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I '44, " “II' '4444 "II" udzlnm 1" . l‘ u' I" | I 14 -d' 535' "L'.|I|.".|'.an 554‘fifl’h‘I In. AM 111111111111 1111111111111111111111111111 TH ESCS This is to certify that the thesis entitled MEASUREMENT OF THE STRUCTURE FUNCTION F IN MUON SCATTERING AT 270 GEV 2 presented by ROBE RT CHARLES BALL has been accepted towards fulfillment of the requirements for Ph . D. degree in Phy51 cs ‘ 1 1w L LIBRARY ‘ Michigan State -, University L’ Major ptofessor Date October 19, 1979 0-7639 OVERDUE FINES ARE 25¢ PER DAY PER ITEM Return to book drop to remove this checkout from your record. MEASUREMENT OF THE STRUCTURE FUNCTION F IN MUON SCATTERING AT 270 GeV 2 BY Robert Charles Ball A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirement for the degree of DOCTOR OF PHILOSOPHY Department of Physics 1979 ABSTRACT MEASUREMENT OF THE STRUCTURE FUNCTION F IN MUON SCATTERING AT 270 GeV 2 BY Robert Charles Ball An experiment has been performed at Fermi National Accelerator Laboratory in which the nucleon structure func- tion F2 was measured. The experiment took place in the muon laboratory using a 270 GeV u+ beam incident on an iron-scintillator calorimeter-target. The primary detection counters used were proportional wire chambers and magneto- strictive wire spark chambers. The spectrometer consisted of toroidally shaped, wire wound magnets interleaved with the spark chambers and plastic scintillator trigger banks. Low angle muons were excluded by the use of veto counters centered on the toroid axis. The data are compared to the QCD (Quantum Chromo Dynamics) model of Buras and Gaemers, wherein least squares fits to several parameters are performed. A disagreement with the model is seen in the region of 25 < Q2 < 50 (GeV/c)2 where the data events are more copious than the model allows. A threshold of production in the variable W2 is considered, and dismissed as unlikely. The effect on the data of a variation of the value of R = oL/oT is quantitatively examined. Possible systematic errors in the experiment are checked. Included in the study are variations of the toroid magnetic field and the incident muon energy, and changes in the resolution of the scattered muon energy. ACKNOWLEDGMENTS I wish to thank many people for their aid in this endeavor. The members of the experiment were few but able, Adam Kotlewski, Larry Litt, Sten Hansen, Phil Schewe, K. W. Chen (thesis advisor), and particularly Jim Kiley and Dan Bauer, without whose work there would be no accurate monte carlo or data reconstruction programs. Larry Litt in par- ticular has been highly encouraging. The neutrino area staff at Fermilab deserve thanks for their cheerful faces, and for their aid during crisis situa- tions. The computer center staff at Michigan State Univer— sity are to be thanked for their unflinching aid (not another box of tapes?!). I wish to thank Tom Pierce for his concern, advice, mathematical software, and general friendship, and with Mary Brake, for their help in editing this thesis. Last, but hardly least, I wish to thank Mehdi Ghods for his time and help, and Delores Sullivan for typing this thesis, particularly the beautiful tables scattered throughout. (iii) Chapter II. TABLE OF CONTENTS THEORETICAL BACKGROUND A) B) C) D) E) F) Introduction: Historical Development Early Theorems and the Deep Inelastic Cross Section The Move Away from Scaling Asymptotically Free Gauge Theories A Particular QCD Model The Structure Function for Iron EXPERIMENTAL EQUIPMENT A) B) C) D) E) F) G) H) I) J) K) L) Beam Transport Proportional Wire Chambers (PWC) Beam Line Scintillation Counters The Target Wire Spark Chambers (WSC) Magnets Trigger Bank Counters (TBC) Halo Vetoes (HV) Beam Vetoes (BV) Target Hodoscope Past Trigger Circuitry The CAMAC Equipment (iV) Page 13 14 18 22 22 28 30 37 39 44 44 53 53 56 S6 65 Chapter III. IV. VI. M) On-Line Computing DATA EVENT RECONSTRUCTION A) Track Finding B) Program Initialization C) The Momentum Fitting D) Program Cross Checks E) Raw Kinematic Distributions THE MONTE CARLO A) Modelling the Data B) Multiple Coulomb Scattering and Energy Loss . . . . . . . . . . . C) Radiative Corrections D) Wide Angle BremSStrahlung (WAB) Background E) Nuclear Fermi Motion F) The Incident Beam C) Computer Program Description H) Event Weighting I) Geometric Acceptance and Resolution ANALYSIS PROCEDURE A) Normalization by Flux . B) The Filter Program COMPARE C) The Comparison of Data to Monte Carlo D) The Construction of F2(x,Q2) E) Systematic Errors CONCLUSIONS (V) Page 71 78 78 86 88 92 95 104 104 104 112 , 115 , 117 , 118 , 122 , 129 , 132 , 142 , 142 , 147 . 15:5 , 160 178 196 APPENDICES A. PROOF OF EQUIVALENCE OF THE STRUCTURE FUNCTION F2 AND ITS MOMENTS . . . . B. PRESCRIPTION FOR 5 AND C MOMENT EVALUATIONS C. SUMMARY OF DETERMINED STRUCTURE FUNCTION VALUES . . D. MAGNETIC FIELD AND INCIDENT ENERGY SYSTEMATICS . . . B. THE TOROID MAGNETIC FIELD REFERENCES . (vi) Page 203 206 208 220 235 239 Table I-l I-Z II I H II-2 II-3 II-4 II-S II-6 II-7 II-8 II I 10 II-lO II-ll II-12 II-13 II-14 III-1 III-2 III-3 LIST OF TABLES Definitions of Kinematic Variables Constants and Best Fit Parameters Used in the QCD Calculation of vW Nominal 270 GeVu+ Beam Line Currents Quadratic Fits to B(I) in 1E4 Some Proportional Chamber Information Z°POSit10nS of all E319 Equipment Calculation of Average Target Density and Radiation Length Some WSC Information Fits to Toroid Magnetic Fields Quadrant Replacements for TBC and RV Alignment Alignment Constants for TBC and BV Counters Some Trigger Signal Definitions CAMAC Scaler Contents Visual Scaler Quantities Contents of the Time-Digital-Converters Primary Tape Event Block Structure Acceptable Three-Point Line Types Single View Line Cuts Vertex Cuts (vii) 2 Page 19 25 27 29 33 38 42 43 SO 51 S8 67 68 68 73 8O 80 80 Table III-4 III-5 III-6 IV-1 IV-2 IV-3 IV-4 IV-5 IV-6 IV-7 IV-8 V-l V-Z V-3 V-lO V-ll V-lZ Track Quality Standards MULTIMU Output Tape Format GETP and GETPZ Output Tape Format Cross Section Corrections Fits to Ionization Loss Beam Tape Format Stepping Process Cuts Accepted Event Tape Format Generated Event Tape Format Smeared Event Tape Format Third Stage Program Cuts Beam Distribution Limits Ordered List of COMPARE Program Cuts Non-Kinematic Information COMPARE Output Tapes . . . . . COMPARE Analysis Tape Format Bounds of Kinematic Planes Used for Analysis Maintained Cell Information . Some Values of Resolution in Kinematic Variables Results of Three Parameter QCD Fits Total Muon Yield in Monte Carlo Systematic Checks . . . . . . . . . Effect of Changing E' Resolution Calculation of Average Nucleus, Nucleon Content . A List of All Examined Systematic Error Possibilities . . (viii) Page 81 83 90 105 109 119 124 125 126 128 128 145 148 151 152 156 156 159 163 183 . 190 192 . 194 Table C-1 C-2 D-l D-Z D-3 D-4 D-S E-l F2 in Parametric Q2 Regions F2 in Parametric x Regions E' Comparison 6 Comparison . Q2 Comparison x Comparison . 2 W Comparison Integral Field Measurements in Spectrometer Magnets (iX) Page 209 215 221 224 226 228 231 . 237 LIST OF FIGURES Figure Page I-l. Muon-Nucleon Scattering Feynman Diagram . . 5 I-2. Early SLAC Scaling Data . . . . . . . 10 I-3. FNAL Exp 26 Data/Monte Carlo Ratios . . . 11 1-4 Sea and Valence Quark Distributions from QCD . . . . . . . . . . . . 20 II-l. N1 Beam Transport System . . . . . . . Z4 II-Z. Experiment Apparatus . . . . . . . . 31 II-3. E398 Hodoscope Diagram . . . . . . . 36 II—4. TBC Diagram . . . . . . . . . . . 46 II-S. Trigger Bank Counter Overlap . . . . . 47 II-6. Geometry of Displacement of Circle of Radius R . . . . . . . . . . . . 49 II-7. Halo Veto Diagram . . . . . . . . . S4 II-8. Target Hodoscope Cross Section Diagram . . 55 II-9. Fast Logic Schematic . . . . . . . . 60 II-lO. FNAL Timing Pulses . . . . . . . . . 63 II-ll. CAMAC and Computer Gating Schematic . . . 64 II-12. CAMAC Crate Contents . . . . . . . . 66 II-lS. DCR Bit Latch Contents . . . . . . . 69 III-l. MULTIMU Program Organization . . . . . 79 III-2. GETP and GETPZ Program Organization . . . 89 III-3. Contour Plot of MULTIMU Inefficiency . . . 96 (X) Figure III-4. III-S. III-6. III—7. III-8. III-9. IV-1. IV-2. IV-3. IV-4. IV-Sa. IV-Sb. IV-6. IV-7. IV-8. IV-9. IV-IO. IV-ll. IV-12. IV-13. V-l. V-Za. V-Zb. Raw Q2 Data Distribution . Raw w Data Distribution Raw E' Data Distribution . Raw 9 Data Distribution Raw x Data Distribution Raw W2 Data Distribution . Radiative Process Diagrams Triangular Region Contributing to Radiative Correction Contour Plot of Radiative Correction Weight Factor Contour Plot of Wide Angle Bremmstrahlung Background Factor . . . . . . Data Beam Distribution and Comparison to Monte Carlo Data Beam Distribution and Comparison to Monte Carlo Contour Plot of l/E' Resolution (percent) Contour Plot of Theta Resolution (mrad) Contour Plot of Momentum Fit E' Shifts (GeV) . . . . . . . . Hardware Acceptance Contour Plot Software Acceptance Contour Plot Contours of Constant Qz Values Contours of Constant x Values Contours of Constant W2 Values FLUXSIM Program Results F2(x ,Q'Z) in Parametric Q2 Regions F2(x ,Q'Z) in Parametric Q2 Regions (xi) Page 97 98 ‘99 100 101 102 113 113 114 116 120 121 134 135 136 1 137 1 138 1 139 1 140 1 141 1 144 .165 .166 Figure V-Zc. V-2d. V-Ze. V-Zf. V-3a. V-3b. V-3c. V-3d. V-4. V-S. V-lO. V1-10 VI-Z. VI-3. E-l. F2(x ,Q'Z) in Parametric QZ Regions - _'z F2(x .Q - - z F2(x ’Q' _ -'2 F2(X ,Q F2(x',Q2) in Parametric x Regions ) in Parametric Q2 Regions ) in Parametric Q2 Regions ) in Parametric Q2 Regions F2(x',Q2) in Parametric x Regions F2(x',Q2) in Parametric x Regions Fz(x',Qz) in Parametric x Regions Ratio of Data/Monte Carlo vs Raw w The Variation of F2 with Target Mass Effects . . . . . . . and Magnetic Effect of Variation of E0 Field on E' Effect of variation of E o and Magnetic Field on O . . . . . Effect of Variation of EO and Magnetic Field on Q2 . . . . . and Magnetic Effect of Variation of E0 Field on x Effect of Variation of E 2 and Magnetic Field on W . . . . O The Normalized Ratio Data/Monte Carlo vs W2. Illustration of Effects of Resolution on the Determined Value of F2 Correction of W2 Ratio by "B High" Results B(R) in Spectrometer Toroidal Magnets (xii) Page 167 168 169 170 171 172 173 174 176 179 184 185 186 187 188 198 199 200 238 CHAPTER I THEORETICAL BACKGROUND A) Introduction: Historical Development Lepton-nucleon scattering theory has evolved consider- ably over the last ten years. A chronological scan of the literature on the subject during this time would first (1) claim that the nucleon (proton or neutron) structure func- tions, the quantities describing the internal structure of the nucleon, scaled in terms of the variable w (defined in Table I-LL Scaling means that as w = 5%? increases beyond a given value, and in the limit that both QZ-Hm and v-rm, the structure functions assume constant values, and can be parametrized for smaller w solely in terms of w. Knowledge of m then determines the value of the structure function independent of the specific value of Q2, the mass of the virtual photon exchanged in the interaction. Since as Q2 increases the effective size of objects which can be probed decreases, an independence of Q2 means all internal structure which is present has been probed. This feature of the data,with its lack of an obvious mass scale,became known as Bjorken scaling.(2) Since the structure functions scaled, the photon was thought to be interacting with Table I-l. Definitions of Kinematic Variables (EO,O,O,k) Incident muon momentum in the Lab frame (E',k'sin6,0,k'cose) Scattered muon momentum in a particular reference frame (M,0,0,0) Nucleon momentum in the Lab frame All interaction products other than u' u - u' Momentum vector of the virtual photon 2&3 = EO - E' Energy of the virtual photon in the Lab frame -Z=- 1-22 Q 4E0E 51D 2 1.12.2. w ZMv v E' _=1-_ E0 E0 2 2 (pm)2 = ZMv + M The total mass squared of X. The missing mass squared - Q The metric used in this table is 800 ’ = +1 '311 = ’gzz = 'g33 0. u r v 00 ll 1.1V 3 essentially free quarks within the bounds of the nucleus,(3) and additionally the quarks were deduced to be pointlike particles. Thus the model of a nucleus built entirely of point-like constituents came into wide acceptance. This model, known as the parton model, also contained the chargeless gluons which bound the whole nucleon together. The gluon presence is necessary because the total fractional momentum carried by quarks and anti-quarks (4) in the nucleus is given by f1F2(x)dx = 0.507 (1) 0 which is significantly less than the value of 1.0 which would have been found if the quarks and anti-quarks were the only particles present. Obviously then, upwards of 50% of the nucleon momentum must be carried by the chargeless gluons. (FZ = sz is defined by Eqn. I-6 below.) As more data became available, scaling began to have problems; it began to appear that the data may not scale after all. At first attempts were made to save scaling; after all, simple theories with simple, easily tested pre- dictions have wider appeal than esoteric, difficult to understand theories. Various new scaling variables, such as w' = (2Mv+M2)/Q2, were tried and all were eventually discarded. Scaling was violated.(5'7) Now attempts were made to parametrize structure functions in terms of two 2 2 variables, e.g., q and v, or q and w, and the asymptotically free gauge theories (Q.C.D. = quantum-chromo-dynamics) began 4 to rise in popularity. These theories bypass nucleons and try to deal directly with interactions between the quarks constituting the nucleons, not just the three valence quarks, but also a ”sea" of quark—anti-quark pairs and the gluon cloud keeping the whole nucleon together. Today evidence is mounting in favor of these theories, of which there are presently several parameterizations.(8’9) In this dissertation I will present measurements taken in the scattering of 270 GeV muons from an iron target, and compare these against the predictions of the QCD model of (9) Buras and Gaemers. B) Early Theorems and the Deep Inelastic Cross Section At the simplest level, deep inelastic muon-nucleon scattering can be represented (Fig. l) by the exchange of a photon between the incident muon and one of the quarks con- stituting the nucleon. The photon is virtual and can there- fore have a mass = Q2. The photon-muon vertex is simply understood. All three particles are point-like and the matrix element squared is easily written from quantum electrodynamics (QED) as [(R'Yvk)*(i'vuk)] (2) .1. 2° ll m2 2 111112 spins This contribution to the cross section can be calculated exactly; however, the second vertex,where the interaction Figure I-l. Muon-Nucleon Scattering Feynman Diagram is strong in characten is less well understood. In terms of the four-current J“, the matrix element describing this vertex is 2 111,11 ~1 (3) where the sum is over all possible intermediate states x. The matrix element as a whole results from combining Eqns. 2 and 3, and also contains a momentum conserving delta function and the photon propagator. It is then J; 2 M = k'v k “ q a4ck+p-k'-x) . (4) The hadronic portion of the matrix element squared can be evaluated by defining the Lorentz invariant, covariant tensor va by E _ 3 p em em Wuv(p,q) - (Zr) 11 1 (5) 3 4 d p35 (p+q-X) (21113 q = k - k' This tensor can be expanded in terms of all possible tensor contractions of p and q as q q . - _ u v J; - ELS ”th,q) - (51w (12 )W1 +112 (p11 q2 C111) (6) . - 219 (p1) q2 q\))W2 7 where W1 and W2 are the (unknown) nucleon structure functions and describe the combined effect of all the strong inter- action vertex corrections. The total inclusive cross section can now be formed by inclusion of a particle flux factor m m' [%¥i], muon normalization factors [EB and E4], the electro- 0 ll 11 magnetic coupling factors ((4na)2), the final muon phase 3 . d k (211) of wuv[EL ——l—§]. Combining these factors and Eqns. 2 - 6, p (211) it is then space [ 3] and the extra factor included in the definition ’73:— ° (7) By a convenient choice of coordinate system and in the high EO limit, we can write _ , 2 _ 1 2 Rukuv - (MonyW1 + Eo(l y)U2)cos 6/2 . (8) The cross section can then be written 2 a2 1 . 0 . MxW 2 dg'oe = 8n 2E251n (vW2){(1'Y) + \flvl y2}cos 0/2 (9) a (Q 1 y 2 or, in terms of the variables x and y 2 z MxW d o 8na .1 2 MX = ____————VW {(l-y) + y }{1"———TXT“Y} (10) didy 4Mon2y2 2 sz 2Eo 1 Y The only remaining problem is to separate the contribu- tions from W1 and W2. Although Wl will contribute most for very inelastic scattering (y ~ 1) at large scattering angles (6), it can possibly contribute in other kinematic regions as well. One methodcz) of separation is to consider the scattering of an imaginary photon of mass Q2 and momentum k (in the nucleon rest frame) from the nucleon p. The total absorption cross section can be separated into two cases, one where the photon is polarized perpendicular (t) to q and p and one where it is longitudinal, or scaler (s). The two cross sections are 2 2 _ 4n e W 0t“ k 1 (11) 4112e2 2 s-—k——[(1 +37) 1112 -wl] Q I Then the ratio of magnetic to electric structure functions is W 2 o W- = (1 + (fin—1,52%) . (12) Defining R = 05/0 the ratio can be written 1:) /Q R . (13) 9 The differential cross section now becomes d o _ SwazE'sinecos2 6/2 Mxy2 1+Vz/Q2 - 2 2 VWZII-y + } (Q ) V V 1+R (14) «2 , . 2 4 2 = 8n E :126cos 9/2 vWZII-y + % 33R} (Q ) y Despite several years of effort, too little information is available on the functional form of R. Although it seems that the general trend is for R to decrease as x or Q2 increases, available data are also consistent with R being a constant (10) R 0.21 t 0.10 . (15) Constant values used in the past have ranged from 0.18 (11) to 0.52.(12) Also advocated(12) is the form R °(l-x) R = —9—67——— R0 = 1.20 f8I§g (GeV/c)2 (16) It is this form which shall be used in the analysis pre- sented in chapter five. C) The Move Away from Scaling Early data taken with an electron beam incident on (13) supported the scaling hydrogen and deuterium targets picture, although to be sure the data did not extend to very high values of w or Q2(Fig.I-2). However, as higher 10 0'40 ' T I l r 0.35 - - 0.30 - 4.8832.3 ml - In: fir Data/Monte Carlo (based on scaling In w') -6.3 1.0- - “up-5.9 ‘3’ 1 1.1 l.2-- I 1' . ; : 1 I 1 _ 1.0.. * Data/Monte COIIO (based on scaling In «H I 2 5 IO 20 50 q‘lt‘nV/c)z Figure 1-3. FNAL Exp 26 Data/Monte Carlo Ratios 12 energy results began to appear, questions concerning the validity of scaling began to arise. The new data did not appear to be constant at higher values of w, nor did it appear to be Q2 independent. Variables such as (14) w' = w + M7, M being the proton mass, were introduced which seemed to give better scaling fits,(15) but doubt over the validity of scaling led to attempts to parameterize the degree of scaling violation in terms of a propagator form in A. The structure function would become(16) 1 (17) 1 + Qz/A2 F2(x,q2) = F2(X)1 With the event rate expected for this experiment (~1011 inci- dent muons), a value of A greater than 50 (GeV/c)2 would have been consistent with scaling at a 70% confidence level level.flj) Finally, u—Fe data at incident energies of lfliGeV and 56 GeV (5) were published which left little doubt that scaling was violated (Fig. I-3). Not only was the data Q2 dependent when compared to a scaling model in the variable w' (18), but the dependence changed with the value of w. For m < 5, the ratio data events/monte carlo prediction decreased as Q2 increased, and for w > 5, the opposite was true. Scaling had been laid to rest. 13 D) Asymptotically Free Gauge Theories During this time the asymptotically free gauge theories (ASFT or QCD) were gaining in popularity. These non-abelian theories, when taken to the asymptotic limit (Q2 + w), approach the free field theories of which the parton models are representative, and could thus conceivably explain the (19) At first apparent scaling of electro-production data. (1970), attempts were made to explain the data using abelian gauge theories, but these were not Ultra-Violet (UV) stable at the origin, so an approach using non-abelian theories was suggested, since these were UV stable. UV stability means the momentum dependent coupling constant, g, for the scalar fields in the Lagrangian (e.g., g in g¢4 where g(g,t==0)==g) approaches a constant value as Q2(=t) approaches infinity, and is strictly increasing as Q2 decreases from there. The origin refers to the approached constant value being zero. In gauge theories, the symmetry is traditionally broken at this point, which is the device through which the intro- duced, vector meson propagators(gluons) would receive a mass. Introduction of Higgs bosons to break the symmetry does not work since either a satisfactory abelian sub-group arises, or the UV stability is lost by attempting the use ).