fig WIWill}!IWIWIEIHWIUIIHWWWWWW! t _ '3 LE V” ’5': EY :1 '3 {If It ,u “-- 3.... - _ 5 I“... ‘5: ("Q E its; a“; Jim—1.! ~51: _ I"? A.“ ck 4‘9 a “B e UEEVQEEfiy THESIS ,i This is to certify that the thesis entitled A STUDY OF THE OPERATIONAL CHARACTERISTICS - 0F LARGE DRIFT CHAMBERS USED IN A PARTICLE SPECTROMETER presented by Timothy Jay Potter has been accepted towards fulfillment of the requirements for Master's degree in Physics MW, \1 Major professor Date September 8, 1981 0-7639 MSU LIBRARIES ”- RETURNING MATERIALS: Piace in book drop to remove this checkout from your record. FINES wil] be charged if book is returned after the date stamped beiow. A STUDY OF THE OPERATIONAL CHARACTERISTICS OF LARGE DRIFT CHAMBERS USED IN A PARTICLE SPECTROMETER By Timothy Jay Potter A THESIS Submitted to Michigan State University in partial fquiTTment of the requirements for the degree of MASTER OF SCIENCE Department of Physics 1981 ABSTRACT A STUDY OF THE OPERATIONAL CHARACTERISTICS OF LARGE DRIFT CHAMBERS USED IN A PARTICLE SPECTROMETER by Timothy Jay Potter A study of the large drift chambers in a particle spectrometer has it been made to improve the spectrometer resolution. The drift distance- ;.‘ drift time relationship has been studied using electrostatics and pub- lished drift velocities as well as using data collected in the drift chambers. The precise position of the drift chambers has been checked and adjusted using a carefully selected set of data. These efforts have yielded a respectable final resolution of 433p in the X view and 380p in the Y view. ACKNOWLEDGEMENTS I would like to thank all of those who helped in the completion of this study, including Brookhaven National Laboratory staff for their assistance in the acquisition of data, Dr. Gerald A. Smith for his guidance and proof-reading, Dr. Robert Miller for his guidance and proof-reading, Dr. Ray Lewis for the information and guidance he gave, Dr. Jim Whitmore for his guidance, Allen Hicks for the help he gave on plotting, and last, but not least, Jeri Ann Zitek, who typed and retyped and retyped this material. TABLE OF CONTENTS LIST OF TABLES ...................... Page LIST OF FIGURES ..................... v Chapter I. INTRODUCTION ................... 1 II. DRIFT DISTANCE-DRIFT TIME RELATIONSHIP ................... 4 III. DRIFT CHAMBER ALIGNMENT ............. 17 IV. SPATIAL RESOLUTION ................ 20 V. SUMMARY AND CONCLUSIONS ............. 26 LIST OF REFERENCES .................... 27 LIST OF TABLES Table Page II-1. Fitted time offsets and coefficients to the distance-drift time polynomial, Eq. II-5 ....... 16 III-1. Fitted corrections to original optical survey positions. . . . . . . . . . . . .......... 19 IV-1. FNHM values, before and after velocity cor- rections, to distributions of the differences between the fitted line and measured coordinates. . . 21 IV-2.' Upper and lower limit estimates of the R00 and PDC resolution .................... 25 iv Figure 1-1. 11-2. 11-3. 11-4. 11-5. 11-6. LIST OF FIGURES Page Experimental apparatus for experiment E-708 at Brookhaven National Laboratory ........... 2 Configuration of E-708 drift cell, including electric field lines (solid) and equipotential lines (dashed). Potential values are given in Volts. ........ 5 Results of a Monte Carlo calculation of perpendicular drift distance, R1, versus drift time, averaged over all track slopes .................. 9 Monte Carlo calculated (solid curves) and measured (data points) drift velocities versus track slope, dX/dZ, in the cell , ................. 10 Results of a Monte Carlo calculation for the difference, ARi, between the actual drift distance and the product of the average drift velocity times the drift time, versus drift time ................. 11 Measured values of R1 versus drift time, averaged over all track slopes .................. 13 Measured values of ARI versus drift time, averaged over all track slopes .................. 14 CHAPTER I INTRODUCTION Experiment E708 was run at Brookhaven National Laboratory in l978, 1980 and l981. The experiment was approved to study gamma ray and char- ged pion and kaon energy spectra produced by 6p annihilations at rest, or near rest. The apparatus was designed to measure these spectra with inproved momentum resolution and higher statistical precision than in previous experiments.