »-» _ l II II — rt rt II II ll f O - i t O I IO KI ^| «*1 n a rx o_ Z Z -£ Z rjr • i i i 5 o 11 II II til T. —i r t LU C > i > C r O O o o O O O O O O L n L O L O N O J J ^ l - O X > ^ > - 0 ^ 0 ^ 1 I I Q Z 88 Appendix B Optical Properties of an Uniform F i e l d The expression for magnification and dispersion from Browne-Buechner1 are as follows •*A«*:4;C*+.'.*= 7 - ^ ^ • d*** <o » I R * + r * ) V i (a1.1) ' _ JL£_ = , a 1 2 ) where R=effective radius of pole face,and r-radius of curvature of t r a j e c t o r i e s as governed by P = 0.3Br The bottom expression (a1.2) i s not d i r e c t l y applicable for t h i s thesis because i t i s n o n - r e l a t i v i s t i c . However, i t i s easy to convert because n o n - r e l a t i v i s t i c a l l y dE/E = 2dr/r, and since dp/p = dr/r i s always correct, we can t r i v i a l l y obtain the desired expression for momentum dispersion D ; , p e r S . o , = ? 7 F = (.**»•- t O l ( d 1 * 3 ) these expressions (a1.1) S (a1«3) are shown as s o l i d l i n e i n Figs 25 S 26 respectively. To obtain the resolution for 1 cm object using the above expressions, we form the expression 89 R * = - W - ( a t . 4) where A h i s the t a r g e t s i z e and i s 1 cm. T h i s r e s o l u t i o n due t o t a r g e t s i z e i s shown as s o l i d l i n e i n F i g 30. For the c e n t r a l ray, the momentum r e s o l u t i o n due to t a r g e t s i z e amounts to 0.01002 P/P, which matches up with t h a t of the measured f i e l d case of 0.0107 p/p ( 1% d i f f e r e n c e ) S i n c e we know the exact p o s i t i o n s of the 24 s c i n t i l l a t i o n c o u n t e r s 9 , we can express t h i s p o s i t i o n as co-o r d i n a t e (x,y) i n the magnet c e n t r e c o - o r d i n a t e system. Then, the bending angle ( 6 = a r c t a n y/x) of the p a r t i c l e t r a j e c t o r y which w i l l f i r e a s p e c i f i c hodoscope counter can be obtained. From t h i s bending angle 0, we can d e r i v e d the r a d i u s of c u r v a t u r e o f t h i s t r a j e c t o r y as from L i v i n g o o d 1 0 , p - e c°* 1 T h i s i n t u r n y i e l d s the p a r t i c l e momentum by f - Ol Bp e 0-3 6,o i and thus we o b t a i n the Q-value 90 In order to obtain good results comparable to the measured f i e l d case, the alignment of the pole face of the magnet had to be accounted for (see section 2 f ) . The re-values of the 24 counters of the magnetic f i e l d were evaluated and plotted as a s o l i d l i n e in Fig 27. The momentum resolution thus obtained equals to 0.00985 Ap/p for the cent r a l anergy counter, This compared favorably to the measured f i e l d case of 0.00957 4 P / P » For the momentum resolution due to aberration e f f e c t , since the angular span was not very large{ < 0.15 rad ), we could use the r e s u l t by Stephans 1 1 : i i a . * . * - Ain't* j. .dU^E , , r-« A b e r r a t i o n - ^ * * ( « O) For central ray, we could simplify t h i s to where d i s half of the angular span. This amounts to 0.00756 _p/p which compared favorably to the measured f i e l d case 0.00645 AP/P ( 17% difference). 91 BIBLIOGRAPHY Browne, C P . and Buechner, W.W. 1956 R.S.I., V o l . 27, No. 11, 899. Enge, H.A. 1958 R.S.I., V o l . 29, No. 10, 855. TRIUMF Annual Rep o f t , 19 72. Robertson, L. UBC Programme L i b r a r y "TRLY: HALLSQFOB". Z'a'cftarov, B. 1965 N.I.M., V o l . 33,136. Z^'charov, B. "C.E.R.N. Momentum F i t t i n g Programme " (unpublished) ¥.ohnson, R.R.%1973v/privateacoimTiuriica'tI6n.?. -Walden, P. L. ,1975vpf.ivat-'eTBeiESifimiaj5idStion. Master son, T. ,l'9^42>^^ i^D'c'<a&ftafi£-<SStion. Li v i n g o o d , J . J . 1934 " The Optics Of D i p o l e Magnets " Academic Press Stephens, W.E. 1934 Phys. Rev.,513. 

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