(19) of larger representations such as SU(8 One possibil- ityczo) is that the symmetry is not broken, but is.exact, and that a ”dynamical” reason exists which does not allow the collision of color singlet particles to produce unpaired color non-singlet particles such as quarks and gluons. 14 Thus the gluons may remain massless, yet unseen. This allows the use of models such as the SU(3)®SU(3) global (21) symmetry model of Cell-Mann with three fermion trip- lets, and an SU(3) color gauge group providing strong interactions. E) A Particular QCD Model The QCD model of Buras and Gaemers(9’22) is just such a model as this. In this model, Q2 independent quark and gluon density functions (q(x,QgD are defined at a value Qg. These are then used as boundary conditions in renormalization group equations to find the functional dependence at any value of QZ > Q3. The most natural functions arrived at by this approach are the moments 1 n = 11dx xn'lth.Q2) ; n 2 2 (18) whereas experimentalists measure the structure functions. However, a complete knowledge of all moments will yield the structure functions, and vice versa (see Appendix A). In general, the moments are quite complicated analyt- ic functions, so Buras and Gaemers have used much simpler analytic functions as approximations. They are, however, accurate to within 2% for x < 0.8 and s < 1.4, where (19) UN g £n[£n(Q:/A:) £n(Qo/A ) 15 The scale parameter A characterizes the strength of the scaling violation and is related to the effective gluon coupling constant 65(Q2) by -2 2 2 _ g (Q ) 124 C ) " (20) as Q 4" (33-2m)£n(Q2/A2) m being the number of quark flavors. For Q2 s A2 this coupling becomes large, invalidating its perturbation theory basis. For this reason, the validity of this QCD formulation is limited to Q2 3 Q3. Buras and Gaemers start by defining quark distributions for the valence quarks p(x,Q2) = pv(x,Q2) + A(X.Q2) n(X.Q2) = nV(X.Q2) + 11x.QZ) (21) fi(x,Q2) = fiCx,Q2) = A(X.Q2) = th.Q2) for the SU(3) symmetric non-charmed sea 2 _ 2 S(X’Q ) ’ 60(X9Q ) (22) and for the charmed sea C(x,Q2) = C(X.Q2) + écx,Q2) (23) where A(x,Q2) represents the strange quark density function. Contributions from heavier quarks are assumed negligible, which means m = 4 in Eqn. 20 above. Gluons (G) are next 16 introduced along with the combinations vgcx.Q?) = pV(X.Q2) + nVCX.Q2)(=uV+dV) (24) V3(X.QZ) pv(x,Q2) - nv(x,QZ)(=uV-dv) and then the analytic approximations at Q2 are used by defining, at Q2 = Q3 2111 Z(i) 2 xVi(x) = A x (l-x) , 1 = 3,8 z xS(x) = As(l-x) 5 (25) ZG xG(x) - AG(l-x) xC(x) = 0 where all z and A are constant. ASFT effects then introduce a Q2 dependence in the form of the variable 3 defined in Eqn. 19, such that the z and A are now functions of s which can be found from the theory. For the valence quarks, a linear dependence of the 2 on s is used, 21(3) = 21(0) + ziG's (26) where the zi(0) are evaluated at Qg using the moments (Vi(Q6)>n' Then the slope parameters 2; are fit to data with the help of the relation - n n = n e'SY (27) 17 with ”NI constant and determined by the theory. The valence quark formulations for Q2 3 Q: are 2 _ 3 zl(§) 22(§) xVscx’Q ) ‘ 8121(3),1+22(s)5 “X *1'x) (28) 2 _ 2 2 Z3(§) 24(g) XV3(X:Q ) " XV8(X:Q ) " B(ZS(S),1+Z4(S)TAX ' (I'X) G' is 33475 = %% which appears in Eqn. 20 for the strong coupling constant. B(zl(s),1+zz(s)) is Euler's beta func- tion, necessary to satisfy the known sum rules 1 L) de8 = 3 = total number of valence quarks in the nucleon 1 (29) L) de3 = l = number of up quarks minus num- ber of down quarks in the proton S, G,and C, describing the sea quark distributions, are strongly decreasing functions of x, becoming nearly insignificant beyond x = 0.3. The functions can thus be completely determined on the basis of their first two moments alone. From ASFT then, - z (5) xk(x,Qz) = Akcs).11-x) k Akcé) = Pk-(2§%; - 1) k = s,c,c (30) Zk(S) =?)—(-;—- Z 18 where Pk is the second parton moment and 3rd k - moment . . . k = nd = average x for the distribution. 2 k - moment The prescription for the moment evaluation is given in Appendix B. Table I-Z summarizes the values of all constants used along with the best fit values for the valence para— meters Z1 and for Q5, and Figure I-4 displays the sea and valence distributions which result. F) The Structure Function for Iron With the evaluation of the various parton densities, a value for the structure function F2 can be calculated. 56 26 Since iron is a complex nucleus, containing for Fe protons and 30 neutrons, an average structure function is found. Weighting by the square of the charge of the quarks, we use the prescription for the proton (p) and neutron (n) p Fin] (x,Q2) = “ng 178(x,Q2)i%)-V3(X.Q2) (31) +3Mm¥)+iamfin 9 9 and then compute the average F2 2 26 151x,421 + 30'F3(x,Q2) F2(X.Q ) = 56 . (32) Buras and Gaemers have tested their formulation on SLAC (18) ep data and Fermilab up data(23) for x < 0.8 and all 0 < 5 5 1.4 and found it to be satisfactorily accurate 19 Table I-2. Constants and Best Fit Parameter Used in the QCD Calculation of 0W2 Contants: n Yn Y? Y? an 8n 2 0.427 0.747 0 0.429 0.429 3 0.667 1.386 0.609 0.925 0.288 Best Fit Parameters Q: = 2.00 21 = 0.70 2i = -l.l = ' = 22 2.60 22 5.0 = ' = - z3 0.85 23 1.5 Z4 = 3.35 Z4 = 5-1 Quantities Derived from Best Fit Parameters = 0 111 = 0 01111 0 2 ' o 3 ° = 0 402 = 0 05738 0 2 ’ o 3 ' = 0 488 = 0 15700 8 o 2 ° 8 o 3 ‘ 20 on _ ' oe - 1 (o) (bl OTI- 07 - I p, (1,!) 051-. l 0, (I13) 05 - .\ - ~:A- ' ' - “~\..1 0 01 02 03 0.6 05. 00 07 0| 0’ ‘0 0 0‘ 01 03 05 05 00 07 0.0 0! 10 I I l (C) (d) i ‘21. 1‘,. lctl.3) l l ! 15".!) 121- . l tor 1 '. 1 1‘ 1 001-“ ‘ \‘ l 1‘ I Q 061- 1‘ 1 1 ‘. 1 \ \ l 00- ‘\""‘.". 2 .: \ ‘ I \L\’!I00 l 01*- \\\‘ ‘.V \ ‘ 0‘ "”353. ' l 0 0' 03 0) . . Figure I-4. Sea and Valence Quark Distributions from QCD 21 (<2% deviation). There is some difficulty for 0.6 < x < 0.75 where their formulation seems to be flatter when plotted (24) as a function of Q2. Speculation against the SLAC data is that this is likely due to target mass effects, higher order corrections and higher twist contributions not treated in their analysis. With the parameterization thus completed, we can proceed to compare the model to the data of this experiment. CHAPTER II EXPERIMENTAL EQUIPMENT A) Beam Transport The muon beam we used was derived from a secondary proton beam at 400 GeV, slowly extracted from the main ring 13 spaced in at Fermilab. These protons,approximately 10 "buckets" over a 1.8 second period,were directed onto an aluminum oxide production target .75" in diameter and 12" long. This target and its associated focusing and bending magnets used to produce muon beams are known as the "triplet train.” The name comes from the mounting of the equipment atop movable cars situated on tracks and the fact that, in this configuration, there are three extra sets of focusing quadrupole magnets. The "buckets" refer to the accelerating peaks of the RF of the main proton ring, 53.1 MHz. This gave a bucket spacing of 19 ns. Most buckets, obviously, were empty; however, when not empty, they usually contained only one muon. Exceptions to this rule which showed up in later analysis were thrown out of the data sample. Preceding this 1.8 sec."live" beam period were approxi- mately 8 sec.of "dead" time during which protons were injected into the main ring and accelerated to their peak energy of 400 GeV. At this point the extraction began. 22 23 Protons not scattered from the production target were deposited in a beam dump, while the remainder of the charged particles (positive or negative, depending on the "train" magnet polarities) were steered into the beginning of the N1 beam line. This first portion of the N1 line consisted of a "decay pipe," a 300 m. long region where the particles (mostly pions with about 10% kaons) could decay into the desired muons. Entering enClosure 100, the first of four momentum selections via dipole bending magnets was made. Only those particles with the desired 270 GeV/c momentum were transported. After a second bend, the beam was directed thru approximately 60 feet of polyethylene, which absorbed all but a minute frac- tion of the pions, kaons, and protons, all strongly inter- acting particles, which had contaminated the beam up to this point. The resultant beam had a u/n ratio of 4x108, and was only very slightly dispersed by its traversal of the poly- ethylene. Two further bends and a bit of focusing brought the beam into the muon lab and to the experimental target. Of 13 the 10 protons which left the main ring, approximately 600,000 usable muons reached the target in what we defined as the beam. We directly controlled the currents in many of the magnets in the transport system and on the triplet train thru a MAC computer connected to a CAMAC serial branch highway. Using this feature, several SWIC's (Single 24 Production 6?} Target / 1W01 Enclosure '1W02 100 0 1w03 1"” . 100 0 F0 0 .\ 101 Enclosure 0 101 I 1E1 l ' 1V1 Enclosure 1W2 102 ’.\1F3 ' E398 PWCl Enclosure BHl 103 113/ E398 chz o C’”fl_’,_,.—av-{BH2 ,11: Bl ‘ B2-*' I 1E4 B3 Enclosure l 1‘_ E398 PWC3 104 "' E398 PWC4 ’f’ BH3 {E319 chs E398 1311105}: 01 Muon E398 PWC6 Lab Figure II-l. N1 Beam Transport System 25 Table II-l. Nominal 270 GeV u+ Beam Line Currents Set Actual Magnet Parity Current Current (AMPS) Triplet Train OUT 0 174.9990 173.0 OVT 0 32.0000 32.0 OHT . 1 120.9990 117.5 OFTl 1 96.2248 92.6 OFTZ 1 95.5998 92.5 ODT 0 3102.0200 2982.0 OPT 0 3077.0200 2955.0 0PT3 0 3227.0200 3110.0-3115. N1 Beam Line 1W01 0 0.0000 4630.0 1W02 0 4332.0100 4180.0-4210.0 lW03 0 4832.0100 4630.0 1V0 0 25.0000 1.9375 lFO 1 370.0120 361.5 lDO 1 370.0120 354.0 lQl 0 4174.9800 4010.0 lEl 0 3862.0200 37l0.0-3720.0 lVl 1 119.9990 67.3-0068.0 1W2 0 3712.0200 3540.0-3550.0 1F3 0 969.9970 947.5 lD3 0 1019.9900 995.0 lE4l 0 4319.9800 4237.48 1E42 0 0.0000 4224.0-4227.0 Measurements taken July 2, 1976 26 Wire Ionization Chambers) and scalers located in the beam, some tuning was available to increase the muon yield. These magnets and their nominal current settings are shown in Fig. II-l and Table II-l. In addition to minor adjustments, the complete N1 line could be "scaled” to use a different beam energy, e.g., calibration runs using beam energies as low as 25 GeV were performed. The magnet currents were "scaled" for lower energies than 270 GeV by a linear decrease from the values used at 270 GeV. Spatial smearing of the beam due to magnet saturation made energies higher than 270 GeV impractical. A11 muons entering the muon lab in the beam passed thru a series of proportional wire chambers and scintillator hodoscopes (see below) on either side of the final bending magnet, 1E4. Using this counter information, if the fB-dl for this magnet as a function of the current in the magnet were known, then the energy of each beam muon could be tagged to an accuracy of 0.1-0.8%, depending on precisely which chamber information was available. Consequently, using a Hall probe with a DVM readout, the magnetic field as a function of depth in the magnet was measured at several current values. The "effective length" of the magnet was calculated on the basis of these measure- ments to be 18.64 m. This length multiplied by the plateau value of the magnetic field in IE4 gives fB-dl. A quadratic fit to the magnetic field as a function of current was used. Two fits were necessary since, after a 27 Table II-Z. Quadratic Fits to B(I) in 1E4. 2 B = a-I + b-I + c I measured in Amps B measured in Kilogauss Coefficients prior to August 23, 1976 a = -o.5964x10'8 b = 0.3289x10’2 c = -0.03107 Coefficients after August 23, 1976 a = -o.1714x10'7 b = 0.3364x10‘2 c = -0.03279 28 one month shutdown in July, the magnet had been slightly modified. The two fits were B = a-Iz+b-I+c. The fit values are given in Table 11-2. Muons with insufficient information to determine their energy were assigned the value 270.00. This occurred on approximately 7% of the raw triggers before August 23, and on less than 1% after August 23. The difference is due to a PWC timing problem corrected during the one month shutdown. The average energies determined by this method were 268.6 GeV for the u+ runs, and 267.6 GeV for the u' runs (those after August 23). B) Proportional-Wire Chambers (PWC) Two different sets of PWC's were used in this experi- ment, each with one bit/one wire read-out. The first, designated E319 chambers, were on loan from Cornell Univer- sity. Construction details can be found in the thesis of Y. Watanabe. (25) The gas mixture used in these chambers was Isobutane (20.0%)-Methylal (3.92%)-Freon 13Bl (0.263%)- Argon (balance), vented to the exterior of the Muon lab after circulation. Since no delay circuitry was used for the signals, a fast-logic pre-trigger was used to initiate the recording, in 16 bit CAMAC latches, of the wire infor- mation. Two signals were involved; the first, called P.C. Reset, cleared previous information from the latches, the second, called P.C. Enable, enabled them to receive new information during the length of the signal. Three of these 29 Table II-3. Some Proportional Chamber Information No. of Planes Size of Planes . ' * Orientation Edge to Edge (cm) PWC Location (cm) A) E319 Proportional-Wire-Chamber (PWC) 5 -3685.54 2 -x,-Y 19.2 4 -517.76 3 x,v',w' 19.2 3 -235.35 3 -Y,V,W 19.2 2 625.32 2 -U,V 32.0 1 649.77 2 -X,Y 38.4 *Sign indicates direction in which the numbered wires increase. Also see Fig. II-2. Wire Spacing = 2.0 mm Reset Pulse Width = 15 ns Enable Pulse Width: PWC1,2,3,4 (X,V') = 120 ns PWC4W' = 86 ns PWCS = 80 ns B) E398 PWC's 1 -15512.95 Y 20.3 2 -8512.30 Y 20.3 3 -6393.49 Y 20.3 4 -6393.49 X 20.3 5 -3294.28 Y 20.3 6 -3294.28 X 20.3 Wire Spacing = 2.12 mm Reset Pulse width = 20 ns Enable Pulse Width = 98 ns 30 chambers were used upstream of the target to record beam information (x-y position and angles and incident energy) and two were downstream, the hadron PWC's, to aid in vertex location and track reconstruction. More information is contained in Table II-3A. The second set of chambers, designated B398 PWC's, were on loan from the University of Chicago. There were six planes, four measuring horizontal and two vertical displace- ments, all situated in the beam line. These were used only in determining the beam energy. Readout was identical to that for the E319 chambers. The E398 chambers were placed precisely on the Fermi- lab beam line (Fig. II-l), which bends thru an angle of 28.68 mR at beam magnet 1E4. Alignment of the two E398 horizontal chambers downstream of 1E4 with the three E319 beam chambers thus aligned our spectrometer with this assumed axis. The gas mixture in these chambers was different than that used in our own. It was gas mixture LK152287 supplied by Union Carbide-Linde Division, and consisted of 24.3% C02, 0.37% of gas 13B1, with the balance being Argon. C) Beam-Line Scintillation Counters Three distinct sets of scintillator counters were used in the muon beam line. The first set were called, "E398 hodoscopes" since they were on loan from the E398, Univer- sity of Chicago people. There were three sets of these 31 maumhwmm< psoEwhomxm .N-HH ousmoa mo; cumwmmwwumgw“ xwwmmuunmwumm I? ”".2 3‘“ Od’ 17€3d hu; co; mo; N>1 $52.83”. no mo .0 \zocoéupz. \ i d LU .d o. ..c. h. L 32 hm meowoocfioon mux< .2 co x¢ \\\\\\\\.moude om moumam om :ommmamxm souoswuofimu oomuwh \\\\\\ \\\\\\\l \\\\\\ \\\\\\\ \\\\\\ \\ \\\\ \\\\\\ H N m mmm232 mmhzaou cow hcfi mad meg 2 = _—§—t— = 5.741 g/cm = 4246 g/cm :1 3 Z t. J‘ 3 2 = = 2.46 cm = 14.1 g/cm 1 1 39 corresponding to few charged particles passing thru this counter. Using these signal pulse heights, the energy of the hadron shower in the inelastic scattering event could be measured directly once suitable calibration measurements were taken. E) Wire Spark Chambers (WSC) The primary devices used to record tracks for scattered muons were magneto-strictive wire-spark chambers. There were 36 planes of these chambers interspersed with toroidal magnets in the spectrometer proper (Fig. II-2), 4 planes making up one chamber. Each chamber was actually made up of two modules, each module containing two planes of wires at 90° to each other. The modules were oriented at 45° with respect to each other in a common aluminum frame, thus giving x, y, u and v planes corresponding to the main spectrometer axes as defined in Fig. II-2. An event trigger would cause a breakdown in a brass sparkgap at each chamber which would cause a chamber break- down along the ion trail of a charged particle which had passed thru the chamber. The chamber wire nearest the ion trail would carry the current to one edge of the chamber where it would intersect an orthogonal magnetostrictive wire encased in a plastic catheter. The presence of the current would induce an acoustic pulse in this wire, which, traveling to the wire end, would be picked up, amplified, and sent on to 40 the CAMAC equipment as a -24 V signal pulse. Two acoustic pulses due to fiducial wireswhich always carried a current were the first and last pulses on the wand, the name for the assembly containing the magnetostrictive wire and its amplifier. Eight sparks per plane were recordable in this manner. Systematic uncertainties due to changing acoustic pulse propagation velocities, dependent on temperature and state of magnetization of the wand, were minimized by moni- toring the fiducial wire locations on a run to run basis and reversing the wand orientations from chamber to chamber. For more information on chamber construction see the Ph.D. thesis of C. Chang.(26) The spark chambers were alignedanby first de-Caussing(28) the toroids and then iteratively fitting sets of straight- thru muon tracks until a change of less than .001 cm was found. Once the relative alignment was set, the toroids were turned back on, and muon tracks of known momentum were fit, iteratively shifting the spectrometer axis and overall orientation until the mean of the X2 distribution of the fit was minimized, and the mean momentum of tracks in each spectrometer quadrant agreed within statistical errors. The first four spark chambers and proportional chambers 1-4 were aligned together using one set of straight thru muons from run 130. The final five spark chambers could not be aligned in this manner because of central dead regions in these chambers 30 cm in diameter. These five chambers were 41 aligned using muons in runs 113-120 where the beam was pur- posely defocused. Lines in the front were extrapolated to find sparks in the back five chambers, and hence a new alignment constant. This was iterated until chamber move- ment became less than .001 cm. One further problem remained in the relative alignment; x, y, u and v relative orientations were now correct, but the match distributions, defined as(27) = ELX - = 51X - AXmatch V2 )c Aumatch f5 11 (1) = 2:1 - = X;£.