(1) The layout of the experimental apparatus is shown in Figure 1-1. There are two Beam Drift Chambers (BDCI and BDC2) on either side of the 9H12 magnet to determine the momentum of the beam particle. The liquid hydrogen target is surrounded by a Cylindrical Drift Chamber (CDC). The one arm spectrometer is made up of the SCM105 magnet between two sets of drift chambers. The Reflected Drift Chamber (RDC) on the tar- get side of the magnet is used to measure secondary particle trajec- tories, both entering and exiting the magnetic field. The Penetrating Drift Chamber (PDC) on the opposite side of the magnet is used to meas- ure particle tracks that penetrate through the magnetic field. The spectrometer is the key to this experiment. The resolution of the drift chambers in the spectrometer is crucial to the resolution of the spectrometer since the momentum of the particles must be derived from the angle and location of their entry and exit through the drift chambers. Figure 1-1. Experimental apparatus for experiment E-708 at Brookhaven National Laboratory. uou NHhmvmux + >~-xm >~-xm Nuom ~_Io _uom 203.5 24% fl $3 \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ \\ no. I\ \\ -5” \ \\\\\\\\\\\\\\\\\\ \\\\\\\\\\\\\\\\\ a.-. N1 V 3 ; There are two important areas of study needed to maximize the res- olution of the spectrometer. The locations of the drift chambers must be carefully determined, especially relative to each other. For this purpose, the drift chambers were optically surveyed. In addition, a check and improvement to these measurements can be obtained with a special set of data. These data were taken with the SCM105 magnet turned off, and special triggers were used to collect events with straight trajectories in the spectrometer. There were two such short runs in the 1980 run period, and one long run in the l981 run period. The second area of study to maximize the spectrometer resolution involves a detailed analysis of the relationship of drift distance to drift time. This relationship is used to accurately locate a particle track within a cell of the drift chamber, using the measured drift time for each track. This is done theoretically, using electrostatics cal- culations, and experimentally, using measured drift velocities from the magnet-off data. \- CHAPTER II DRIFT DISTANCE-DRIFT TIME RELATIONSHIP A drift chamber is an enclosed volume of gas with a set of wires ar- ranged in planes or cylinders of repeating cells. When a charged par- ticle passes through the gas, it produces about 30 ion pairs per centi- meter, which in turn produce on the order of 70 secondary ion pairs per centimeter along the track.(2) A high voltage is maintained between the wires so that electrons and ions will drift toward the wires. As the electrons enter the high electric fields near the anode, they are multiplied in an avalanche effect. Sensitive electronics are connected to the anode to detect the arrival of the avalanche. An E708 RDC or PDC cell is shown in Figure II-1. The plane of the paper is perpendicular to the electrode wires. The central wire is the anode and is held at a potential of about 2600 volts. The rows of five field wires bordering the cell on each side at a distance of 2 cm from the anode are kept at 0 volts. In E708, signals from the charge de- tectors were fed into LeCroy 2770 Multichannel Time Digitizers. The dig- itizers measure the time of the drift chamber signal relative to the time of a trigger pulse generated from scintillation detectors. The least count of the time digitizers is approximately 2 nanoseconds. If the trig- ger conditions were met, this information was passed on through a CAMAC crate to a Data General Eclipse computer, which in turn wrote it onto magnetic tape. Figure 11-1. Configuration of E-708 drift cell, including electric field lines (solid) and equipotential lines (dashed). Potential values are given in Volts. ----—---- ‘—-‘----------‘---~-------‘--- -‘~~ ” ““ I ’0 ‘ \‘ I, I” -‘-—“’"--------_-----------~‘---- ‘\ \\ I I I' ~\ \ I I ’ 0-- a--~ - ‘c-D- ‘ \ ’ I I ‘-‘~-”’ ~--o‘ “ - .0 -““-”-~‘ “ \' I I I I‘ ’-‘ ’ ’ I‘ \ | | : g l \ \ l “, I ‘ | ‘ I I : I I I \ \ ‘- -~I - I, ‘ ‘\ \ - - - - d I [I \ ‘ .