- Aymatch y? y' AVmatch /7 V should have a zero mean, but as it turned out, did not. This was solved by offsetting x and y views in a linear fashion by minimizing the expression 9 2 2 WSE=1 (Aymatch) + (Axmatch) az+b+c2+d 2 (Ammatch + /Z ) (Z) az+b—cz-d 2 (Avmatch + /f ) where z is the chamber longitudinal position, and. yshift = az+b, xshift = cz+d are the chamber offsets necessitated by the minimization. 42 Table II-6. Some WSC Information Active area 73" x 73" Be-Cu wire 0.005” diameter spaced 0.7 mm apart Fiducial wire separation 184.15 cm in WSCl-S 182.88 cm in WSC6-9 Gas mixture Ne-He 78-80% Ar 2-3% Alcohol 0.7 SCFH @80°F Wand catheters contain Ar Spark gaps contain Nz Time from trigger signal to spark gap breakdown 220 ns Wand Reversal* and Chamber High Voltage Chamber Reversal H.V. Chamber Reversal H.V. 1X + 6X - 1Y + 6Y - 1U _ 8.6 kV 6U + 8.6 kV 1V ' 6V + 2X ' 7X + 2Y - 7Y + 2U + 8 4 kV 7U _ 7.6 kV 2V + 7V - 3X + 8X - 3Y + 8Y - 3U _ 8 4 kV 8U + 7 8 kV 3V - 8V + 4X - 9X + 4Y - 9Y + 4U + 7 6 kV 9U _ 7 4 kV 4V + 9V - ' 5X + 5Y + 5U _ 7 2 kV 5V - * + means increasing time counted with increasing displacement - means decreasing time count with increasing displacement 43 Table II-7. Fits to Toroid Magnetic Fields Coefficients a c d f M1,M3,M5,M7 12.20 19.92 -0.08357 0.0004346 M2,M4,M6,M8 12.07 19.71 -0.08270 0.0004301 Current = 35 Amps Average Field = 17.09 kG M1,M3,M5,M7 17.27 kG M2,M4,M6,M8 B(r) = a/r + c + dr + fr2 B in kG r in cm 44 As it turned out, there was one problem missed before all the data tapes were momentum analyzed; systematic, linear offset between fit tracks and actual sparks in the y direction for the last six spark chambers were discovered to exist. Refitting all the data would have been exhorbitantly expensive, so this shift was simulated in fitting Monte Carlo tracks and determining the final resolution (Sec. IV—I). F) Magnets The momentum analyzing magnets in the spectrometer, eight numbered from upstream to downstream and denoted Ml-M8, were wire wound iron toroids operating at saturation current levels (~35 A). The toroids were ~80 cm long with an inner diameter of 30.5 cm and an outer diameter of 172.7 cm. Lead loaded concrete filled the inner hole volume not occupied by the wiring. The radial field dependence was known to within 1% and fit by a polynomial in r, the radial coordinate (Table II-7L Two power supplies were used, one for even, the other odd numbered magnets, necessitating two sets of fit coefficients. For construction and field measurement details, see the Ph.D. thesis of 5. Herb (28) and Appendix E. G) Trigger Banks Counters (TBC) The trigger banks were hodoscopes of scintillation counters roughly circular in shape with a diameter of 180 cm. They were three in number, positioned in the spectrometer 45 behind magnets M2, M4 and M5, and were centered on the spectrometer axis. The primary function of the trigger banks was their contribution to the hardware trigger; coincident signals from all three banks were required to form ”8" in B-S-BV (Sec. II-K). The three banks were designated SA, SB and SC, and consisted of two planes of hodoscopes, one oriented verti- cally with phototubes at the t0p and bottom of each counter and designated with a superscript (e.g., SA'), the second oriented horizontally with phototubes at either end and designated with no superscript (e.g., SA). The latter were also know as SAV, SBV and SCV because of their vertical displacement measurements. The primed TBC's were also known as SAH, SBH and SCH because of their horizontal displacement measurements. They were constructed specifically for this experiment. Spatially, the H TBC's were always upstream from the V TBC's. The V trigger banks contained eight separate scintil- lators, six of which gave the bank its roughly circular shape, but left a 14" square hole in the center, and two which corrected the center to a circular shape with a 13 1/2" diameter. (Fig. II-4) The H trigger banks contained only six scintillators since the two center counters were designed to give a central hole diameter of 12”. In addition all counters were rectan- gular in shape so that the primed banks were not circular in cross-section. (Fig. II-4) 46 112121 'f' Trigger Banks SA' 235 235 SB' and SC' Overlap k" , e. Counters .. n 14%" wide 68 56 x 3/8" thick 12" . diam. 171717 All phototubes are Amperex 56AVP Trigger Banks SA, SB, SC 59.94" C:j 23" 14" 23" \\&::> Correction Counters 76:] 14" [:15.5" <3}.s" diam. CI: [::> F*-35.5"-———4-; <\_/> ”<2 Figure II-4. TBC Diagram 47 14 inches e p r \ / \ \ / \ __/ Figure II-S- Trigger Bank Counter Overlap 48 The software (on-line and off-line) and hardware (after the initial discriminators) treated each bank (primed and un-primed) as if it were only five counters, numbered either from top to bottom, or east to west. Individual phototubes were numbered from 1 to 5 (east and bottom) then from 6 to 10 (west and top) for each bank. The two correction tubes on the center section of the V banks were numbered 11 and.12. The holes in the trigger banks were present so that.non-SM deqyinelastically scattered muons traveling down the spec:m trometer axis would not trigger the apparatus. Since the trigger banks were placed on the rails with no special attempt to center them, it was necessary to determine their positions relative to the axis with some 'alignment procedure. The procedure settled upon was as follows (Fig. Il-6): the spectrometer was divided into four quadrants with the intention of measuring the circular coor- dinates of point B, the trigger bank center, with respect to point A, the spectrometer axis. If a counter in a quad- rant recorded a hit, the track radius at the nearest spark chamber to the counter struck was entered in a histogram for that counter/quadrant. The counter position would thus be "shadowed" in the histogram, and the minimum radius in the distribution would correspond to the length of the rays dl thru d4. 49 UP x VIEW FROM A UPSTREAM Figure II-6. Geometry of Displacement of Circle of Radius R 50 Table II-8. Quadrant Replacements for TBC and BV Alignment (for quadrants other than four in Equations 113-5, the measurements d1——+ d4 and 0 should be replaced as follows): With Quadrant Replace a < 45 e > 45 measured from c.c.w. quadrant boundary axis 2 d1 d3 d3 d2(unused) dl(unused) d4(unused) d3 d4 d1 d4 d2 d2 9 90 + 6 90 + 6 3 d1 d2 d2 d2(unused) d4(unused) d1(unused) d3 d1 d4 d4 d3 d3 9 180 + 6 180 + e 1 d1 d4 d4 d2(unused) d3(unused) d2(unused) d3 d2 d3 d4 d1 d1 6 270 + e 270 + e 51 Table II-9. Alignment Constants for TBC and BV Counters 1/30/79 Counter Displzgement §?§:§. (figs SA3 0.53 180,n -0.53,0 SAZ,4 0.44 180,4 -0.44,- SA'3 1.03 0,0 1.03,0 SA'2,4 -- -- -- SB3 1.10 10.1,0.176 1.08,0.19 SBZ,4 0.65 0,0 0.65,- SB'3 2.13 3.5,0.061 2.13,0.13 SB'2,4 0.38 90,n/2 -,0.38 BVl 1.16 18.5,0.324 1.10,0.37 SC3 0.72 -24.5,-0.428 0.66,-0.3O SC2,4 0.69 0,0 0.69,- SC'3 2.20 4.96,0.087 2.19,0.19 SC'2,4 0.41 90,n/2 -,0.41 BV2 1.58 12.3,0.214 l.54,0.34 BV3 0.31 15.2,0.266 0.30,0.08 Measurement error is 2.5 mm 52 From the law of cosines, for 0 < 45°, - 2fd cos(90+6) (3) 3 and 2 4 - 2fd4 cos(90-e) . (4) Eliminate R from the equations using R = f + <11 and it is found that, purely in terms of the measured quantities, e < 45° 2 2 f = d4d3+d4d3 - El 2d1(d4+d3) 2 (5) . di+2fd1-d§ 5199 = 2fd 3 For 6 > 45° 2 2 f g d4d2+d4d2 - El 2d1(d;+d2) 2 (6) di+2fd1-d§ cose = 2fd 2 The center displacement for the non-circular counters, which is also a measure of the T.B. displacement, is easily determined. For the V trigger banks d +d -d -d 3 4 1 2 = 4 (7) xDisp 53 and for the H trigger banks d +d -d -d _ 4 2 1 3 yDisp - 4 . (8) Theta can not be determined. Table II-8 summarizes the variable replacements in the equations for quadrants other than no. 4, and Table II-9 summarizes the displacement determined in this way. H) Halo Vetoes (HV) Located between E319 PWC3 and 4 was a bank of scintil- lator counters equipped with Amperex 56 AVP phototubes known as the Halo Vetoes. Halo, muons outside of the useful beam muon area and extending over the entire cross section of the spectrometer, could fake the hardware into believing a deep inelastic trigger had occurred. Consequently, this bank was installed to introduce a 20 ns dead time for coincidence in the "B" telescope whenever a halo muon was detected by the bank. Muons passing thru the 7" diameter central hole in the bank were considered to be beam muon candidates and were not vetoed in this manner. (Fig. II-7). 1) Beam Vetoes (BV) Situated on the spectrometer axis behind Magnets M4, M6, and M8 were three 12 1/2” diameter scintillator counters equipped with 56 AVP phototubes known as the Beam Veto counters (BVI-BVIII). These counters were designed to S4 15" square dab 7" diam. CORRECTION COUNTERS 7" diam. (::f//20” square Figure II-7. Halo Veto Diagram 55 4- 7.51n. , 9% +7111 '- 4 TH3 '9". "0.4-1- ;.. 7.5 in - .4 := 'fL /\ ~ ~~ \‘ C/////// \N \\\1 .1 1 1 1 1 I J [/on- \\\\\\\1 ~\\\\\\‘ . 7.5 in THG \\\\\\\‘ ~\\\\\‘~ 7 in ,1 1 7 17 1 1 r? *7 1- £;;;:::& L:\\\\\~ ///J .,1 7' 41* 1 fr 1' 1 .1 ;‘ \\\\~ 31 THE ' 7* THE 4-1 re TH4 a: 3 -3" 3. ‘UP EAST ALL COUNTERS 2 in WIDE 1/4 in OVERLAP Figure II-8. Target Hodoscope Cross Section Diagram 56 prevent a trigger from muons which merely went straight down the spectrometer axis. A coincidence between BVIII, the furthest downstream, and either of BVI or BVII was sufficient to set this veto in motion. To prevent accidental vetoes due to shower pions striking the vetoes, the hadron shields were set behind the target and surrounding WSC9, and the toroid holes were filled with lead loaded concrete. The hadron shields also served to protect the front spark chamber information from being washed out by too many sparks. Since the placement of these counters was done visually, they were aligned in the same manner as the trigger banks were aligned, except that histogram entries were made if the counter did not show a hit. J) Target Hodoscope The target hodoscope is the second piece of equipment from which informatiOn was never utilized. The hodoscope consisted of eight ~2" scintillator counters, four hori- zonally and four vertically oriented, overlapping each other by upto half their width (Fig. II-8). The counters could have been used, for example, to eliminate extra beam tracks when more than one beam track was present. K) Fast Trigger Circuitry With the target hodoscope, the summary of the data measuring equipment is complete. This equipment would be useless, though, without fast—logic hardware to trigger the 57 measurement procedure and a means to collect the measurements. The trigger procedure was accomplished with several racks of NIM-standard electronics located next to the spectrometer itself. Using various "AND" and "OR" logic modules, several signals were defined. (Table II-lO). From these signals the primary single-muon trigger, B-S-BV, was formed. This trigger meant a beam muon was present along with a spectrometer track, but nothing was detected in the toroid hole region. Once formed the trigger signal started the spark-chamber firing sequence, closed the event gate (see below) and scaler gates, and signaled the CAMAC elec- tronics to begin processing. In addition to this primary trigger, two other triggers were used. The first of these was B-SD-BV, a multi-track trigger. This trigger required the appearance of more than one track with penetration necessary only as far as the second trigger bank. This trigger did not work well. See (29) and J, Kiley(30) (both to be published) the theses of D. Bauer for details. The second additional trigger was used for random beam sampling, defined as B-(POSA). This random beam sample was later used as input to the Monte Carlo programs, because the accepted sample of scattered deep-inelastic events depends on the beam distribution used. These three triggers were treated identically in all respects. Two further triggers were hardware defined, but not used to trigger the data-collection procedure. These were 58 Table II-10. Some Trigger Signal Definitions HV--the logical "OR" of the twelve halo vetoes [HVIOHV20...OHV12] C--the logical "AND" of the telescope counters C1, C2 and C3 [C1-C2-C3] B104--the "AND" of the telescope B1, B2 and B3 [Bl-B2-B3] B--B104-C'HV; if B is present, then a beam muon is entering the target. BV--BVIII°(BVIOBVII); the beam veto signal, if present, indicated a muon traveling down the spectrometer axis. P--the output of a square wave signal pulser SA--the "OR" of the signals from the ten segments of trigger bank A. SB,SC--defined similarly to SA. SAH--the "AND" of any two segments of the V SA counters. SAV--the "AND" of any two segments of the H SA counters ‘ SBH,SBV--defined similarly to SAH,SAV. SLH--SC2@SC3®SC4;zalow angle trigger indicator from V trigger bank C SLV--SC'2©SC'3OSC'4; a low angle trigger indicator from H trigger bank C SCH--SC1@SC5©SC'1®SC'5 a high angle trigger indicator S--SA-SB-SC; presence of this signal means a muon has penetrated all three trigger banks; self- vetoing for 200 ns after firing SL--SA°SB-SLH-SLV; the low angle signal, indicated the muon trajectory was near the spectrometer axis; also self vetoing for 200 ns. -continued on next page- 59 Table IIr10 continued SH--SA-SB°SCH; the high angle signal, indicated the muon trajectory was near the outer edge of the spectrometer. SD—-(SAHOSAV)°(SBHOSBV); the multi-track signal, indicating simultaneous firing of several counters in trigger banks A and B PCS--C~(POSA); the fast pre-trigger for collecting P.W.C. information O is the logical "OR" 60 owumEozom uflwoq ummm 5-: 92%: mm . . S R\:\m .2\cn\n.l uxwoq kutk mfmfxxru puls- 61 Q~..a uo~3w Si‘ .N. U4..\ 9. .Q5CI .;a~a~ A who‘d -~Kn @ D ® .Iui‘ua ,. .54.. b..0\u¢.. .ut\|! n\ ~n. o0 *- OI. ‘l‘ s' . 0 oh... accq. “a. '6‘. o 5‘ §°s§ Ul‘J‘u ~n‘. \ud Qh\n1orh + Nh“ cc t-o u. 002...: i-¢mm ~13...» HQ. Ue xm \fl .3 s» .r «.e u .6 and W“ n. m :51 — ‘ NJ «as. I \ 2.5 e3: has.“ he uxuog kva m\m .4Xw Avozcwucoov .m-HH unnumm 61 vhxuztrr 8.... : 33.x...“ . s ‘0. dr. ... .oE» .8..- 33.4 . ‘3'». U($(» «~23; d .3 .q. u... @ .3... T I. . 4. It. ‘1u\\' «6.... «(3.7; 4:! k 0 ~3 ~ 4.“ 3.51; s; -§v§fi 3% .L . . E. 2!—»\ Q s n I b i...) . \a . s. 1.“... ‘Na :3- ‘dm 1; 2.5 m3: the.“ ewe. 633 :5 Emma: Acozcwucouv .a-~H unawam 62 B-SL-BV, a low-q2 trigger, and B-SH-BV, a high-q2 trigger. These triggers were scaled, and, if in coincidence with one of the active triggers, entered into a CAMAC bit latch (see below). The electronics were gated on/off (active/inactive) in several different ways. The major gate, called the spill gate, was activated by the Fermilab timing signal T2 (Fig. II-lO). This gate which activated all electronics, came on then just before the muon spill was due in the muon lab, and stayed on until just after the muon spill had ended. The remainder of the gates were active only within the time range supplied by the spill gate. Logical signals S and SL initiated their own 200 ns veto gates. These gates kept the logic from overloading with too high of a signal rate. In addition to setting the spark-chamber firing sequence in motion, the trigger signal also set up three gates, the scaler inhibit, the event gate, and a trigger veto gate. The scaler inhibit shut down all scalers, CAMAC and NIM, for a period of 5 us, thus preventing spark chamber noise from disturbing them. The trigger veto gate prevented a second trigger signal from being formed for a period of 2 us, until the event gate was fully closed. The event gate was a 42 ms gate which closed down only trigger-related electronics after a trigger signal was sent out. This dead time allowed the spark chamber capacitors to recharge in preparation for a new trigger. 63 ob All mZHH mh woman; wcwsfle q5) pull spark to get best improvement of x I_. Write track to output tape Iterate for all tracks until run ends Figure III-2. GETP and GETPZ Program Organization Table III-6. 90 GETP and GETPZ Output Tape Format 150 Words/Track Words Content 1 Same as for MULTIMU 31 32 0.0 33 Fitted x-coordinate HPC 34 Fitted x-coord. WSC9 42 Fitted x-coord. WSCl 43 Same as words 32-42 for y coords. 53 Word 46 set = -1024.0 if track is not fit 54 Same as for MULTIMU 58 59 x-component of scattered muon momentum E' from the fit 60 y-component of E' 61 z-component of E' 62 Extrapolated x-position of scattered muon track at z = 0 63 Extrapolated y-position of scattered muon track at z = 0 64 X2 of the momentum fit 65 Degrees of freedom 66 ZADC 67 Shower Energy . 68 e , angle of scattered muon track from z-axis in x-z plane 69 By, same as word 68 for y-z plane 70 Same as word 62 71 Same as word 63 91 Table III-6. Continued Words Content 72 Chamber shut off by GETPZ or zero 73 Chamber shut off by GETP or zero Value in words 72 and 73 is l6-J where WSCJ or HPC (J=10) is turned off 74 Same as for MULTIMU 156 92 It is similar to the output tape from MULTIMU in many respects, but also contains the information from the fitting procedure. D) Program Cross Checks The reconstruction of tracks would have been worthless without verification that it was being done correctly. This confirmation was done in four ways. The first was performed during the debugging of the program. Extensive computer dumps of information on many triggers were taken to verify the program was choosing the correct lines to start with, and then following them correctly to the downstream end of the spectrometer. Finding the correct front lines was no easy task. The hadron shields protected the front chambers to quite an extent, but often there were still many possible lines which could be formed through the many existing spark positions. In addition, there was no shield other than the target itself to protect the hadron proportional chambers. There will be more on this problem shortly. The second verification was obtained through comparison with results of a program called VOREP used earlier in this experiment3zn. This program began with lines found at the downstream spectrometer end, and worked upstream searching for more track indicating sparks. Event tracks found by the two programs wEre passed through a filter program de- scribed in chapter five. On an event by event basis then, these resultant tracks were examined and differences determined. 93 Two regions in particular were examined, Q2>'40 (GeV/c)2 and Q23<8 (GeV/c)2. The large Qz region was found to be inefficient by 914% while the small Q2 region was approxi- mately 1.5% inefficient when compared to the combined event sample of the two programs. These results will be summa- rised at the end of this section when the final check is discussed. The comparison with VOREP still left a class of events not found by either program. This class was determined by visually scanning events randomly chosen from the data tapes. From a subset of tapes scattered over nearly the full range of u+ data taking, 50 triggers per tape were checked for program efficiency. A total of 2298 triggers were checked in this way. The triggers were checked against the fitting program results; if the fit seemed reasonable, no further checks on the trigger were performed. If the fit results were poor (e.g., XZ/DOF too large), or the track was not fit or even found, then the trigger was flagged for visual display using a program which would display a pictorial view of the entire target-spectrometer system in any of the four X,Y,U,V views. These could then be checked to see if a track was actually present. In this way, 1251 triggers were visually scanned, of which 117 seemed to con- tain good muon tracks. This good category did not'include very low energy,high angle tracks or very low angle tracks which the filter program would not pass. For these triggers the VOREP reconstruction was tried, and if this failed, 94 Sparks were chosen from those availableand manually inserted into a MULTIMU type output file for fitting. When finally filtered, 49 events were found to have been missed. This corresponds to an overall inefficiency of ll.5:l.9%. In addition, there seemed to be a Q2 dependence to the ineffi- ciency; it was higher (~20%) at higher values of Q2, although 49 total events did not allow an accurate deter- mination in this kinematically unfavorable region. With these results in hand, all the checks performed were reexamined. It was determined that the inefficiency, for the most part, stemmed from the demand for the presence of three of the front four chambers in any one view to find a line candidate (sec. III-A). Since the hadron PWC's, the most efficient chamber used (99.5%), were only 40 cm across as opposed u>190 cm for the spark chambers, large radius tracks would miss them completely, thereby demanding all of chambers WSC7,8,9 to be present. Chamber WSC8 was the most inefficient chamber used (efficiency 81%), so for these large radius tracks, real problems in reconstruction existed. This was also true for triggers in which the maximum allow- able number of hadron PWC sparks was found, in which case the information contained by them was not used at all. In this case it was as if the chamber was missed completely. These large radius tracks were primarily low to middle E' tracks at large angles, which means that large Q2 tracks were preferentially excluded by the reconstruction. 95 To solve this a special version of MULTIMU was used which required only two point lines be found and allowed information from saturated hadron PWC planes to be consid- ered. This obviously meant more line possibilities, but also an increase of execution time and core storage by the program. The program was therefore restricted to triggers from which tracks were not previously found. Additionally, triggers cut by the filter program's cuts on beam track dis- tribution (sec. V-B) were ignored. In this way, 28 data tapes representing 21% of the 270 GeV u+ data sample were reconstructed. Events successfully passed through the filter program were weighted by this fraction and the result was applied as a weight factor to all events from the origi- nal MULTIMU which passed the filter program cuts. This weight factor is shown in Fig. III-3 as a function of the reconstructed variables Q2 and x. E) Raw Kinematic Distribution With this weight factor the reconstruction of the data is complete. The modified version of MULTIMU was able to find nearly all of the 49 trigger tracks found in the visual tape scan. The remaining events are assumed to be uniformly distributed kinematically, and as such can be treated as an overall normalization factor of 11n9. Some of the-raw kine- matic distributions of events are shown in Figures III-4 thru III-9. These distributions reflect the true cross section for deep inelastic scattering of muons modified by 96 nucoouonv >ocoMo«mwo:~ Dzmbqaz mo “OH; psowcoo .m-HHH mesmHm X mam.c mom.c mN¢.O mvm.c mo~.o me.O mcm.o mNc.c . _ n _ n — b — p n p 3 . b .moN l m m mm 1 .m.~e - Nae\>eov 1 Tm.mc Hm ._ .m.~w 97 oo.o~N oo.cm~ eewusefieemflo Eben No see .e-HHH beam“; Nxxao\>mc. ommmsam a oo.o a co.o a oo.om oo.oe oo.cm oo.pu 00’ OO'UI 00'08 00‘07 98 oo.om :euuseMeomfla eeea 3 28¢ m-HHH bezmfle oo.om co. v mouse co. N 60.3 oo.c I 0 0 0 8 0 0 0N / 3 .13 .I m... .0 0 Z .7 0 0 0 x I O z OO'OZS 99 :ewuabfleumea been .m zed .e-HHH beamfie A>mo. MEHmmm oo.oem oo.oom oo.o N oo.o N oo.om_ co.oez oo.oe2 ae.om oo.omu 00 100 .00 28.00 *r ’ F T 48.00 58.00 8000 SINITHETR] X 1000. Figure III-2 Raw Sin(e) Data Distribution. 1108.00 101 :ofluznwpumwn mum: x 3mm .m-~HH beam“; 102 00.0 v 00. v 00.0 0 00. :60626oeumoo ewe: N3 zen a-HHH beam“; ANS->666 mmmczam 3 00.0 00. 00.0 n 00. u 00. 6 00.0 aeguuu DD‘ 00' .01:- 00' 103 both acceptance and resolution effects. To deduce the actual cross section, or alternatively the structure functions, the acceptance and resolution effects must be precisely deter- mined. The monte carlo program discussed next was designed for this purpose. CHAPTER IV THE MONTE CARLO A) Modeling the Data The spectrometer used in the experiment, unfortunately, did not have 4n steradian acceptance. Therefore a means of modeling the expected kinematic distributions was needed. The monte carlo program MUDD, originally written at Fermilab by A. Van Ginnekan to model D5 production, fulfilled the required acceptance and resolution modeling. Our analysis would have been easier if Eqn.(I-9) developed in chapter I for the differential cross section was the only contribution to events measured. Corrections were necessary though, and the ones eventually applied to the cross section are listed in Table IV-l. Conveniently for the user, the program was structured in such a way that these corrections could be applied as easily after the pro- gram was run as during the actual running. B) Multiple Coulomb Scattering and Energy Loss As a muon traverses the length of a dense substance, there is a large probability that it will be deflected by the concentrated charge of a nucleus without undergoing a deepeinelastic scatter. This type of scattering, called 104 105 Table IV-l. Cross Section Corrections 1) Wide Angle Bremmstrahlung background 2) Radiative Corrections 3) Fermi Motion effective nucleon mass 106 (32) multiple scattering, results in an angular scatter which is approximately gaussian with standard deviation epTzne for rms rms angles ems < eplane and falls as a power for ems > eplane’ rms . . (12) where eplane 15 defined by rms = .015 GeV/c . eplane Zproj p8 6x/LR radlans (l) and describes the net. distribution of scattering angles projected onto any plane. 6x is the thickness of the medium, LR is the radiation length of the medium, p is the projectile momentum, z is the projectile charge and Bc is its velocity. THIS The non-projected angle is distributed with arms = fieplane' The formula is accurate to within 10%. This distribution was simulated in the monte carlo by the equation 0.00045-5 x ) ) V 2 2 p 8 LR where RV is a uniformly-distributed random number in the - l/Z ems - (-Zn(R (2) interval [0,1]. A broad, single-scattering tail on the dis- tribution was ignored since, in the region where the tail could begin to be important, it is damped out by the nuclear (25) form factor. As the muon traversed the target and spectrometer it was also losing energy thru various processes. Energy 1055(34) for three of these processes, ionization (or col- lison), brem strahlung, and pair production, was modeled in 107 the monte carlo. A fourth process, energy loss due to nuclear interactions, is much smaller than these three processes and was ignored. The ionization loss is due to the interaction of the muon with atomic electrons. For particles heavier than electrons, the average energy loss is given by the Bethe- Bloch equation, dfi = Znnzze dx 2 [MET—M) - 232 - a - u] (3) where n is the material electron number density, I the material ionization potential, W the maximum energy trans- max fer to the electrons, u a screening function for inner shells important at low energies, and 5 a density function due to polarization of the material by the passage of charged particles. For passage thru a material of finite thickness, a broad distribution of energy loss is formed, known as straggling. This distribution is shown in Figs. 2.71 and (35) 2.72 of Rossi. Following Rossi's prescription and using the values of Sternheimer(36) and Joseph(37) a most probable energy 1055 8p was thrown: a =5-t-9-[B+1.06+21n-P-+ Znfl- 82 - (5 +5] p 2 m 2 B u 8 ZHN e482 -5 GeV A = —_on—_ = 7.15x10 ——7 [for iron] (4) me\ o g/cm me B = Zn —— = 15.64 I2 108 r’ o 0443 82(fii)1°73 ; £1 < 4 U U 10[0.043 - 0.730(x-l) - 0.188(x-1)2] 5 =4 + Z‘ba 31 - 4.3 ; 4 < 31 < 100 (s) m m U U 2 bag: - 4.3 + (3.8 + 3.33x10'4(fiE—)2)'1 U U , E , E— > 100 K H _ E x log10 E- b = A = 1.48 p 1 NZ Bi 2mu Eu 5:17 T~=m—/(—m +fi‘) (6) Eu u e u 0.3Zm btp A = '3 Fits to the distribution in Rossi in four regions in terms of the variable x = 3%;5 (Table IV-Z) have been made and the integral probability i: the regions computed. Based on these normalized integrals the region in which an energy loss would be made was chosen (randomly) and the ionization energy loss for the desired material thickness was computed. Table IV-Z. 109 Fits to Ionization Loss Region 1: Region 2: Region 3: Region 4: Probability fraction 0.094 f(x) = 0 4662 e(x+1.878-10‘4)-o.4662 -4.0 S x < -0.75 Probability fraction 0.378 ( _x2 ) 2 fo) g 1. e2-0.9703 (210* -0.75 S x S 0.75 Probability fraction 0.453 f(x) = 0.003672-x - 0.08318 + 0.61186/x - 0.25295/x2 0.75 < x f 9.0 Probability fraction 0.075 1 f(X) = 7 X E -ep-m 9 < x < Ao E-c x = ———E. e and A defined in A p ° 0 the text 110 The energy loss due to muon bremsstrahlung, the radiation of a real photon by the muon, was modeled after (38) the work of Tsai. The probability of a bremsstrahlung is a function of the muon energy, given by the equation NA P(E) = 7X- 01: GB (7) where do 3 = __B = 2.. l OB jdy dy m, [y Fmdy u 4 4 pm = (g - ‘3‘Y* yznzzwl - gm - mm (3) + 204 4 §an)) 1 3 6 1.008(—5—) f(z) = 1°202(T%7)2 - 1.0369(T%7)4 + 213; (Del-37) ) y = photon energy muon energy 01 and 01 are Bethe-Heitler screening functions. To shorten computing time, a fit to P(E),which is accurate to within 1.2%,was used rather than this exact expression. If a thrown random number was greater than this probability, no bremsstrahlung loss was calculated. Otherwise a photon energy in the distribution %IW50 was chosen. This energy,loss occurred about 2-3% of the time. Pair production of electrons by the incident muon was calculated on an average basis only since straggling is 111 highly suppressed (~v'3) and what is known about the strag- (39) gling is only approximate. This contribution to energy loss is always present and is 2 dB _ N me (azre) Eo a? -' Km—--—-1T—-—Eo(19.31nm—' - 53.7 f) U U ; Eo < 20 GeV (9) E . L9 183 .12 .2-u f (9 1712?? + l)/(9 anu 9 +Zn2) ; E > 20 GeV 0 where f is due to screening by atomic electrons. The total energy lost by the muon in traversing a thickness t of material was the sum of the contributions from these three processes. Multiple scattering and the muon energy loss were computed together for a given step thru the target or spec- trometer. Before the interaction point, the muon was straggled, then multiple scattered and straggled again over the course of a full step. After the interaction point it was multiple scattered and then straggled for the first and only time for each step. Both these methods gave the same distributions of multiple scattering and energy loss for traced muons. 112 C) Radiative Corrections Several higher order corrections to the measured cross section must be performed in order to arrive at the final parametrization of F2. These processes, the most important of which are shown in Fig. IV-l, disguise themselves as an interaction at given B0 or E', when in reality they may have been a lower E0 or higher E' process associated by the emission of a photon away from the virtual photon-muon vertex of the primary deep inelastic scatter. Collectively, the corrections to the measured cross section due to these pro- cesses and necessary to arrive at the true deep inelastic cross section are known as radiative corrections. If the cross section is measured at a point A in Fig. IV-Z, the value measured will be larger than the true cross section because of radiative contributions from events in the shaded triangle ABC.(40) The region is bounded by the lines coséLY= :l and the elastic scattering peak, where BY is the angle, in the scattered muon-virtual photon plane, of the emitted photon with respect to the virtual photon. Rather than calculate the exact correction(41’42) for every simulated muon, the value of the radiative correction on a 30x30x10 grid in E', 6 and E0 respectively, was calcu- d(41) lated by the effective radiator metho and then linearly interpolated to the specific values needed for a given muon. This method was compared to the exact method of Mo and (41)and the ratio was found to be accurate to one or two (43) Tsai percent with deviations becoming larger for very low E'. 113 Figure IV-l. Radiative Process Diagrams c/ “-cos 0 = -l ( Y) A B ‘cos(ey)=+l '— V Figure IV-Z. Triangular Region Contributing to Radiative Correction 114 l 145 r 105 ' zzs ' 265 E' (GeV) Figure IV-3. Contour Plot of Radiative Correction Weight Factor 115 The value of the radiative correction was R = (10) where the subscripts el, inel and QCD imply the elastic peak and inelastic radiative contributions and the Buras QCD(9) parametrization of the cross section, respectively. The event weight was multiplied by this factor to get the corrected weight. Fig. IV-3 shows the value of R as it contributes to the Monte Carlo event weight, as a function of E' and 0. D) Wide Angle Bremsstrahlung (WAB) Background (44) D. Yennie pointed out that wide angle bremsstrahlung can be an important background process in deep-inelastic lepton scattering. In this process a bremsstrahlung photon is emitted at a larger angle than usually considered (t >> mfiEY/(EO-E')). Momentum conservation requires the muon to also undergo an angular deflection which can then be misinterpreted as a deep-inelastic scattering. The ratio of the cross section for WAB to the deep- (45) inelastic (DI) cross section can be written as, 2 2 l 7 - _ dGWAB 2 220 )F' G(QII) (l yw'zy') (11) doDI 0A l-y vW Fe 1 2 2 (1'y+21yfi) 116 .01 - 26 42 - 0 _ (mrad) 58 - Contours are values of DS in the equation: - 74 WTnew = WTold-(1+DS) _ 90 b 25 T 65 ' 105 ' 1'45 ’ 105 r 525 ‘ 2255 ' 3' (GeV) Figure IV-4. Contour Plot of Wide Angle Bremsstrahlung Background Factor 117 where 2 ”deQi'Fszmfl) GCQII) =1 2 2 2 (12) O (Q,+Qll) F2 is the nuclear form factor and directions are defined with respect to the beam direction. The nuclear form factor integral was approximated by the fit C(Ql|)2 = ('68'362 - 2910107 + .70671 + .0119690 - .49948-10‘402) (13) b.) e which is accurate to less than 1% for omega between 5 and 100. The effect of this background correction is shown in Fig. IV-4. E) Nuclear Fermi Motion Since the target iron nuclei are not at rest in the lab frame, but the cross section is calculated in an assumed nucleon rest frame (NRF), the effect of the nuclear motion on the measured cross section must be considered. This was done by allowing the nucleon a Fermi gas motion described by the probability distribution P2 W(P) = 2 2 ' (14) ((P -PF)/2MkT) 1+e 118 where kT==0.492/56 GeV, PF==O.259 GeV and M is the proton mass. The spatial orientation of this Fermi momentum was randomly applied, and the momentum range was limited to |P| S 2-PF. Once the transformation to the nucleons rest frame had been performed, the cross section could be calculated. This was done using an effective nucleon mass of 0.9086 GeV which was used to determine the energy at which the inter- action occurs (15) This effective nucleon mass conserves the atomic weight of iron for this Fermi momentum distribution.(28) F) The Incident Beam Because of the holes in the toroidal magnets, the spec- trometer acceptance depended on where the beam entered the target. To best simulate the data then, a beam representa- tive of that actually entering the target was used in the monte carlo. The pulser trigger information helped to accom- plish this. These triggers, collected from the data tapes and merged into three beam tapes (Table IV-3), were input to the MUDD program at event generation time. Study of the beam distributions from data events and monte carlo events showed that for beam track radii less than 8 cm, angles less than 2 mR, and extrapolated positions at BVIII less than 10.5 cm in radius the beam distributions 119 Table IV-S. Beam Tape Format. Total Record Length 100 words Words Written/Event 10 words Events on Each Record 10 events 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) Breakdown of 10 words Data run number Data event number Track $10pe in x-z plane Track slope in y-z plane Track x—position at upstream target edge Track y-position at upstream target edge X2 for fit in x-view X2 for fit in y—view Contents of data event DCR #3 Beam track energy 120 DD H9 x§_ 1 L 1 1 1— 8 #**‘=~.‘~: o w‘ x“: O- 7‘ N: _ " N f ‘5. f N .J _J MD _ _ L) \ 2 ca 9 Lo .00 2.00 4.00 5.00 a'.00 10.00 “J- 1 1 L ' 1 1 '- 1-15 __9_ 1++++++M4i++ M ET 1 + 4, 4) ++ + _ 44 +11 .00 2.00 6T0 00 10.00 DHTH BERN RRDIUS (CH) c035 Figure IV-Sa. Data Beam Distribution and Comparison to Monte Carlo 121 40.00 110' N/CELL 10 00 20.00 39.00 T“}\_ 1’*/7. ,zd”' 1/4”r O 0 =0.00 0.40 0.30 f 1120 1.00 2.00 09TH BERN HNGLE (HRHU) 3. L L l l l 3 .:q L- RRTIO 1:00 _9._. 4r 4» 3. t O + c0.00 '0100 0240 0100 {T20 {:00 DHTH BERN HNGLE (HRHD) Figure IV-Sb. Data Beam Distribution and Comparison to Monte Carlo 122 agree with each other (Fig. IV-Sa-b). Events with beam tracks outside of this acceptable region were rejected and the incident fluxes for both data and monte carlo were corrected. C) Computer Program Description The program itself was run in three stages. The first stage used as input the target density, length, and radia- tion length as well as the position and composition of all spectrometer elements. A beam tape containing information from the experiment's beam sampling pulser trigger was read to determine where an incident muon would enter the target. A random number was thrown to determine where in the target this muon would undergo a deep-inelastic scatter. The muon was then traced to this z-position in a maximum of ten steps, each of a minimum length of 5 cm. While stepping to this point, and during all later target stepping, the muon would undergo simulated multiple scattering and energy loss (sec- tion IV-B). If at any time the muon lost all its energy, it was deemed to have failed a hardware cut and rejected. At the interaction point, a Fermi-momentum vector (sec- tion IV-B) was determined, and quantities were relativisti- cally transformed into a reference frame where the interacting nucleon was at rest. In this frame, an energy and'an angle for the scattered muon were chosen (section IV-H), the currently favored weights were applied, and a trans- formation back to the lab frame was performed. 