- ‘ s __ I ’ I s ‘ - - - .. - o ~‘ ~- _ - - ‘ — - - - n - — O -' - - ’P‘ - - - I. - - o ‘ - - -‘ ’-.. o.- G' - - - b - -- --- ‘--- a " O -~‘§ o“. o ‘ ’ «I! ~ 1 ’ «- ‘ ’ ’ ‘-P ~ ‘ ‘ ‘ a ’ ‘ a a ‘ s ’1’ a, .- ‘ ‘ ‘ ‘s I I ’ p \ \ \‘ \\ I’ I, I I ‘ ‘ \\ I I I I \ \ ‘ \ ’I ll ’ , ' \ \ 1 I . I \ I ~ , ‘ i t : : I ‘ 1 J t | l 1 ’ ' l I : L1 E t 'r : : I I I I‘ I I I I I g i i ‘ \ I I , I‘ ‘ \ , I I ’l f \ \ \ \ I I ’ I \ s . .. I ’ I \ \ r \ s s ’ I I ‘ \ \ ' 'l I , \ ’ I ‘ s‘ ~ ‘ "i- - a O. — ’ ’0 - bd - ' a ‘h- h ‘ "" ~- ---- ‘- - — - ' -- —’ -p—O :- -- ~ - ‘ ’— ~ - - . . - __ O ’0 0" - - - - - 0-. "' - - - - ~~ I \ [I I ‘ - .. .- “1-0- - - - - ~ ‘\ I \ I ’ - q - c0- '1 — - ~ ‘ II ’ a ’ - - ‘ ‘ ’ ‘ " T \ \\ ’ I ’ - ' ' "" ‘ a \ | I I - I I I I ' I I I i | I I I 1 | ‘ \\_, ‘\-’ \‘z, \-, ’ \a’l I l I I \ ‘s-d"-‘ J’ ~~"" I I ‘ \ \ ~~--“"~Q---"-~---O’ ’ I I \ \\ \\ s----- "‘--O I, [I \ \‘ ----—---------——------C’ I \ I \ ~ ’ \\ ~~~--------—-—-----------------‘-“‘ J” \‘~ ” ‘Q ," -~---------““-o--------------------"- Figure II-l 1 -| o-r-¢rz-r-aimr.. ¢-.-~ . -. 6 The time that is recorded for each cell hit can be used to recon- struct the coordinate of the charged particle moving through the cell, if the location of the cell and the drift distance-drift time relation- ship are known. The magnitude of the drift velocity is a function of both the gas used in the drift chamber and the magnitude of the electric field at each point. Since the electrons follow the electric field lines, the shape of the electric field is also important. Both the mag- nitude and the direction of the field are strongly dependent on the con- figuration of the wires in each cell. The basic equations fOrthe two-dimensional electric field perpen- dicular to the fine wires are . V and E 2x ln r ‘ (11-1) 21 fyr , (II-2) where V = potential, E'= electric field, A = charge per centimeter, and r = the distance from the wire. A description of the field requires that the A for each of the wires be known. The relative values of A are determined by using the known differences in potential between the sur- faces of the wires as boundary conditions. One additional boundary con- dition is required.to solve for the A's. If there is a net charge in the cells, there will be an overall potential difference relative to in- finity. In reality, there are ground connections on the field wires and in the high voltage power supply for the anode wires. These provide cur- rent paths to neutralize any charge imbalances. This gives the last- lboundary condition, namely the sum of the charges in a cell must equal zero. It should also be noted that cells share field wires at the boun- dary between neighboring cells. This means that only half of the charge on the field wires should be included when calculating their contribution 7 to the electric field at some point in a cell. Including only the nearest neighbor cells, a good description of the electric field has been obtained. If as many as eight cells are in- cluded on either side of the cell of interest, the magnitude of the elec- tric field varies by only one-half percent. The electric field and equi- potential lines calculated for the RDC and PDC cells are shown in Fig- ure 11-1. The numbers refer to calculated potentials. This method of determining the field has been confirmed on the wire configuration of the Mark II drift cell which is similar to that of E- 708.(3) Our calculations give field values that agree with these pub- lished results, with one exception. The abscissa on the graph of the_ electric field as a function of the distance from the cell boundary given in ref.3 must be changed to correspond to the distance from the anode. Presumably this is due to a simple error in definition in these published results. The magnitude of the drift velocity as a function of the magnitude of the electric field for several gas mixtures has been taken from a published empirical graph.