123 With the event fully generated the beam cuts could be performed. These cuts, also performed on the data, ensured that the beam distributions (angles from the axial direction, energy and position) of the accepted events were identical. Deviations between MUDD and MULTIMU beams would result in a systematic deviation of the determined structure function from the true value. The total weight of events cut in this manner was used to correct the usable flux by the equation wCUT usable flux = thrown flux x (l - W———) (16) TOT where WCUT = sum of beam cut event we1ghts and wTOT = sum of all thrown event weights. Events with an acceptable beam were then passed to further tracing. From interaction to the face of the first spectrometer magnet, 5 cm steps were used, while for the remainder of the spectrometer, a new step size was calculated. A check of the muon energy was made to determine if a smaller step was necessary, since low energy, large bend tracks need to be stepped more carefully to avoid a systematic positional deviation which would increase with distance into the spec- trometer. For this purpose the E-26 monte carlo criterion was used which determined a step small enough that a 5 GeV (5) muon in the Fermilab E-26 spectrometer configuration no longer deviated in position when it reached the downstream spectrometer end, yet large enough that computation time was not excessive. A step determined by this method which was 124 Table IV-4. Stepping Process Cuts A. Hardware Flagged Cuts 1) 2) 3) Not all three trigger banks struck u st0ps or leaves spectrometer before SC trigger bank BV3 and either of BVI or BVZ struck Software Flagged Cuts 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) Scattered 0 energy less than 10 GeV at hadron shield ZMIN < Zabs = -250.0 cm ZMIN > Zabs = 600.0 cm DMIN = distance of closest approach of incident and scattered (smeared) muon track DMIN > MINIMUM (2.0+15-R8, 10.0 cm) R8 = track radius at WSC8 R = track radius at a given point (z) R < 16.8 cm at SA' R < 17.9 cm at SB' R < 18.0 cm at SC' R < 17.5 cm at BVl and R < 16.7 cm at BV3 R < 18.0 cm at BVZ and R < 16.7 cm at BV3 Muon crosses spectrometer axis between SA' and BV3-- check for change to 180° Opposite quadrant as com- pared to that at SA' Beam Cuts 1) 2) 3) 4) 0) BVIII) Track radius < 10.0 cm (2 Track radius < 10.5 cm (2 Track angle < 0.002 Radians IE0 - 270.0l f 27.0 GeV 125 Table IV-S. Accepted Event Tape Format Total Record Length 510 words Words Written/Event 17 words Events on Each Record 30 events 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) Breakdown of 17 Words Event weight 0 0 Beam track angle, x-projection Beam track x-position at 2 Beam track y-position at 2 Beam track angle, y-projection Incident muon energy at upstream target edge (E0) ZMIN--z position of closest approach of incident and scattered (smeared) muon tracks Angle scattered muon makes r.e. incident muon, measured at the hadron shield E of scattered muon measured at the hadron shield Energy of incident muon in the target nucleons rest frame (EO(NRF)) Angle of scatter in the NRF (0(NRF)) Energy of scattered muon in the NRF (E'(NRF)) Radiative correction weight factor Run x 100000 + event from which beam track came F2 = 0W2 used in the event generation Incident muon energy in the lab frame at the inter- action point (EO(LAB)) Scattered muon energy in the lab frame at the inter- action point (E'(LAB)) 126 Table IV-6. Generated Event Tape Format Total Record Length 17 words Words Written/Event 17 words Word Breakdown 1) Event weight 2) Beam track x-position at z 0 O 4) Beam track angle, x-projection 3) Beam track y-position at z 5) Beam track angle, y-projection 6) Incident muon energy at upstream target edge 7)*Scattered muon energy at the hadron shield 8)*Angle scattered muon makes r.e. incident muon, measured at the hadron shield 9) Thrown interaction position, z-direction 10) EO(NRF) 11) E'(NRF) 12) 0(NRF) l3) Radiative Correction weight factor 14) EO(LAB) 15) E'(LAB) 16) Logical variable, FALSE, unless muon cut on a hardware cut 17) Integer flag indicating cut classification 1 = accepted 2 = software cut 5 = hardware or energy of u too low *Variable set to zero for a beam cut event 127 smaller than 5 cm was used instead of the previous 5 cm stepping which was otherwise retained. Cuts applied during the stepping process are listed in Table IV-4. Events which were not cut were sent back to the main program and pertinent information was written onto an output tape, as shown in Table IV-S. In addition another output tape was maintained containing information from all events generated, including those cut because of their beam track. This tape format is shown in Table IV-6. The whole process was now repeated for a new muon until the maximum specified at the program beginning was reached, whereupon this first program stage terminated. The second program stage calculated the radiative cor- rection to the event weight (section IV-C), filled in the appropriate word for either the generated event tape or the accepted event tape, and wrote the block out to a new tape. The method used was a three-dimensional (E0, E', 6) linear interpolation of a table previously written for this purpose. The table contained the radiative correction value at spe- cific values of the variables E0, E' and 0, measured in the rest frame of the target nucleon. If values were returned with the radiative event weight correction less than 0.84, then the correction was calculated by the same method used to generate the table, since for these events the inter- polation was not accurate. The third program stage performed the requested resolu- tion smearing on the accepted events, applied kinematic cuts, 128 Table IV-7. Smeared Event Tape Format Total Record Length 459 words Words Written/Event 153 words Events/Record 3 events 1) 2) 3) 4) S) 6) 7) 8) 9) 10) 11) 12) 13) 150) 151) 152) 153) Breakdown of 153 Words I:2 EO(NRF) E'(NRF) 6(NRF) E'(LAB) EO(LAB) ZMIN Eo Number of resolution smears (max 36) this event E' at hadron shield, first smear sin(0) at hadron shield, first smear x = Qz/ZMv for first smear WEIGHT for first smear E' at hadron shield, 36th smear sin(0) at hadron shield, 36th smear x for 36th smear WEIGHT for 36th smear Table IV-8. Third Stage Program Cuts 1) 40 < E' < 325 GeV 2) 0.010 < e Radians 3) 1 < 02 < 500 (GeV/c)2 4) Beam Track Radius < 8.0 cm 129 and wrote an output tape in preparation for comparison to the data. At this stage reweighting the events could be performed, if desired, or additional cuts applied. The out- put tape here was similar in structure to the final data event output tape, but contained additional information on the actual kinematic variables of the deep-inelastic scatter (Table IV-7). Using this procedure a total of 450,000 muon paths were traced, of which 206,900 passed the tracing cuts of Table IV-Z. The resolution smearing and other third stage cuts (Table IV-8) allowed a final total of 200,150 events with up to 36 smears each to be written to output tapes. The analysis used in chapter five is based upon these tapes. H) Event Weighting There are two basic methods of simulating a distribu- tion such as that of a cross section. The first is to actually select events in proportion to the cross section. For example, if one wishes to throw an exponentially decaying distribution, one normalizes the kinematically accessible region of the exponential by the maximum ordinate value it attains in this region, then for a given value of the abscissa, which is randomly selected, a second random vari- able is thrown. If the height of the normalized curve exceeds the random value [range 0 to 1] then the event is deemed acceptable and the track tracing can proceed. Otherwise, a new abscissa value is chosen and the process is repeated. 130 This method, used by the E-26 monte carlo program, results in all scattered muons having a unit weight just as for the data, since the rate of occurrence of these simulated muons is exactly proportional to the cross section. The second method, which was used by MUDD, is to ran- domly throw kinematic variables, and then to weight the event by the probability of its occurrence (the cross sec- tion). In this way there was no iterative repetition in search of acceptable values of E' and 6. To aid statistical stability, speed execution time, and approximate the kine- matic acceptance, the scattering angle was not thrown uniformly over the experimenfls usable kinematic range. It was instead thrown uniformly from 8 mR to 20 mR and then as a falling distribution from 20 mR to 150 mR. This approxi- mated the cross section somewhat which falls as 0-3. The procedure was to throw one random number to determine which angular region will be used. A value less than 0.409092 meant the low angle region would be used, a higher value chose the large angle region. For the low angle region, a second random number was thrown (VRAN) and the angle used was 0 = 0.008 + V 0.012 - (17) RAN Since any value of theta in this region had the same proba- bility of occurrence as any other, all such events were assigned a weight factor (We) of 0.0293333. For the high 131 angle region a second random number was again thrown, but the assigned angle was e = 1.0 (18) 1.0 _ V 1.0 _ 1.0 0.020 RAN 0.020 0.150 The weight factor for such an event was W = 73.3332°02 - (l9) 0 The weight factor in either case was the inverse of the probability of occurrence of the chosen value of 0 so that the total probability of choosing some angle to scatter thru was exactly 100%. The scattered muons energy (E') was thrown as a uniform distribution between 20 GeV and the energy of the incident muon (E0). A weight factor (WE, = E - 20.0) was then 0 assigned such that probability was again conserved, i.e., there was a 100% probability of throwing some value of E'. The form of the cross section used was a variation of Eqn.(I-l4) of chapter I which took into account the average density and total length of the target (3 and L) and the total flux of muons the program was expected to simulate (F). This cross section formula was (NO is Avogadro's number) 2 d o _ _ 2— _ . 2. E'-sin(0)' ddee ' OTHROWN ‘ SWF“ pLNo (fic) ( 2 2 ) y-(q ) (20) 2 1 . r . - _ _JL__ V“2 (1 Y + 2 (1+R)) 132 The final event weight used, at this stage of the program, was just WEIGHT = WE"We'GTHROWN' This weight, modified by the wide angle bremmstrahlung correction (section IV~D) was the weight written out on the first stage output tape. Since by this formulation each thrown event was weighted as if it were the only scattered muon to result from the total input flux, a normalization factor equal to the total number of interactions thrown must be applied to the results. I) Geometric Acceptance and Resolution Previously, monte carlo events were tracked thru the spectrometer, recording spark positions at all spark cham- bers. These sparks were then smeared by a spark resolution function, and then turned off or left on depending on the comparison of a random number to the measured efficiency of the chamber module. Sparks appearing in regions where the reconstruction program would not look, such as the toroid hole region in the back five spark chambers, were also dis- carded. The remaining sparks were then fit as if this were a real data event. The MUDD program does not fit events, but rather takes an event which has passed all geometry cuts and then smears this event in E' and 0 using a gaussian function. Using the E-26 monte carlo, suitably modified, events were started thru the spectrometer with specific values of E' and e as measured at the hadron shield. These tracks were fit using the standard event fitting program and the resulting 133 distributions in 1000/E'FIT and 0 were fit to guassian forms. This procedure was followed for E'- 0 pairs on a rectangular grid for 81<6'<100 mR and 20iH opswww 92: .m com . 66m _ o- . 66— _ cq— . 66— _ om _ w (1. {)1 {iv - . Xe.— O N \0 O5 -\r_l.__r__r 33 1 ,/ I A \\\\\ // We» M \ \\ P2 ._ S: _1 M .14. v l.(\\l‘l|v . .r m we, .re 35. f\e / 3:5 6 135 mmev :ofiu3H0mom «Hock mo uofid pseucou .n->H opzwwm :68 .m com 63 oww 66” 6: '.o N .‘-_ $0 0‘ I}. /. t. 3 V j/\ \— in N m \ "5 ‘\~>\- .“l... T- .5— 1.;1- -§"T “55 5:"). _§'_T"-é-'T- m 6 a: E V ‘ --—’-‘#’ 1 g, F - N A,. - ..-. —.- I— N ('1 ¢ — i L}. 1"». 1...- . ° 2 136 fl>oov momfiem .m one seoeoEo: do eoHd esooeou fl>oov .m 66m 66~ 6- 62 6: 66— 66 . .m->H ensued fleev 137 uoHd psoucou oocmuaooo< oumzwam: .m->H ouswwd 9o”: .m can omw 6- . 66— . 6: 'Ar 66- 25 138 poHd wsoucou museumooo< opmzumom fl>oov .m we _ exN 1 cm em, 1. em_ (in em. _ ow w oN . A fi#.em n -8 1. I g. \|\I\ Wow \‘ I 1 rfi b\\\\\\I\\\\mm .3 V 1...) L - |I\\\\\\ % +3 1; .3 . a: u A A “cox :\ 6 \ m _ _ JL.__l .oH->H oesmfid flmsv 139 mosHm> No acmumcou mo musoucou .HH->H opsmwm :68 .m CNN owH ova - — . F p 1 1\\ .11111111111111) \\\\\\\\1\\\ o N\ \\ j . \11111111\\\\\\\\\\\ e\H\\ ,2. 1.111111 \‘111111111111‘1111‘111‘ \\\.\ T. .11)....) )\ 3 1 11111111111111)“ fin 140 260 300 220 (GeV) E' Contours of Constant x Values Figure IV-12. 141 monam> CNN _ N n>ouv OMH _ z ucmumcou mo muscucou .m 3: ! 1 l l L .mH->H deeded mmev CHAPTER V ANALYSIS PROCEDURE A) Normalization by Flux As was noted previously, the number of deep-inelastic interactions which can be expected to occur is dependent on the flux of muons onto the experimental target, i.e. on the total number of muons transported to our target during the entire course of the data taking. This quantity, labelled BEVG’ was counted continuously by a scaler during the experiment, and updated on the data tape as each event was written onto it. The value in this scaler was too large, since it did not allow for the incidence of two muons in the same r.f. bucket (sec. II-A), one of which scatters, the other not, nor did it allow for the non-zero (27) of the beam veto (a BV signal when accidental rate BV should not really be present). Either of these con- ditions would veto a good deep-inelastic interaction, pre- venting it from being recorded. For this reason, the quantity BoEVd’EVG was used as the flux. The BV signal was delayed (d) 60 ns, equivalent to three r.f. buckets. If this signal was not present when a muon next set BEVG’ then the muon was also counted in the flux. The delayed BV 142 143 signal was used since any bucket had the same probability of containing a muon, and it could therefore simulate the two occurrences listed above while avoiding problems with hard- ware dead time. The scaler was gated live by the event gate since deep—inelastic interactions could only be successfully recorded when the event gate was live. For the 135 runs considered here, the total quantity of muons scaled was 1.2859 x 1010. Two types of hardware failure were not detected in the same on-line manner as the BBVG correction, but were accounted later. These were complete failures of the spark chambers and the so-called branch diver CAMAC errors (BDERR) discussed in section II-M. These types of errors completely invalidated the trigger on which they occurred, so the flux was corrected downwards by the rate of their occurrence. This 1.51% correction changed the flux to 1.2665 x 1010 incident muons. Three types of error other than these have been cate- gorized,but there is no correction to the flux for their occurrence. The three types are pulser triggers (B-PEVG), triggers due to random electronic noise, and a category including triggers via halo muons and interactions outside of the target bounds. These types can be ascertained to be uninteresting, and a monte carlo simulation has shown that the occurrence of such triggers does not affect the total number of good deep-inelastic interactions when normalized by the computed value of the muon flux. 144 mofiemom anemone szxagn .H-> ensued xcsw ou< sows: mHoMMMHH Hence we w ecu ed em on so em oe em ON CH c a _ _ . _ _ . q . m .oom w.H u someog Hfieem eooefioeflm ++ 1 . 1 + 1. + a e l 1 r 1., 1 1641111: «:1. 1..-: 8. 3mm - - ea .uomE me n 0569 ween wouwazewm . . L . . p . _ _ ll 0 .uom\: 666.6HH ll «1 .oom\: eee.emm oom\: oco.ooN.N u > .oom\= ooe.- ll 4 .oom\: coo.kem 145 Table V-l. Beam Distribution Limits Limit Description 1 Angle from axial less than 0.002 Radians 2 Extrapolated track radius from axis, measured at target center, less than 8 cm 3 243 GeV < E < 297 GeV (10% deviation from nomina? value) 4 Extrapolated track radius from axis, measured at position of BV3, less than 10.5 cm 146 The reason is that such triggers close the event gate, and so do their own correction of the incident flux. Two final types of occurrence also resulted in correc- tions to the flux. The first was a category of triggers where the incident muon track did not satisfy the criteria listed in Table V-l. The second was that class of triggers where zero or more than one beam muon track was found. Both these categories contain events wherein a portion of beam muons was rejected as unsuitable for use, and so the inci- dent flux was corrected for that fraction rejected. These two brought the flux down to the value finally used, 9.186 x 109 incident muons. For the monte carlo program MUDD, several of these cor- rections were unnecessary. Pulser triggers with bad inci- dent energy or without just one beam track were not written onto the pulser trigger tapes, and the triggers with hard— ware failures were always ignored. The only flux correction necessary then was due to the same cuts on the beam track muon listed in Table V-l for data events. These tracks, in order to simulate the effects of the chamber measurement error on their characteristic angles, positions, and energy, were smeared by gaussian functions for each monte carlo event. The angles were smeared 0.01 mrad, the x and y posi- tions by 0.01 cm, and the energy by 0.3 GeV unless.the energy was exactly 270 GeV. In this case only positions and angles were smeared since it was assumed this was a trigger where the energy was assigned to 270 GeV, the real energy 147 being unavailable. Since the monte carlo was assigned to generate a number of scattered muons equivalent to 1.0 x 1010 incident muons, the beam cut correction of 6.2% corrected 9 this to 9.380 x 10 incident muons. B) The Filter Program COMPARE With the reconstructed and momentum-fit data events in hand, the task of comparison to the monte carlo predictions could begin. To do this, it was first necessary to elimi- nate from the event sample (a total of 860,037 events of the 1,364,000 taped triggers) those triggers in unstable geo- metric and kinematic regions. For example, scattered muon tracks which were too near the toroid holes were eliminated because the positions of the trigger bank counters SA, SB, SC, and the beam veto counters BV were uncertain by 1/4 cm. For safety then, circular cuts on the track radius were applied which radii were at a minimum 1/2 cm removed from the counter edges. Additionally, the number of triggers cut because of their beam track was tallied. All applied cuts are listed in Table V-Z. The order in which the cuts were applied is important. Triggers were first checked for proper beam distributions. These cuts were applied first since, in the cut accumulation, once a trigger failed, higher numbered cuts were not con- sidered; the program simply skipped to the next trigger. To then get an accurate determination of the amount of beam lost in this way, these cuts must be applied first. This 148 Table V-2. Ordered List of COMPARE Program Cuts . . Triggers Cut Descr1pt1on Cut 1 6,037 2 ' 163,767 Cuts as listed in Table V-l 3 683 4 1,452 5 Junk Triggers* 34,358 6 Z-position (fit) of interaction (ZMIN) within range -250 cm < Z < 600 cm 10,797 7 Minimum distance of approach of incident and scattered muon tracks (DMIN) less than 5 cm 9,832 8 XZ/degree of freedom <10 2,689 9 Track radius at trigger banks less than 16.8 cm (SA), 17.9 cm (SB), 18 cm (SC). Cut applied to fit muon track radius at nearest spark chamber to trigger bank (WSC 6-4 respectively). 334,647 10 Track may not cross axis into opposite spectrometer quadrant between WSC6 and WSC3 7,408 11 Scattering angle >0.01 Radians 6,639 12 Energy of scattered muon in the range 40 GeV < E' < 325 GeV 20,988 13 QZ > 1 (GeV/c)2 0 Total Reconstructed Trigger Sample 860,037 Total Triggers Cut 599,297 Total Accepted Events '260,74O J 1 *A junk trigger is one which could not be successfully fit, or which had fewer than one degree of freedom. 149 was also true in the monte carlo generation, where, once it had been determined that a beam track was in an unacceptable region, no further tracing thru the spectrometer occurred. Geometric cuts were next applied. This corresponded to the monte carlo where, as muons were traced, information was accumulated on the track position at the various points in the spectrometer where cuts were applied to the data triggers. Failure to pass the cuts resulted in the monte carlo event rejection at this stage. The final cuts applied were those on the track kine- matics. In the monte carlo, the third stage, resolution program was where the simulation of the GETP track fitting was performed. This assignment of energy and scattering angle to the muon occurred after the monte carlo performed all other cuts, and this order was preserved in filtering the data triggers. The order in which the geometric and kinematic cuts were applied was not important as far as the final data to monte carlo comparison was concerned. It did make the comparison easier to perform, and allowed optimal cut values to be determined without undue delay, for instance, due to a necessity to rewrite monte carlo output tapes, a tedious process. One obvious difference between these two approaches is in how the cuts are applied. In the monte carlo, the exact position of the muon was always known, and was the quantity used in the position dependent cuts. For data triggers, the 150 most reliable quantity was the predicted track position from the momentum fit, since the muon position was not always known at every chamber. These two quantities are obviously not identical. A given track may pass a set of criteria based on the fit position, but fail on actual position, or vice versa. For a large number of scattered muons though, this difference minimizes, leaving no detectable trace in a comparison between the two methods. This is a result of the careful alignment of the spectrometer in which the distribu- tions of actual minus fit spark positions were adjusted at all chambers to give mean values as close as possible to zero. The deviations from a mean of zero discovered for y-positions in the downstream-~most six spark chambers (section II-E) were simulated in monte carlo runs so that no systematic error would be introduced. Once a trigger was past the set of cuts, it was con- sidered to be a good deep-inelastic event and its kinematic content could be accumulated. Three output tapes were main- tained for this purpose. The first tape was merely a list containing the run and event numbers unique to each event. It was written in a computer output format, 13110, which is thirteen run-event pairs per line written, 10 columns per pair, right justified in the field. Each pair was one num- ber, equal to the run number multiplied by 100000 then added to the event number. The second tape contained various non~kinematic bits of information on the event which were not really pertinent 151 Table V-3. Non-Kinematic Information, COMPARE Output Tape Word Information 1 E0, measured at upstream target face 2 ZMIN (cm) 3 DMIN (cm) 4 Xz/degree of freedom 5 Extrapolated beam track position at target center, x-z plane (cm) 6 Extrapolated beam track position at target center, y-z plane (cm) 7 Extrapolated beam track radius at target center (cm) 8 Projected beam track angle, x-z plane (mrad) 9 Projected beam track angle, y-z plane (mrad) 10 Axial angle of beam track (mrad) ll Eé, if event is in SLAC data region. This region is defined as 02 < 15 (GeV/c)2 w < 9 QZ«(n-+0.5) < 29.1 152 Table V-4. COMPARE Analysis Tape Format Record Length 500 words Event per record 100 events Words per event 5 words Details for a Single Record Word Content 1 E0, lSt evenntmeasured at upstream target face 2 E', lSt event measured at the hadron shield 3 SIN(6), lSt event 4 x, lSt event 5 Zint’ 1St event 496 E0, 100th event 497 E', 100th event 498 SIN(e), 100th event 499 x, 100th event 500 z 100th event int’ 153 to the determination of the structure function. These data are listed in Table V-3. This output tape was rewound and compacted by a separate program at the termination of COMPARE, and then released. The third tape written was the basis for all subsequent analysis. For each event, four words of kinematic informa- tion and one word of non-kinematic information were packed into a 500 word buffer and written onto the tape when the buffer filled. The tape format is shown in Table V-4. With the information included, any other kinematic variable desired could be calculated. In this way, accepted event information from 135 data tapes was packed into 45 files and written onto one, 2400 foot magnetic tape, ready for the next stage of analysis. C) The Comparison of Data to Monte Carlo To compare the 45 files of data events to the monte carlo distributions in a meaningful way, the difference between real and reconstructed kinematic variables must be understood. The reconstructed variables for the data events are those assigned through the GETP fitting process, wherein the value of scattered muon momentum was assigned which mini- mized a calculated X2. The spectrometer E' resolution was about 10%, so there was a 70% probability that this assigned momentum was within 10% of the value which the muon actually scattered with. Only this reconstructed momentum, and simi- larly, the scattering angle , could be known for the data events . 154 For monte carlo events, the actual momentum was known at every point in its trajectory, thus, the real value of Eo and E', the incident and scattered muon energies, and 0, the angle through which the muon scattered, could be known, and were in fact retained in the information recorded for the muon. On the basis of the values of E' and 0, measured at the hadron shield, reconstructed values (Eé and OR) for both variables were assigned using the known resolution functions. These resolution smeared values corresponded to the fit values determined for data events, without undergoing the time consuming fit procedure. For the monte carlo events then, both the reconstructed and real kinematic variables were known. In the ideal situation, the momentum fitting procedure would determine values of momentum and angle which distribu- tions would be centered at the actual values of these vari- ables. An attempt was made to do this, but the shifts in mean induced by the fit were complicated functions of muon energy and angle. Consequently, this effort fell short of its goal of zero shifts in the means of the distributions. For purposes of comparison between the samples of data and monte carlo, the only variables in common were the reconstructed energy and angle of the scattered muon, Efi and GR, and the energy of the incident muon at the.upstream face of the target, E0. In addition, other variables such as Qé and xR could be constructed using these values. It was decided, then, to group events into cells in a kinematic 155 plane which axes were Q; and x The size of the cells R' depended on the production rate of data events in that region of the plane. Generally, as many as a thousand events per cell were obtained in high yield portions of the plane, without exceeding a maximum memory storage require- ment imposed on the complete program. Other regions were maintained with up to several hundred events per cell. A summary of the six kinematic planes used is given in Table V-S. For these planes to be useful, it was necessary to accumulate many pieces of information for each cell. In addition to the total event weight for both data and monte carlo, the information necessary to compute the average values of several real kinematic variables in the cell was maintained for each cell. This information is summarized in Table V-6. It is these accumulated monte carlo predic- tions of real kinematic variables against which all data will be plotted. To bin the data for plotting, a program was told what the parametric variable for the plot would be, e.g. Q2 between 10 and 15 (GeV/c)2, and then which variable against which to plot, for example x. A sequence of input lines would then give minimum and maximum value pairs of this variable for a given data point. The program searched all cells of all six planes for averages in the correct range, printing the list of cells for later use. In an attempt to optimize the data presentation, this process became an Table V-5. 156 Bounds of Kinematic Planes Used for Analysis 2 2 Plane Xmin max AX Qmin 2 Qmax 2 2 (GeV/c) (GeV/c) (GeV/c) 1 O .06 0.003 0 30.0 .0 2 0.06 .14 0.005 0 70.0 .0 3 0.14 20 0.010 0 90.0 .0 4 0.20 .30 0.010 0 112.0 .0 5 0.30 .60 0.020 0 152.0 .0 6 0.60 .00 0.100 0 136.0 .0 Table V-6. Maintained Cell Information Word Information 1 2 Data Event Weights 2 Integer number of data entries 3 2 Monte Carlo Event Weights 4 2 Monte Carlo Weight - Q2 5 2 Monte Carlo Weight - x 6 2 Monte Carlo Weight - W2 7 2 Monte Carlo Weight . (Q: - Q ) 8 2 Monte Carlo Weight - (xR - x) 9 2 Monte Carlo Weight - (w; - W ) 10 2 Monte Carlo Weight - (Q; - Q ) 11 2 Monte Carlo Weight - (xR - x)2 12 2 Monte Carlo Weight . (Wé - W ) 157 iterative one. Parametric variable ranges were used which approximated, or were larger than, the resolution in that variable at the center of the parametric range. Sizesof the dependent variable range were adjusted to give approximately 3% statistical error to the data/monte carlo ratios deter- mined by this approach. To determine the ratios, a two step process was used. In the first step, the monte carlo was broken into fifteen pieces, each of equal size, and the list of cells was read and the total event weight for each data point accumulated separately for each piece. All of the data events were treated together as a sixteenth piece. The second step read all sixteen pieces and computed the flux normalized ratios of data event weight divided by monte carlo weight, Data Weight/Data Flux (1) Ratio = Monte Carlo Wt./Monte Carlo Flux The error in this ratio was the standard method of errors in quadrature, 93: A_D_2+A_M_21/2 (2) R D M D = Data Weight M = Monte Carlo Weight R = Determined Ratio 158 For data events, normally distributed errors were assumed, which gave the error as AD _ ‘0‘ - um (3) DN = integer number of data entries. For the monte carlo events, this was not true. Each monte carlo muon was weighted by the total cross section for that muons E0, E', and 6, making this a bias-type calculation as opposed to the real data above, which is an analog calcu- lation. It is for this reason that the monte carlo was (47) broken into fifteen pieces. Each piece then gave an independent prediction of the muon yield in the various cells. These predictions were normally distributed, so the error in the mean value of the fifteen predictions could be computed as 2 .Z a? - N 52 [9114!] = [1 :2 1:1(N—1) ] (4) a = average of the predictions ai with N = 15. This N was chosen as convenient because the monte carlo was written on fifteen output tapes. Associated with each of these ratios, the average values 2 were calculated. These of the real variables 02, x, and W values were put into a common file together with the ratios and statistical errors, while at the same time computing the model dependent F2(x,02)(see section V-D). In addition 159 Table V-7. Some Values of Resolution in Kinematic Variables Q2 2 X 0192) 2 001232 coo (GeV/c) (GeV/c) (GeV) 11.5 0.04 3.7 62 0.05 12.0 0.10 2.6 37 0.06 11.8 0.20 2.3 33 0.12 21.8 0.10 4.1 38 0.05 22.4 0.20 3.6 36 0.09 22.1 0.30 3.2 30 0.13 33.0 0.10 7.8 53 0.04 33.5 0.20 5.1 34 0.06 34.5 0.29 4.7 33 0.11 34.6 0.40 4.3 29 0.14 45.1 0.15 9.1 49 0.05 45.3 0.24 6.6 35 0.07 44.9 0.33 5.7 32 0.10 44.5 0.44 5.0 29 0.15 59.5 0.19 11.4 46 0.05 62.3 0.29 9.0 35 0.07 61.3 0.40 7.5 33 0.11 87.9 0.30 13.7 39 0.06 93.1 0.40 12.5 37 0.09 97.0 0.54 11.4 39 0.15 124.0 0.38 19.0 41 0.08 129.0 0.54 16.0 40 0.12 160 to the average values, the shift in the means of these vari- ables due to the fit process could be determined, along with the associated resolution in the variable. Table V-7 shows some values of the resolution calculated in this way. The non-zero shifts in means which were found indicate that the raw distributions shown in chapter III, which are plot- ted as functions of variables assigned through the fitting process, cannot be directly used to obtain structure func— tions or cross sections. D) The Construction of F2(x,Qz) With the determination of the data to monte carlo, flux normalized ratios and associated errors, the next step was to determine structure functions. This was done using the Buras-Gaemers prescription detailed in chapter I. The values of Q2 and x were those average real values computed using the accumulated quantities of Table V-6. The structure functions and associated errors, then, were just these QCD values multiplied by the data/monte carlo ratios, ll 7U A )4: O cmi,Qz) QCD (S) l D N f"'\ >41 0 AF2(R.QZ) The Euler Beta function used in F2 was calculated using the gamma function MGAMMA on the IMSL library of programs,(48) so thatc49) 161 F(21)F(z2) B(zl,zz) = r( ) . (6) zl+zz The data points determined in this way could next be manipulated in various fashions. The method chosen here was to do a least squares minimization fit to three para- meters using the programcso) KINFIT4. The three free para- meters were A, 03, and an overall normalization factor N. For each parametric Q2 (x) region, the structure functions could then be shifted to the central value of Q2 (x), while still being plotted at the same values of x (02). The prescription used for the shift was -2 11,02,012), - - 2 - p(x,0')=F(x.Q)° _- 2 2 F2(X,QZ)B or (7) - -2 - - - - F (X'.Q ) F2(x',Q2) = 130,92) - 2 - -2 B F2(X.Q )B where i and 02 were those accumulated, average kinematic values found above, x' and 0'2 were the central values to which the shift was being performed, and the subscript B indicates the best fit values of A and 0: were used in the evaluation. Lack of the subscript B implies the same values of A and 0: (0.5 and 2.0) were used as in the monte carlo. The values of the structure function found in this way are shown in Figures V-Z and V-3, while the data point values are tabulated in appendix C. 162 6x6 mHo.ono~o.H oa.HHme.~ em.onHe.o em H em A No eH 0x6 woo.onmmm.e ex.onam.~ eH.onoe.o HNH H OH A No a . eon meo.onemm.o ea.onem.m ON.onee.H we N em v No v oH m uxm eoo.onomm.o om.oHeH.~ eH.onem.e mm N CH v No A 6x6 eoo.enenm.o NH.Hnee.m m~.onee.o mm H OH v No v OH e 606 moo.onweo.o we.onoo.m HN.onHN.H we H em v No v eH m oxa eoo.onmam.o oe.OHHH.H mH.ean.e mm H oN v No e HeeHv Hooo Hoe eeo.oneHo.H mm.enee.H mH.OHNe.o amH N one m Hoe.onomm.o mm.onee.~ mH.onHe.o meH H oge N woo.onaed.e He.enem.H HH.owmm.o omH H owe H o z o < . N wwwwom .H.o\a_o.1.~7n mcofluufiwumom “Hm mpouoEmHmd awn .6 + Hee. oflnmb 163 606 6m A NO coo.onmwm.o vm.onmm.N NH.onmc.o mmH H om v No v oH mH :oHpuswumcouoH «new :H HocoHonmocH Epocha Nm.H om: HN-mH muHm- 606 cm A NO coo.owmmm.o mm.onNe.N mH.owom.o mmH H om v No v oH NH ecowHo>cou-:oz N on A No OH 606 wmo.owmmo.H mm.HHHm.o ee.oHNN.o oo H om A No mH 6x6 Nmo.onnoH.H NH.ono.o eH.ono.o em H om A No eH 006 acowpo>cou-:oz 66H N ON A No mH 6x6 NHo.oweNo.H ww.onom.o 6N.owHo.N em N oN A No NH 606 «Ho.oano.H Ho.HHom.N mm.owmo.o ooH H oN A No HH 0 z o < N mwcHom HO\Aoum« mcoHuuHHumom uHm H6O + mpouoempmd uHm wochpcou .m-> oHneH 164 ll U 0’ D. mHHou N0 UHHuoEmpm6 mN.o m AH N EeoH mHHou x uHHuoEmHm6 u ux6+ N0\Hx-H6.N.H u z mH H EHomx U06 omo.onmoo.H mn.Nnmw.H No.onmm.o oo H cm A N0 HN U06 mHo.cHNoo.H on.HHmm.m em.owom.o OOH H 6N A N0 ON 606 coo.cnmmm.o Ho.owmm.N mH.onwo.o wcH H CH A N0 6H 2 oo < moeHod e H N m m 6\ came mcoHpoHHumom 6H6 mwouoEmpm6 HH6 u 6 + noscHucou .m-> oHnme 165 — - + A=0.3, Q§=1.8 ——— + A=0.60, Q§=2.s3, N=l.0 0 + R=l.2-(l-x)/Q2 * + R=0.25 A-+SLAC Fg/Z (ref 24) ° .. L L L L. 1 j_ L L 4_ L . '1 , 1. _ +\ .1+\‘ 3:) \ ‘ ' \ 8 \ e ‘\\\ . \ T .00 0200 0110 0.00 T I W T 0100 0740 0140 0.00 0.00 0.0: 0.00 X 100 :0-16) J-Ol Figure V-Za. F2(x,QZ) in Parametric Q2 Regions 166 0,00 go“ ‘ 2"" 7' . o“ .30. Ha;- *1 F£GUCLE£§O 10 000 1 + +/ + Cl.- + -o- -I- / / / O L \ \ 8 $.00 0100 0.10 0100 0' 0200 0200 021: 0'.00 .00 0100 0200 X (00 “-201 Figure V-Zb. F2(x,QZ) in Parametric Q2 Regions 167 0.50 0-40 F2/NUCLEON 0.30 0.20 0:10 0220 0130 0140 X (00 20-25) cp.10 a Q l 0.50 l 0.40 l F2/NUCLEON 0.30 0.20 l O ‘50 .00 0110 0320 0'.30 0140 x 100 25430) Figure V-Zc. F2(x,Qz) in Parametric Qz Regions 168 8 c0.00 0110 0220 0230 0340 0350 X (00 30-40) Figure V-Zd. F2(x,Qz) in Parametric Q2 Regions 169 1 F2/NUCLEON 0.20 10 0. _l .00 01100.20 0.30 0140 0.50 X (00 40- 50) cp.00 F2/NUCLEON 0.20 0410 cp.00 .10 0120 0.30 0140 0150 0180 X0 (00 50- 80) Figure V-Ze. Fz(x,Q2) in Parametric Q2 Regions 170 0.30 0.40 0.20 I F2/NUCLEON 0.10 cp.00 .10 0120 0'.30 0140 0150 0100 X (00 80-120) O 9 . :0.30 014 0150 0150 X (00 120- 200) Figure V-Zf. F2(x,Q2) in Parametric Q2 Regions 171 O ‘7 L 1 1 o- - O ‘9 0' ‘ — — + 130.3, 2 Qo=l.8 0 —2+ A=0.60, 0.10 Qo=2.53, N=1.0 .1”, _ 2 ¢J._ o + R—l.2 (l-x)/Q :6 ~ _ z * + R—0.25D \ A 4 SLAC F /2 3 (re% 24) O * o- - D “3 50.00 10.00 20.00 30.00 am (X 003-008] “3 1 1 1 1 D Z“? 00" " LLJ ._l L) D 22:: \V‘ N '- - L0 D 1'2 . “0.00 10.00 20.00 30.00 40.00 00 (X .06-.10] Figure V-3a. F2(x,Qz) in Parametric x Regions 172 mconom x oHHuoEmHm6 :H HN0.x6N6 .nm-> opzmHm 8.1. x. 8 8H? 8.6m 8.9m 8.9.. 85% 85% 8.8 850 z 4 4 .. Z 356 6%» l? w .. 6 n . nun mm. N 173 mconom x quuoEmpm6 :H HN0.x6N6 .om-> opsmHm 3.16. x. 66 66.6H.H 66.66% 66.36 66.6.6 66.6.6 66.6.6 666.6 66.6% 66.6.6 666% 66.6.— 66.9u . V 9 u” n F P L. . . - . . a I. I D a i I i l G D M . .rm 4 00. HI 3 0 L JON S O u q q u u u q u n - q . 26.16. x6 66 66.66.— 664% 666.6 66.56 66.6.6 66.56 66.6.0 66.6.6 66.6.6 66.6.— 66.9» m J. 2 . now 0. HI 3 0 1 -ON ..8 O u u u u u u - n u - D 9: 1 L 1 L 1 L 6" . ID 32? Oo‘ " m 1 .J U D 26 at 7 - 15° '— 0D 9 “0.00 20.00 40.00 60.00 00.00 100.00 120.00 110.00 00 (X .4-.5) L 1 1 L l J._ l 2 z6" .. O E'.“. Um ~-~ + 3°. "" 0 ~— ——-—- '\° + '—+¥-_ N u. + 6 O ‘3 “0.00 20.00 70.00 50.00 80.00 100.00 100.00 100.00 00 (X .5-.7) Figure V-3d. F2(x,Q2) in Parametric x Regions 175 This fitting process has revealed several problems. First of all, the values of the fit parameters A and Q: are dependent on the kinematic region of the data which is being considered. Consistent results of A, Qg = 0.60, 2.5 are found with the restrictions QZ > 10 or QZ > 20 (GeV/c)2 applied, but when lower Q2 data are added to these sets, or the restriction QZ > 30 is used, A and Q: eaCh undergo a substantial decrease in value. Additionally, the QZ > 30 fit finds a 10% rise Uq== l.l)in.the data over the model prediction. A second problem is the dependence of the results of the fit on the form of R = oL/oT which is used. When a constant(27) R = 0.25 is used, the fits do not exhibit the consistenty of results found previously for fits which include the 10 < Q2 < 30 data, and in fact the fit is non- convergent in a few instances. This constant R does not seem to be consistent with most recent experimental results, although it cannot be ruled out with certainty. Structure functions with this value of R are therefore also plotted in Figures V-Z and V-S, although the associated error bars are left off. Data at low x are difficult to interpret. The possibility of the suppression of the cross section due to nuclear shadowing effects has been neglected, since it appeared to be a less than 5% effect(51) applicable only for x < 0.1. A clear fall in the data/monte carlo ratio in the raw w distribution for large m can be seen, though 176 8.2: 2% 09% 8.“. 8.... 85.. 3 3mm m> ofipmu oucoz\mumm mo cwumm acute 2...». 85» 8...“ 8.... 8... 8.3.. 8... .q-> otsmflm j l l *: E * * +++++ + + 176 3 3mm m> ofipmu oucoz\mumo mo owumm .¢-> opswwm conga 8. _ 8.8 86.. 8.9» 8.8 8.8 8.... 8...» 8.2 8.8 8.... 8.3.. 8... 41 J * + + + + + + + + T a .'o —.— “‘0 01188 T 177 (Fig. V-4). This may be due to shadowing, but a definitive statement would require data on targets with several values of atomic weight. It does place low x (<0.1) data in poor light, though. Resolution at x = 0.05 can throw such events as high as x = 0.15 (0.05+Zo) and at x = 0.10, x = 0.15 is only removed by one standard deviation. From the 0.03 < x < 0.06 data, this would imply that data with Q2 as high as 30(GeV/c)2 in neighboring, small x regions should be viewed with suspicion. Similarly, data points with Q2 < lOCGeV/c)Z should be viewed with suspicion. These data are in a region of rapidly changing acceptance for the most part and even a small mis- calculation here could throw the results off significantly. Exceptions are those muons which suffer large energy losses before the deep-inelastic interaction, resulting in an E0 at the interaction point << 270 GeV, and a significantly smaller Q2 than would otherwise be construed. This is not a highly probable circumstance though (<2%), and these events cannot be separated from the rest of the sample. With these points in mind, the values of A and Q: equal to 0.60 and 2.53 respectively have been chosen as most representative of the data sample with a data normali- zation factor (N) of 0.992. These values have been used in equation V-7 to arrive at the shifted data tabulated in appendix C. The results of all of the fits have been sum- marized in Table V-8. 178 E) Systematic Errors Besides the statistical errors resulting from the finite number of scattered muons, there were many possible sources of systematic errors which contributed to inaccu- racies in the determination of F2(x,Q2). In this section, several of the most important will be discused along with their effects on F2. First of all, strictly speaking, Q3 should not be allowed to vary without also changing the values of the Z1 and 25 used in the evaluation of the valence quark density functions. This cannot be done using this data, because there are no events at the low values of Q2 which are neces- sary (l < Q2 < S), and nearby regions are unreliable as has been noted above. J.G.H. de Groot et al.(52) have performed a fit with Q: = 5.0(GeV/c)2 and found values for 21, 22, zi, and 2% which are within quoted errors of the values used by Buras and Gaemers. For these reasons, Qg has been allowed to vary while retaining the same values for all Zi' Scrutiny of the high x regions (0.4-0.S in particular) of Figure V-S show that the model prediction of F2 is approxi- mately constant, while the data start below the curve and rise above it as Q2 increases beyond ~80(GeV/c)2. This is (53) of target mass effects on the likely a consequence structure function, at least where the data are below the curve at lower Q2 values. These low points are seemingly an extrapolation of the SLAC data of reference 24 at high x and low Q2(<20(GeV/c)2). Figure V-S illustrates the way in 179 025 r- 018 b \ wnmam 1.01. ms: 1101.01.71 1.05 02‘ '- 01s - 0.22 - ASF I (A303) r 08 - 020 - “-012 - l 1 l 1 l 1 F 10 4 s a 10 12 2 0.07 L x a 06 88057 0.11 b + I— . .- \ °"° ‘~ ASFI (11.031 °°6 ASP 1 (11.03) 009 - ~ 000 - 0.05 - _~_- 007 L- f “" 006 ' 00:. - 1 1 J 1 1 l l 1 1 1 6 8 1o 12 1t. . 6 0 1o 12 11. 16 0' [6011’] 01:12 h x .075 0017 - 1.0.0 0000 - 0016 - 0020 -\ MASF 0 (11:01.7) 0015 - MASF 0 (A: 0.1.7) \ F 0026 r- \\ 0014 L- / 2 ~ 002‘ '- / 0°13 " \‘ 0.022 1- ASP 1 (11.03) a 0012 ~ / 0020 - 0011 - 255115.03) T l l l l l L l L L a 10 12 1:. 1s 16 12 1:. 16 01 [Govt] Figure V-S. The Variation of F2 With Target Mass Effects 180 which these mass effects can modify F2. The curve labeled ASF I is the parametrization used here, except with A = 0.3, and the curve labeled MASF II illustrates the change in F2 as the target mass effects are included. Q: is l.8(GeV/c)2. The target mass effect is primarily a high x phenomenon, and should have little effect for x < 0.4. This has not been checked with this data set because of the difficulty in calculating the required single and double integrals in the formula, 2 F2(x,QZ) = K3 if F2(£.Q2) M2 4 3 1 F2(5"Q2) Q E 5' M4 5 4 1 1 cha".Q2) + 12 ’1 K x f dg' f d5" 2 Q E 5' E" where K = 1 (1 + 4xzMz/Q2)1/2 1 g=——-——— (9) 1 + K‘1 M = nucleon mass Two sources of systematic error are unavoidable. 7HE fhst is the fact that the analytic forms used by Buras and Gaemers are only accurate at a 1-2% level. The second is that the radiative correction weight factor 181 is accurate only to 1%. These two sources of error will be added in quadrature with another source to determine uni- versally applicable systematic error. This third source is the PASSOUT correction for the MULTIMU reconstruction inefficiency. This correction is applied as a function of the reconstructed values Q; and xR which are found. When it is applied to the data events of MULTIMU, a comparison of the corrected sample to the uncorrected sample should appear exactly as does the comparison of PASSOUT events to the uncorrected sample, since this is the comparison it is designed to mimic. It is not perfect though, suffering from deviations as large as 2% in such integrated distributions as Q; or Eé. Consequently, a 2% systematic error should be added to the quadrature summation. These three error sources, when added,give a 3% total contribution due to systematic errors which would be added in quadrature to the statistical error bars. These error sources are applicable in all kine- matic regions, as Opposed to those still to be considered, which apply only in limited kinematic regions. Because of this, they have been totalled separately. It is to coincide with this 3% error that statistical errors of ~3% were used as a criterion to bin the data. A problem related to the MULTIMU inefficiency is the inefficiency of PASSOUT itself. The overall efficiency of MULTIMU-PASSOUT is 98.1%, leaving 1.9% of events unrecon- structed. Three possibilities exist: one, this 1.9% is kinematically similar to the PASSOUT plus MULTIMU event 182 sample, or two, the 1.9% is kinematically similar to the PASSOUT sample, or three, it is dissimilar to everything previously found. One and two are most likely, but the few events available cannot distinguish between them. All analy- sis performed here is presented with the use of possibility two, that the 1.9% is kinematically similar to the class of events already found by PASSOUT. The change in F2 expected if case one is the truth is shown in appendix C, but not included amongst the quadratically summed error sources. Samples of monte carlo have been run to test three possibilities, E0 is 1% too low, the toroid magnetic field is 1% too low, and the toroid magnetic field is 1% too high. The first is important because an independent measurement of EO using, say, the University of Chicago Cyclotron Magnet in the muon lab, was never performed. The second two are important because the magnetic field measurements performed are acknowledged as accurate to only 1%. The integrated effect of these corrections on the scattered muon yield is shown in Table V-9, while the ratio of resultant distribu- tions from these runs to the full monte carlo sample is shown in Figures V-6 to V-lO. None of these ratios appears exactly as does the corresponding data/monte carlo ratio, 2 distribution in the although some features of the Q2 and W E too low runs show similarities to it, such as the slow rise to a plateau as Q2 increases. This similarity could imply that the computed value of 0 in the toroids is too small, 183 Table V-9. Total Muon Yield in Monte Carlo Systematic Checks Run Scattered Muons* Reference 295,290f860 (full Monte Carlo sample) Raise B field by 1% 294,650 (B High) Lower B field by 1% 296,660 (B Low) Raise EO by 1% 287,311 (EO High) *Based on flux of 9.4x109 incident beam muons. 184 Eo Higher F3 1—8 €§~1 D C c25.0 00 20. 00 125. 00 10s. 00 225. 00 2 5.00 83 E PRIME (GEV) L l J_ J_ J B Lower j D O c25.00 70.00 105.00 105.00 25.00 205.00 a E PRIME (GEV) D Q c25.00 70.00 105.00 105.00 205.00 205.00 E PRIME (GEV) Figure V-6. Effect of Variation of EO and Magnetic Field on E' 185 c: 3.1 L L 1 "' + Bo Higher o 1’ H F-Ei a .q 1- m“ D O c110.00 20.00 40.00 80.00 ‘70.00 D THEIR (NRRDJ N | I '7 B Lower .. CD 1—8 #1 c”110.00 20.00 40.00 50.00 70.00 THETR (MRHD) L 1.20 B Higher +1 1 RHTIO 1.00 .80 c10.00 20.00 40.00 50.00 70.00 THETH (NRHD) Figure V-7. Effect of Variation of EO and Magnetic Field one 186 2 r- r r- 1-0 Eo Higher 00 _e_.. RHTIO l 1 —9— D D c0.00 {0.00 30.00 40.00 00.00 70.00 D Q SQURRED (GEV/CJIIZ N 1 1 L 1 'I‘ B Lower - 3 1 ES. W 1' #‘0 L. 05" + + 3; c0.00 10. 00 40.0 60.00 70.00 C Q SQUHROEDo (GEV/CJIIZ N 1 1 1 1 B Higher RHTIU 1:00 __..9_ —e— —e— —e— -e— ~9— 80 c0.00 M0 40. 00 60.00 70.00 10 SQURRbED0 (GEV/CJXXZ Figure V-8. EffeEt of Variation of Eo and Magnetic Field on 1.20 RHTIO 1.00 -e— ‘1’ E0 Higher 1 O “3 °0.'00 0.12 0124 0'.36 0140 0.60 o X ‘1’ 1 1 L 1 - "‘ B Lower O H 1—8 ‘1’ <1) 0013“ + ' 5’: 1 1 =0 .‘00 0'.12 0124 0136 004a 0'.60 O C‘: 1 1 1 1 "‘ + B Higher ° 1 H 1—8 11 c: '- L 05" + 8 1 :0.00 0'.12 0'.24 0'.36 0'.4a 0.60 X Figure V-9. Effect of Variation of E0 and Magnetic Field OHX 188 5? c0.00 10.00 20.00 30.00 10. 1 D N SQUHRED (GEVJunZ 010 N L 1 L 1 "— B Lower P RRTIU 00 l 1 _e_. ._e_ ._e.... .00 10.00 20.00 30.00 40.00 N SQUQRED (GEVJxe xIO‘ L l 1020 63080 11 .00 10.00 20.00 00.00 40.00 N SQUHRED [GEVJxe ulO‘ RHTIO 1.00 cp.00 Figure V-lO. Effegt of Variation of EO and Magnetic Field onW 189 but there is no independent evidence such as improved toroid measurements to support this. The background curves in these ratios are B-spline (S4) of the trends of the measured points, approximations and are intended not only as a guide for the eye, but also to get an approximation for the value of the systematic deviation which these changes would induce. Consequently, both the values of the ratio and the approximation of the spline have been tabulated in appendix D for all five plots. To check the resolution in E', new monte carlo calcu- lations were not run, but a sample of the full monte carlo was used with different resolution algorithms. This required only the third program stage to be rerun. The two algorithm changes used were to raise the E' resolution everywhere by the addition of one percent, and similarly with the subtraction of one percent (e.g., 9% became 10% or 8%). These re-smeared runs were then compared to the sam- ple of events which had undergone the usual smearing process. Raising the resolution 1% should smear more events out of the kinematically acceptable regions, and thereby would raise the overall data/monte carlo ratio. The total rise is quite small though, only 0.05%. When a comparison of W2 distributions is performed, the rimais seen to be restricted 2 primarily to lower values of W , rising a maximum of 1.8% at W2 = 30 GeVZ. Similarly the net effect of lowering the resolution is to lower the overall normalization by 0.03%. In W2, the region near 250 GeV2 is essentially unchanged, 190 Table V-lO. Effect of Changing E' Resolution The Ratio column represents how the Data/Monte Carlo Ratio would change given the resolution change Q2 Raise 1% Lower 1% Ratio Error Ratio Error x = 0.045 5.7 0.957 0.127 1.035 0.137 7.2 0.994 0.028 1.007 0.028 8.4 1.004 0.028 1.006 0.028 9.3 1.002 0.026 0.998 0.026 10.4 1.003 0.024 0.994 0.023 11.3 1.009 0.022 0.997 0.022 12.2 1.001 0.025 1.006 0.025 13.4 1.003 0.024 0.990 0.023 14.3 1.046 0.039 0.998 0.037 15.2 1.012 0.033 1.021 0.033 16.1 0.973 0.041 0.994 0.042 17.1 1.013 0.050 0.987 0.047 18.3 1.005 0.054 1.011 0.055 19.3 1.014 0.064 0.967 0.059 20.4 1.024 0.088 1.077 0.094 x = 0.25 7.1 0.797 0.174 0.823 0.174 8.5 1.039 0.119 0.896 0.103 9.3 0.990 0.136 1.095 0.154 10.8 1.034 0.086 0.948 0.076 12.5 1.042 0.058 1.014 0.055 14.5 1.003 0.057 0.949 0.053 16.5 1.044 0.047 0.997 0.044 18.2 1.015 0.044 0.961 0.041 20.5 1.014 0.039 0.985 0.037 22.2 1.019 0.047 1.009 0.046 24.5 1.001 0.034 0.952 0.032 26.5 1.021 0.050 0.989 0.048 28.7 1.014 0.034 0.981 0.033 30.1 1.018 0.042 0.956 0.039 32.7 1.031 0.041 1.044 0.041 33.7 1.001 0.038 0.958 0.036 36.9 1.009 0.032 1.003 0.032. 39.9 0.977 0.037 0.986 0.037 41.7 1.008 0.036 1.012 0.036 45.5 1.017 0.031 0.997 0.030 49.6 0.982 0.030 1.012 0.031 54.8 0.998 0.028 1.007 0.028 Table V-lO. 191 Continued QZ Raise 1% Lower 1% Ratio Error Ratio Error 61.2 1.019 0.028 0.993 0.027 69.3 0.989 0.025 0.986 0.025 78.8 0.996 0.033 0.992 0.033 90.9 1.018 0.051 1.005 0.050 x = 0.45 24.7 1.042 0.142 1.073 0.138 29.0 0.849 0.067 1.243 0.100 32.9 0.908 0.054 1.125 0.071 37.3 0.960 0.050 1.052 0.055 42.4 0.948 0.044 1.075, 0.052 47.4 0.970 0.041 1.005 0.043 54.9 1.016 0.035 1.035 0.035 64.3 0.985 0.033 0.994 0.033 79.6 1.009 0.029 1.000 0.028 100.8 1.004 0.033 1.013 0.033 124.5 0.959 0.052 1.000 0.054 192 Table V-ll. Calculation of Average Nucleus, Nucleon Content Material 2 N 0(g/cm3) Fe 26.0 29.85 7.87 H (CH)[C = 1.10] 7.1 6.02 1.032 Vinyl = Mylar (C5H402) 50.0 46.09 1.39 A1 13.0 13.98 2.70 Air(N2) 14.0 14.02 0.001205 12.993205 ZO'Z- Zp.N. 2= 11=4. N= 11= .0 + A = 50.76 30-= 0 4799 26 = EEE-= 0 4643 K 56 AFe 193 while for W2 = 30 GeVZ, the change again maximizes, the ratio falling by 2.2%. I When the effects of these resolution changes on F2 are examined as functions of Q2 and x, more detail can be seen than from the integrated W2 distribution. Specifically, low x regions (<0.2) are nearly unaffected except at values of Q2 < 7 (GeV/c)2. As x increases, the perturbations in the comparison increase, affecting larger values of 02, until for the highest x region, all values of Q2 are affected. These effects in some representative regions are shown in Table V-10. Neither raising nor lowering the E' resolution has any effect on the regions near Q2 of 40 (GeV/c)2 and x of 0.2 where the data seem to be higher than the monte carlo prediction. A final possibility considered is that the use of a nucleus with 26 protons and 30 neutrons as representative of our target in obtaining structure functions, will result in the wrong relative mixture of neutron and proton in com- puting F2 per nucleon. To check this, an exercise is per- formed (Table V-ll) in which 2 and h, respectively, the average proton and neutron content in the nucleus, are calculated. When these averages are used, structure func- tion values are returned which differ at most by 0.6% from those obtained with the original Fe values. In summary then, the systematic effects listed in Table V-12 have been examined to determine their effects on the data. Several (3,4,5) were assumed as universally 194 Table V-12. A List of All Examined Systematic Error Possibilities 1) Variance of Q: 2) Target Mass Effects on F2 3) Analytic Approximations from QCD 4) Accuracy of Radiative Corrections 5) Reconstruction Efficiency 6) Inefficiency of Reconstruction Efficiency Correction 7) EO Assumed 1% Higher 8) Toroid Magnetic Field Assumed 1% High 9) Toroid Magnetic Field Assumed 1% Low 10) E' Resolution 1% High 11) E' Resolution 1% Low 12) Effect of R = oL/oT 13) Calculations Using an Average Nucleus, not Iron 195 applicable, and result in an overall variance of 13% in the structure functions. Three (7,8,9) were determined to be highly probable, and their individual and summed contribu- tions to the systematic error were tabulated in Appendix D. Two (6,12) were tabulated with the determined structure functions in Appendix C, and two (1,13) were rejected. Two (10,11) were determined to be unlikely, but within the realm of possibility, and the last one (2) affects primarily regions with x > 0.4, although resolution spreads this to slightly lower values of x. CHAPTER VI CONCLUSIONS How accurate, then, is the QCD model of Buras and Gaemers in describing this set of data? It is clear that lower Q2 data (Q2 < 20 (GeV/c)2) can be accurately described by this model. However, an increase to the Q2 region 20-25 finds the data beginning to rhxeabove the model prediction for x < 0.15. Further increasing Q2 to the region 25-30 (GeV/c)2 finds this excess beginning at x 2 0.27 and lasting II as low as x 0.12. This high region persists to values as 50 (GeV/c)2, above which it is not in evidence. large as Q2 These statements apply to the model predictions using A and Qi nally found by Buras and Gaemers for A and Q5 (0.3 and 1.8 equal to 0.60 and 2.53 respectively. The values origi- respectively) do not describe this data very well even at low values for Q2, close to those which were originally used by Buras and Gaemers. (55) which has been suggested is that One explanation this excess above monte carlo represents a threshold of production in the variable W2, starting at the value W2 = 80 GeVz. These data do not support this contention. The regions of excess mentioned above correspond to minimum 196 197 W2 values between 85 and 130 GeVZ, depending on the Q2 region, but the excess ends at values of W2 between 180 and 300 GeVz, again depending on the Q2 region. This can be seen more clearly in Figure VI-l where the data/monte carlo ratio with A, 03 = 0.60, 2.53 is plotted. This shows that the ratio is plateaued at 95% at low W2, rising to a high 2 2 of 180 GeV . The ratio then drops back of 105% around W to approximately 92% , belying any possibility of a threshold. These low values for the ratio below W2 ~ 90 GeV2 cor- respond to the low data points at higher values of x in Figures V-2. These data, as was discussed in chapter five, are likely low because of the failure to include target mass effects in the calculation of F2. Consequently, the appearanceofzithreshold in the ratio may be artificial. The values plotted for F2 are correct, though. The reason is that small changes in the model, especially in a region where F2 is not rapidly changing, have a negligible effect on the determined value of the structure function which is 2 _ Data F2(Q .x) - 111—"1 C] 032.0210 (1) where Fm is the model prediction. The monte carlo weight 2 (M.C.) is linearly dependent on this F2, so except for resolution effects, the determined F2 is independent of the model used. If F is rapidly changing, the resolution will 2 smear more events away from peak regions in F2 than are smeared in from nearby regions. 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Z t __ + + on u a d 1 fl 02‘! 201 cancellation of F? in Eqn. 1 will not give the same deter- mined results for F2 when F? is changed by more than a small amount. Figure VI-Z shows some very early results of this experiment where F? was changed from the Stein modelafi)to the model of Buras and Gaemers,resulting in a shift of the determined F2 due to resolution. Of the systematic effects considered in chapter five, only raising the magnetic field by 1% gave results which could simulate this rise in W2. The values tabulated in Appendix D reflect the manner in which the ratio would change given the indicated monte carlo change. Multiplying these values by the determined W2 ratios of Figure VI-l will then simulate this monte carlo change. The result, plotted in Figure VI-3, gives ratios consistently closer to unity than in Figure VI-l. This cannot be done for the data regions of Figures V-Z and V-3 because of insufficient statistics, but it casts significant doubt on this high ratio region of W2. One serious problem is the indeterminate status of R = oL/oT. For high values of x, this causes little prob- lem for the two models attempted here, R = l.2-(l-x)/Q2 and R = 0.25 (plotted without error bars in Figures V-Z and V-3). However, for x < 0.2, serious disagreement occurs, particularly with increasing Q2. Changes in F2 aS'large as 12% occur solely due to this change in R. Although, as was shown, the fitting process does not always converge using R = 0.25, this cannot be used as a criterion for rejecting 202 this form of R. Only better experimental results can be used for this purpose, and these do not currently exist. Systematically accurate results at higher Q2 values, particularly at low x, await this occurrence. In conclusion, with the form of R assumed here, the parameters A 0.60 and Q3 = 2.53 (GeV/c)2 most accurately describe the data. Values of F2 for Q2 between 25 and 50 (GeV/c)z seem to be higher than these parameter values allow, but as has been noted, this corresponding region in W2 can be lowered to a tolerable level by a one percent rise in the assumed value of the toroid magnetic field. Higher values of x, where the data are too low r.e. this prediction, are likely explained by the target mass effect, which is not calculated here. Therefore, this model of F2 would seem to give a satisfactory explanation for the trend of the values of F2 measured here, but does not seem to adequately predict the observed details. APPENDIX A APPENDIX A PROOF OF EQUIVALENCE OF THE STRUCTURE FUNCTION F AND ITS MOMENTS 2 Problem: Define a function f(x) in terms of its moments. {Pi} is a set of orthogonal functions such that fPi(x)Pj(x)dx = 5. 1j We can make the expansions: f(x) = .2 aiPi(x) i=0 k x = 2 b P (x) i=0 1k 2 From equation A-3 then, blk = kaP£(x)dx 203 (A-l) (A-Z) (A-3) (A-4) 204 Now let us experimentally measure the moments Cj’ defined as Co = ff(x)dx ‘ (A-S) C1 = Ix f(x)dx (A-6) c2 = fx2f(x)dx (A-7) Substitution of A-2 into A-S gives c = f E aiPi(x)dx = .E ai‘fPi(x)dx i 0 1 0 (A‘S) = 120 ai510 = a0 Similarly, from A-2, A-3, and A-6, C I( E P C ))C i b P C ))d = a..X X X 1 i=0 1 1 i=0 “1 2 = a0b01fP0(x)PO(x)dx + albllfPl(x)Pl(x)dx = aob01 + albll (A-g) + a : C1‘aob01 1 b 11 205 In this way, a matrix equation is established, BA = c 0r ’ boo 0 o 0 0 d / a0\ / c0\ b01 b11 0 0 0 a1 C1 b02 blZ bzz 0 "' 0 a2 = C2 \ bOn bln b2n "' - bnn / k an/ \ Cn/ and so the ai are uniquely determined. Knowledge of the moments Ci is then sufficient to determine the function f(x). APPENDIX B APPENDIX B PRESCRIPTION FOR 8 AND C MOMENT EVALUATIONS To evaluate the moments n where q = S, C, G, and n = 2,3, the following procedure is followed. Define: 0(“)(QZ) = e'Yng l ’ o n (B-l) -Yng Dgn)(QZ) s {cl-an)n - snn} e * 'YnS + {ann + enn} e n- - n e”.Y S (B-Z) where «15023,»n a n + n (B-3) and the parameters on, B y? and Yn and moments evaluated n’ - at Qg are given in Table I-2. In SU(3) gauge theory with four flavors then 206 207 n = g D§“)(Q2) + % Df“)(QZ) n = % (D§“)(Q2) - Dfn)(Q2)) (B-4) 2 2 (l'an)an s 2 'Ygé n = {ann ' ———§———’ n} e n (l-a )a -Y?s +{n + -—-§§——E n} e APPENDIX C APPENDIX C SUMMARY OF DETERMINED STRUCTURE FUNCTION VALUES This appendix contains lists of all the structure func— tion values which have been determined. There are two sections, one for data binned (parametric) in Q2 regions, and one for data binned in x regions. The first three columns contain the average Q2, x, and 2 W values determined for the data point, while columns four and five are the determined value of F2 and its error at this point. Column six is the central value of Q2 (or x) to which the point has been shifted, as explained in Chapter five; columns seven to twelve refer to this shifted value. Columns seven and eight are the resultant, shifted F2 and its error, while columns nine and ten are the obtained re- sults if R=0.25 is used instead of R=l.2-(1-x)/Q2. 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Nomm. 88 8.88 888.8 8.88 888.8 8.88 888.8 8.88 888.8 8.88 888.8 8.88 888.8 8.88 888 8 8.88 888.8 8.88 888.8 88.8va88 8.888 888.8 8.888 888.8 8.888 888.8 8.888 888.8 8.888 888.8 8.888 888.8 8.888 888.8 8.888 888.8 8.888 888.8 8.888 888.8 8.888 888.8 8.888 888 8 8.888 888 8 8.888 888.8 8.888 888.8 8.888 888.8 8.888 888.8 8.888 888.8 8.888 888.8 8.888 888.8 8.888 888.8 82 x nomzcwucouo.~-u .o 218 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 88888 88 888.8": 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 88888 88 88.8u8 88:88> 8888888 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 88888 88 8888888 omm.o omm.o omm.o omm.o omm.o omm.o omm.o omm.o omm.o omm.o omm.o omm.o omm.o omN.o omN.o omN.o omN.o omN.o om~.o omN.o omN.o omN.o omm.o omN.o omN.o omN.o omN.o omN.o omN.o omN.o x omoo. mcoo. emoo. 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NmNo. coao. 8oma. emoo. vooo. 8moo. mono. omoo. ovoo. omoo. oo8o. nmoo. mono. mmoo. mooo. mmoo. vooo. oooo. mono. mnoo. aaoo. 8oao. vooa. aNao. 8N8a. oooo. cama. aooo. mama. aooo. amva. vooo. cova. oooo. coma. mnoo. mmva. ouoo. omva. 888: 88 oao.auz 8888. 8888.. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 88888 88 88.8u8 monam> woum8cm 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 8888. 88888 88 8888888 omm.o omm.o omm.o omm.o omm.o omv.o om8.o om8.o omv.o omv.o omv.o omv.o om8.o om8.o omv.o omv.o omm.o omm.o omm.o omm.o omm.o omm.o omm.o omm.o X omao. mmoo. mvoo. aooo. whoo. ooao. omoo. emoo. oooo. amoo. mcoo. oooo. omoo. nooo. nooo. amao. vmao. aooo. emoo. Nooo. emoo. Nooo. emoo. omoo. 888 ovuo. coco. hamo. oomo. ono. omma. ammo. NNoo. emoo. oooo. sumo. muoo. aomo. cooo. onoa. mNNa. omva. omma. aoma. 888a. mmva. onea. aama. nmva. m 8.888 ~8m.o 8.88 888.o m.88 8mm.o o.8o amm.o 8.88 Nam.o o8.ovxvom.o o.wma 888.o m.ama 888.o 8.8oa 888.o 8.8m 888.o 8.88 888.o 8.o8 am8.o 8.88 Nm8.o 8.88 888.o o.mm am8.o 8.88 aa8.o 8.88 mo8.o om.onvo8.o 8.888 amm.o 8.88a 88m.o m.oma 88m.o a.ama omm.o 8.888 88m.o 8.888 omm.o m.coa 88m.o ~.mm m8m.o 82 x 8.888 8.888 8.88 8.88 8.88 888> 8.888 8.888 8.88 8.88 8.88 8888 8.88 8.88 8.88 8.88 8.88 88>. a.8oa c.co 8.88 a.oo v.oo o.mm m.om o.88 80 88888888888.8-8 88888 APPENDIX D APPENDIX D MAGNETIC FIELD AND INCIDENT ENERGY SYSTEMATICS This appendix contains the summary of systematic shifts induced on integrated distributions through changes in incident muon energy (E0) or toroid magnetic fields (E). The three cases considered are underestimating E0 by 1%, raising the monte carlo magnetic field by 1% (B high), and lowering the monte carlo magnetic field by 1% (3 low). The actual point by point ratios for all three situations are (54) approximation first summarised, along with the B-spline of the ratios, and then the approximation is evaluated at regular intervals to find the maximum summed shifts away from R=l.0 in either direction. The ratios represent how the data/monte carlo ratios would change, given the indi- cated change in the monte carlo. The value of the smoothing parameter, S, used in the approximation is indicated for all sets of ratios. In situations where both 3 high and B low deviate in the same direction from 1.0, only the worst case (largest deviation) is summed. Values in parentheses are continuations of the curves beyond the available data, or result from the use of such. 220 Table D-l. I E Comparison 221 A. 50 Higher 1% ($831.58) E'(GeV) 30.12 50.37 67.57 82.57 97.56 112.50 127.50 142.50 157.40 170.10 180.00 190.00 200.10 209.90 219.90 229.70 242.60 Ratio .948 1.101 .973 1.035 1.050 1.094 .973 1.067 1.063 1.027 .913 1.011 1.022 1.084 1.046 .888 .961 Error .041 .038 .033 .034 .033 .035 .029 .033 .034 .040 .035 .040 .042 .048 .049 .046 .058 B. Magnetic Field Higher 1% 32.88 55.24 72.51 87.60 102.50 117.60 132.60 147.60 162.60 175.00 185.00 195.00 205.00 217.60 232.10 245.80 1.048 1.099 .983 1.027 1.074 .952 1.041 .985 1.013 .868 .997 .936 .860 1.081 1.029 .960 .040 .037 .032 .033 .034 .029 .032 .030 .031 .032 .039 .037 .035 .041 .047 .089 Spline Approximation (5'35. .967 1.033 1.033 1.032 1.037 1.040 1.039 1.032 1.0 1 1.009 1.003 1.004 1.008 1.010 1.002 .979 .920 80) 1.061 1.046 1.035 1.026 1.017 1.009 1.000 .986 .967 .947 .934 .931 .943 .983 1.029 1.029 2222 Table D-l.(continued) Spline E'(GeV) Ratio Error Approximation C. Magnetic Field Lower 1% (S-ZS.75) 30.02 .932 .040 .927 47.68 .914 .036 .944 62.66 1.037 .037 .963 77.59 .928 .030 .978 '92.48 1.040 .034 1.002 107.60 .968 .030 1.020 122.50 1.088 .034 1.028 137.60 1.050 .033 1.027 152.50 1.010 .031 1.019 165.00 .939 .035 1.011 177.50 1.028 .032 1.003 190.00 .998 .040 .994 200.10 1.051 .044 .971 210.00 .881 .038 .929 219.90 .913 .043 .901 229.80 .906 .049 .929 242.40 1.148 .077 1.115 D. Individual and Summed Maximum Contributions _ Maximum E'(GeV) 5° High 8 High 3 Low Below Above 1.00(%) 30.00 .97 1.06 .93 10 6 35.00 .99 1.06 .93 a 6 40.00 1.01 1.06 .93 7 7 45.00 1.03 1.05 .94 6 a 50.00 1.03 1.05 .95 5 e 55.00 1.04 1.05 .95 5 9 60.00 1.03 1.04 .96 4 7 65.00 1.03 1.04 .97 3 7 70.00 1.03 1.04 .97 3 7 75.00 1.03 1.03 .97 3 7 90.00 1.03 1.03 .98 2 6 95.00 1.03 1.03 .99 1 6 90.00 1.03 1.02 1.00 - 5 95.00 1.04 1.02 1.01 - 6 100.00 1.04 1.02 1.01 - 6 105.00 1.04 1.02 1.02 - 6 110.00 1.04 1.01 1.02 - 6 223 Table D-1.(continued) _ - Maximum E'(GeV) Ho High B High B Low Below Above 1.00(%) 115.00 1.04 1.01 1.03 - 7 120.00 1.04 1.01 1.03 - 7 125.00 1.04 1.00 1.03 - 7 130.00 1.04 1.00 1.03 - 7 135.00 1.04 1.00 1.03 - 7 140.00 1.03 .99 1.03 1 6 145.00 1.03 .99 1.02 1 5 150.00 1.03 .98 1.02 2 5 155.00 1.02 .98 1.02 2 4 160.00 1.02 .97 1.01 3 3 165.00 1.01 .96 1.01 4 2 170.00 1.01 .95 1.01 5 2 175.00 1.01 .95 1.00 5 1 180.00 1.00 .94 1.00 ‘ 6 - 185.00 1.00 .93 1.00 7 - 190.00 1.00 .93 .99 7 - 195.00 1.01 .93 .99 7 1 200.00 1.01 .93 .97 7 1 205.00 1.01 .94 .95 6 1 210.00 1.01 .96 .93 7 1 215.00 1.01 .97 .91 9 -1 220.00 1.00 .99 .90 10 - 225.00 .99 1.01 .91 10 1 230.00 .98 1.02 .93 9 2 235.00 .96 1.03 .98 6 3 240.00 .93 1.04 1.06 7 6 245.00 (.93) 1.03 (1.06) (7) (6) Table D-Z. 224 Theta Comparison A. Eo Higher 1% (SI19.35) 6(mrad) 10.62 13.00 15.00 17.00 19.00 21.00 23.00 25.00 27.89 31.91 36.77 51.41 B. Magnetic 10.62 13.00 15.00 17.00 19.00 21.00 23.00 25.00 27.90 31.90 36.75 45.09 62.47 'Ratio .960 .926 1.003 1.061 1.143 .980 1.030 1.007 1.033 .949 1.061 1.054 Field Higher 1.041 .925 .972 1.005 1.006 .999- 1.061 ..953 1.057. 1.017 1.009 1.014 .968 Error .104 .062 .048 .042 .040 .031 .033 .033 .026 .027 .028 .019 .122 .064 .046 .038 .034 .032 .034 .030 .027 .028 .026 .024 .027 Spline Approximation .950 .997 1.021 1.032 1.035 1.032 1.026 1.018 1.009 1.005 1.022 1.055 1% ($818.10) .956 .971 .982 .992 1.000 1.007 1.013 1.017 1.021 1.023 1.021 1.008 .968 C. Magnetic Field Lower 1% (5-18.10) 10.69 13.00 15.00 17.00 .985 .951 .986 .962 .105 .064 .047 .036 .961 .973 .982 .989 9(mrad) 19.00 21.00 23.00 25.00 27.90 31.89 36.77 45.06 62.39 0. Individual and 6(mrad) 10.00 12.00 14.00 16.00 18.00 20.00 22.00 24.00 26.00 28.00 30.00 32.00 34.00 36.00 38.00 40.00 42.00 44.00 46.00 48.00 50.00 52.00 54.00 56.00 58.00 60.00 62.00 Table D-2.(continued) 5° High B .93 .98 1.01 1.03 1.03 1.03 1.03 1.02 1.01 1.01 1.00 1.01 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.07 1.06 (1.06) (1.06) (1.06) (1.06) (1.06) (1.06) Ratio .981 1.067 .984 1.032 1.029 .969 .988 1.002 .938 Summed High .95 .95 .98 .99 1.00 1.00 1.01 1.02 1.02 1.02 1.02 1.02 1.02 1.02 1.02 1.02 1.01 1.01 1.01 1.00 1.00 .99 .99 .98 .98 .97 .97 225 Error .033 .035 .031 .035 .026 .027 .026 .024 .027 Spline Approximation .995 1.000 1.004 1.006 1.008 1.007 1.002 .986 .940 Maximum Contributions 8 Low .96 .97 .98 .99 .99 1.00 1.00 1.01 1.01 1.01 1.01 1.01 1.01 1.00 1.00 1.00 .99 .99 .98 .98 .97 .97 .96 .96 .95 .95 .94 Maximum Below Above 1.00(%) 12 lHl—‘NU‘ I A gmqmqmmmkuwuuwbfiuuuwu I (5) (5) (6) (6) (6) (anUIh»bI»LoR)Nn-rdn Table D-3. Q2 Comparison 226 A. Eo Higher 1% (5-10.4S) 2 Q 2 (GeV/c) 4.42 7.00 9.00 11.00 13.00 15.00 17.00 19.00 21.89 25.91 30.77 39.82 64.67 B. Magnetic 4.46 7.00 9.00 11.00 13.00 15.00 17.00 19.00 21.89 25.91 30.77 39.07 62.50 Ratio .964 .967 1.059 .954 1.061 1.073 1.091 1.062 1.024 1.003 .987 1.010 .992 Field 1.106 .925 .964 1.098 .989 .930 1.055 .955 1.046 .973 .992 .968. 1.022 C. Magnetic Field 4.42 7.00 9.00 11.00 13.00 .979 .960 .965 .938 .998 Higher Lower Error .074 .053 .049 .037 .042 .042 .043 .043 .031 .032 .029 .024 .018 Spline Approximation .975 .998 1.011 1.022 1.029 1.033 1.036 1.037 1.035 1.028 1.017 .993 .992 1% (S-23.20) .088 .051 .043 .044 .038 .034 .042 .037 .031 .031 .029 .024 .018 1% (5'16.90) .074 .050 .043 .036 .039 .980 .988 .992 .996 .998 .999 .999 .998 .996 .991 .984 .973 1.022 .941 .959 .971 .981 .990 2 Table D-3 (continued) 227 Q . S line (GeV/c)2 Ratio Error Apprgximation 15.00 1.075 .042 .997 17.00 1.015 .040 1.003 19.00 .948 .037 1.007 21.89 1.024 .031 1.012 25.91 1.022 .033 1.015 31.58 1.011 .026 1.015 51.81 1.012 .015 1.012 D. Individual and Summed Maximum Contributions Q2 E H h ' h ‘ L bMaXigum 2 ig B Hig B ow A ove elow (GeV/c) o 1 .00 (9‘) 4.00 .97 .98 .94 9 - 6.00 .99 .98 .95 6 - 8.00 1.01 .99 .97 3 1 10.00 1.02 .99 .98 2 2 12.00 1.03 1.00 .99 1 3 14.00 1.03 1.00 .99 1 3 16.00 1.03 1.00 1.00 e 3 18.00 1.04 1.00 1.01 e 5 20.00 1.04 1.00 1.01 - 5 22.00 1.03 1.00 1.01 - 4 24.00 1.03 .99 1.01 1 4 26.00 1.03 .99 1.01 1 4 20.00 1.02 .99 1.02 1 4 30.00 1.02 .98 1.02 2 4 32.00 1.01 .98 1.01 2 2 34.00 1.01 .98 1.01 2 2 36.00 1.00 .98 1.01 2 1 38.00 1.00 .97 1.01 3 1 40.00 .99 .97 1.01 4 1 42.00 .99 .97 1.01 4 1 44.00 .98 .97 1.01 s 1 46.00 .98 .97 1.01 5 1 48.00 .98 .97 1.01 5 1 50.00 .97 .97 1.01 6 1 52.00 .97 .98 (1.01) 5 (1) 54.00 .97 .98 (1.01) 5 (1) 56.00 .97 .99 (1.01) 4 (1) 58.00 .97 1.00 (1.01) 3 (1) 60.00 .98 1.01 (1.01) 2 (1) 62.00 .98 1.02 (1.01) 2 (2) 64.00 .99 (1.02) (1.01) 1 (2) 228 Table D-4. x Comparison A. Eo Higher 1% (5-29.15) 8 line x Ratio Error Apprgximation .01 .850 .061 .927 .03 1.048 .031 1.002 .05 1.000 .027 1.005 .07 .948 .028 .989 .09 1.091 .036 .999 .11 .973 .035 1.028 .13 .984 .038 1.055 .16 1.185 .038 1.067 .20 1.005 .035 1.039 .25 .986 .032 .990 .32 .977 .033 .974 .48 1.120 .022 1.120 B. Magnetic Field Higher 1% (S-S7.90) .03 1.044 .029 1.027 .05 1.009 .027 1.002 .07 .979 .029 .988 .09 .910 .029 .980 .11 .933 .033 .974 .13 1.037 .041 .970 .16 1.140 .035 .964 .20 .946 .033 .960 .24 .834 .032 .965 .30 1.065 .035 .991 .47 1.023 .019 1.024 22E) Table D-4.(continued) Spline Approximation C. Magnetic Field Lower 1% (5-21.80) x Ratio Error .01 .834 .058 .904 .03 .997 .029 .969 .05 1.005 .027 1.005 .07 1.027 .030 1.019 .09 .962 .031 1.017 .11 1.052 .038 1.005 .14 .991 .028 .977 .18 .975 .033 .945 .22 .844 .031 .935 .28 1.038 .032 .968 .38 1.015 .030 1.029 .57 1.124 .027 1.123 D. Individual and Summed Maximum Contributions . Maximum x 50 High 3 High 6 Low Below Above 1.00(%) .01 .93 1.05 .90 17 5 .02 .98 1.03 .94 8 3 .03 1.00 1.02 .97 3 2 .04 1.01 1.01 .99 1 — 2 .05 1.00 1.00 1.01 - 1 .06 1.00 .99 1.01 l 1 .07 .99 .99 1.02 2 2 .08 .99 .98 1.02 3 2 .09 1.00 .98 1.02 2 2 .10 1.01 .98 1.01 2 2 .11 1.03 .97 1.00 3 3 .12 1.04 .97 1.00 3 4 .13 1.06 .97 .99 3 6 .14 1.06 .97 .98 3 6 .15 1.07 .97 .97 4 7 .16 1.07 .96 .96 4 7 .17 1.06 .96 .95 5 6 .18 1.06 .96 .94 6 6 .19 1.05 .96 .94' 6 5 .20 1.04 .96 .94 6 4 .21 1.03 .96 .93 7 3 .22 1.02 .96 .94 6 2 Table D-4.(continued) .23 .24 .25 .26 .27 .28 .29 .30 .31 .32 .34 .35 .36 .37 .38 .39 .40 .41 .42 .43 .45 .46 .47 .48 .49 .50 .51 .52 .53 .54 .55 .56 E0 High L01 L00 .99 .98 .98 .97 .97 .97 .97 .98 .98 .98 .99 L00 L01 L01 L02 L03 L04 L05 L06 L08 L09 L10 L11 (103) un1u uu1n “”1” (101) “HIM (101) ”film U.UJ 23() B High B Low .96 .97 .97 .97 .98 .98 .99 .99 1.00 1.01 1.01 1.02 1.02 1.03 1.03 1.03 1.04 1.04 1.04 1.04 1.04 1.04 1.03 1.03 (1.03) (1.03) (1.03) (1.03) n.0n (1.03) (1.03) (1.03) (1.03) (1.03) .94 .94 .95 .96 .96 .97 .98 .98 .99 1.00 1.00 1.01 1.01 1.02 1.02 1.03 1.03 1.04 1.04 1.05 1.05 1.06 1.06 1.07 1.07 1.08 1.08 1.09 1.09 1.10 1.10 1.11 1.11 1.12 Maximum Below Above 1.00(%) II-‘NNN-‘SUIUIO‘G‘O‘U‘U'AO‘ I I a>\la\¢-#c»0063rih‘l IIIII r‘r'Hudrd umbO-‘O - (18) - (l9) - (l9) - (20) - (20) - (21) - (21) - (22) - (22) - (23) Table D-S. W 2 Comparison 231 A. Eo Higher by 1% (S-24.00) 2 w (GeV)z 15.33 35.29 54.83 72.45 89.97 110.00 129.80 149.80 169.90 192.50 217.40 242.40 267.40 292.40 319.70 349.70 383.90 420.90 B. Magnetic 15.35 35.14 52.38 67.33 85.01 104.90 125.10 145.00 164.90 187.50 210.00 232.50 257.50 282.50 310.00 Ratio 1.031 1.188 1.077 .924 .970 .993 .990 .968 .995 1.070 1.084 .994 1.054 1.007 1.023 .979 1.038 .816 Field Higher .985 1.088 .987 .938 1.037 .899 .978 .896 .983 1.062 .955 .987 1.032 1.038 1.019 Error .039 .046 .042 .041 .035 .038 .038 .039 .038 .038 .038 .035 .037 .038 .035 .036 .040 .049 Spline Approximation 1.083 1.063 1.035 1.009 .987 .974 .978 .994 1.015 1.034 1.041 1.036 1.028 1.022 1.023 1.017 .974 .846 1% (S-Z4.00) .036 .042 .043 .041 .040 .034 .037 .033 .037 .037 .037 .034 .037 .037 .034 1.013 1.013 1.000 .984 .965 .949 .946 .952 .965 .981 .995 1.006 1.015 1.021 1.024 W2, (GeV)° 339.90 372.00 410.50 C. Magnetic 18.59 37.51 52.47 67.30 82.44 99.95 120.00 142.60 164.80 184.90 207.50 232.40 257.50 282.50 307.40 334.80 367.10 408.20 Ratio .989 1.083 1.035 Field Lower .981 .994 .984 .967 .835 1.057 .929 1.079 .983 .964 1.098 1.026 1.066 1.000 .978 .965 1.000 .904 Table D—S.(continued) 232 Error .037 .043 .051 1% (S-30.00) .033 .044 .044 .043 .036 .041 .035 .037 .038 .036 .039 .037 .039 .036 .036 .035 .037 .039 Spline 1.028 1.034 1.052 .978 1.021 .965 .912 .918 .960 .988 1.000 1.016 1.030 1.036 1.033 1.024 1.011 1.000 .986 .963 .914 Approximation D. Individual and Summed Maximum Contributions w2 . _ . _ Maximum (Gev)z Eo High B High B Low BelogoAggve 15.00 1.08 1.01 .94 6 9 20.00 1.08 1.01 .99 1 9 25.00 1.07 1.02 1.02 - 9 30.00 1.07 1.01 1.03 - 0 35.00 1.06 1.01 1.03 - 9 40.00 1.06 1.01 1.01 - 7 45.00 1.05 1.01 1.00 - 6 50.00 1.04 1.00 .98 2 4 55.00 1.04 1.00 .95 5 4 60.00 1.03 .99 .93 7 3 65.00 1.02 .99 .92 8 2 70.00 1.01 .98 .91 9 1 75.00 1.01 .98 .91 9 1 80.00 1.00 .97 .91 9 0 233 Table D-5.(continued) 2 Maximum (583)z Eo High B High B Low Below Above 1.00(%) 85.00 .99 .97 .92 9 - 90.00 .99 .96 .94 7 - 95.00 .98 .96 .95 7 - 100.00 .98 .95 .96 7 - 105.00 .98 .95 .97 7 ‘ 110.00 .97 .95 .98 8 - 115.00 .97 .95 .98 8 ' 120.00 .97 .95 .99 8 ' 125.00 .98 .95 .99 7 “ 130.00 .98 .95 .99 7 ’ 135.00 .98 .95 1.00 7 ' 140.00 .99 .95 1.00 6 ' 145.00 .99 .95 1.00 6 - 150.00 .99 .95 1.00 6 ' 155.00 1.00 .96 1.01 4 1 160.00 1.00 .96 1.01 4 1 165.00 1.01 .96 1.02 4 3 170.00 1.02 .97 1.02 3 4 175.00 1.02 .97 1.02 3 4 180.00 1.02 .98 1.03 2 5 185.00 1.03 .98 1.03 2 6 190.00 1.03 .98 1.03 2 6 195.00 1.04 .99 1.03 1 7 200.00 1.04 .99 1.04 1 8 205.00 1.04 .99 1.04 1 8 210.00 1.04 1.00 1.04 - 8 215.00 1.04 1.00 1.04 - 8 220.00 1.04 1.00 1.04 - 8 225.00 1.04 1.00 1.04 - 8 230.00 1.04 1.01" 1.03 - 7 235.00 1.04 1.01 1.03 - 7 240.00 1.04 1.01 1.03 - 7 245.00 1.04 1.01 1.03 - 7 250.00 1.03 1.01 1.03 - 6 255.00 1.03 1.01 1.02 - 5 260.00 1.03 1.02 1.02 - 5 265.00 1.03 1.02 1.02 - 5 270.00 1.03 1.02' 1.02 - 5 275.00 1.03 1.02 1.02 - 5 280.00 1.02 1.02 1.01 - 4 234 Table D-S-(continued) W2 . _ . - Maximum (GeV)2 Eo High B High B Low Below Above 1.00(%) 285.00 1.02 1.02 1.01 - 4 290.00 1.02 1.02 1.01 - 4 295.00 1.02 1.02 1.01 - 4 300.00 1.02 1.02 1.00 - 4 305.00 1.02 1.02 1.00 - 4 310.00 1.02 1.02 1.00 - 4 315.00 1.02 1.02 1.00 - 4 320.00 1.02 1.03 .99 1 5 325.00 1.02 1.03 .99 1 5 330.00 1.02 1.03 .99 1 5 335.00 1.02 1.03 .99 l 5 340.00 1.02 1.03 .98 2 5 345.00 1.02 1.03 .98 2 5 350.00 1.02 1.03 .98 2 6 355.00 1.01 1.03 .97 3 4 360.00 1.01 1.03 .97 3 4 365.00 1.00 1.03 .96 4 3 370.00 1.00 1.03 .96 4 3 375.00 .99 1.03 .96 S 3 380.00 .98 1.04 .95 7 4 385.00 .97 1.04 .94 9 4 390.00 .96 1.04 .94 10 4 395.00 .95 1.04 .93 12 4 400.00 .93 1.05 .93 14 5 405.00 .91 1.05 .92 17 5 410.00 .90 1.05 (.92) (18) 5 415.00 .87 (1.05) (.92) (21) (5) 420.00 .85 (1.05) (-92) (23) (5) APPENDIX E APPENDIX E THE TOROID MAGNETIC FIELD A short summary of the toroid magnetic field is pre- sented here, as excerpted from the Ph.D. thesisczs) of Steve Herb. Each toroid magnet consists of four toroids flame cut from ~7 5/8" thick, hot-rolled, low carbon, steel plate, bound together by tack welds and welded straps. The coils are #8 PVC insulated wire, about 460 turns spaced uniformly about the circumference. B(H) was measured for a smaller toroid cut from the same material. From this measurement, B(R) for the full toroids was predicted with the aid of the relation H = 6615737 ' 13A) “5‘13 Next, B(R) was measured by monitoring flux changes through coils wound through 1/4" holes drilled at different positions in one of the four toroidal pieces of the larger magnet. The effect of the holes on the measurement was measured to be less than 0.1%, and the results of the measurement agreed with the predictions from the small toroid to within 1%. 235 236 Finally, the total flux in each magnet was measured using a single coil wound around the entire magnet. These measurements were uniform to within 11%. They were repeated for this experiment with the same resultant uniformity (Table E-l). The previous experiments test measurements with 33A current are shown in Figure E-l, and are 0.9% lower than measurements at 35A, independent of the radius. The measure- ments were performed in a "standard" situation with two magnets connected in series. With four magnets, the measured field was 1.0% to 1.5% lower. The single loop, total flux measurements for this experiment were an average 1.3% lower than those from the previous measurements, so that the values measured with all eight magnets powered are predicted to be 1.4% to 1.9% lower than the curve of Figure E—l. Calcula- tions using the coefficients of Table II-7 yield results which are 1.0% to 1.4% lower than the curve predictions, values within the measured range of uniformity. 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