(4) These data were fit to a polynomial of the form, ' V(E) C1IEI'1+C2+C3|EI+CuIEI2+clel3+C6|Elu+c7|5|5+CelEI6+CelEl7 (11‘3) for E > 200 volts/cm, and V(E) cmlel . (II-4) for E < 200 volts/cm. The RDC and PDC gas mixture ratio was determined from samples taken during both the 1980 and 1981 run periods, giving miXtures fOr both chambers of~70% Argon and m 30% Ethane. ' With the above informatiom, a Monte Carlo calculation was carried 8 out. Tracks were generated with intercepts at 50 different positions along a line segment connecting the anode and middle field wire, and with.9 values of track slope (dX/dZ) from 0.0 to 1.0. Electrons were drifted toward the anOde from 100 points per centimeter on the tracks over one quarter of the cell. The drifting was done in one nanosecond in- tervals in time. The resultant perpendicular distance (R1) versus drift time (t) coordinates were plotted separately for the nine slopes, as well as combined into the single plot shown in Figure II-2. These data were then fitted to a straight line, the slopes (de/dt) of which are interpreted as drift velocities for a particular track slope. These vel- ocities are plotted in Figure 11-3 (solid curves) versus the slope of the track. Using the same method, velocities for 60-40 and 80-20 Argon- Ethane gas mixtures were also calculated and plotted in Figure 11-3. It should be noted that these curves vary only slightly (m 6%) over the en- tire range of slopes. The velocity calculated for all slopes may be in- terpreted as an average velocity. The difference (ARl) between the per- pendicular distance to the track and the distance calculated using this average velocity has been evaluated for each of the nine slopes. A typ- ical set of points with slope = 0.250 is shown in Figure 11-4 plotted versus drift time. We note that ARl is non-zero for t 6 200 nsec, sug- gesting a correction to R1 as a function of t is required. Magnet-off data were then used to produce plots similar to those calculated in the Monte Carlo. Both the RDC and PDC contain three X and three Y planes of wires. Fits to six planes in each coordinate were used to remove the two~fold ambiquity (R1 = :velocity x t) inherent in drift chambers. In order to remove spurious solutions, the sum of the squared differences between the fitted trajectory and the actual hits were Figure 11-2. Results of a Monte Carlo calculation of perpendicular drift distance, EL, versus drift time, averaged over all track slopes. II HE V-szSIPASSIMG IE5?! -N=ALL I TS FOR T 0” V8 9) II I a),nu THE x-Avxs Inc uoaoc I c c. ' ' ' 10 VII" "9RD! 0 I ' 0 ‘ 7860 ’ Q 1 scaiicg PLOT uunacn no- “In“. no- PLO? STAYXSIICS 3' :L' U N o p. U)” - hIo— GNWCMMWhflwnnhh4WHMHn~«whmmrnwfiowqubhawcmhumncnlfiwMmauo¢ 376K IUNUDO‘WOMF‘IMO‘VOM NnWOflNMOu-IMOONFO'DIDNVIHC‘U‘»OOno-I ( u r-fi .: - ‘2‘ :2. ,__,_“‘“‘-‘_“ 2:; “-_ ‘ -“ff; tv‘tr‘w4-‘::fi‘:: ‘ - ‘ . 1 I I :~ “~:.. - .49 -r‘rfifi: * La- 2 CHI-unwouo-w-II-II: Tee—w v..: r.- v-Or“: f - ‘ ‘ w 3‘3 3 GI . ; x 2 i ; . I :0 amen uI . l ' ; . chIoo "‘ ' i ' : ' ’ g . 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Measured values of ARL versus drift time, averaged over all track slopes. ' DI 0N INC Y" SLOPESI250 rug x-Igls Iuo ucsuc 0) 0L 0 93 0 I 3 911" HOROI I SCITYER PLOT NUMBER = 0 0 0 no- “go“. on- ’LOI IIIXSIICS C710" 3-“ U to '288 00‘ ¢ P A 1111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111101000000 Ono-ooocooo-o-------.--u.--.O-uuuuuuoufi cc--.-0--.--.-----O-.--..-” "—— A X 0X15 ’ROJEC'IOI ON 14 O"ooc-OC‘OOOOUOOUO‘Iv-uaHNNQNOONNQ.HdflNOULIg' "~60C'F‘GOUOCDI‘I - M “H toanpwmu—mnuounannouuwwmuQuad-hwuuuwuuounmnuwu—kuuOuwmnuwmuuwuuo I. d F. N I. '4 I. d O" '0 '0 H I. I. d '0 d '0 III I. o .0 CI. .0 H '0 '1 u. H '0 I0 d d d '0 H I. '0 '0 .0 '0 '0 «I d ' H H '0 ' d I. CI '0 d I. II I. O. '0 H a. CI. '0 '0 CI. d '0 g. '0 GI II '0 .0 d an '0 fl Cl. C. .0 III H d d u d d d H ' H II '0 0'0 nun—wumnuwm-o-um-uumnuwmu044-‘-‘7*- ”QOQQFFflfiflomIflCC tnmummaaodaumnncq Q mnmoJMOQ‘ 0090 744-9444. - --- - A 3‘ O'CMHHNHQ '07,“- ).aO'fiuOOGQQOOUQOOQISQOQC30€° |3°O°°\3°°OOO-“:'Glc‘fl JOOO" (mo) Tuv 12233005566170899 50505050505050505 33% -n--.-------O------o- IBIIIIIIIIIIIIIIIIIIII .o--------.------ q 5 t (nanoseconds) Figure 11-6 15 track to the anode, in terms of t, the drift time. The fit was found to be only slightly dependent on slope. The best fit takes the form RL = c1+czt+c3t2+cut3+c5th+c5t5 . (II-5) where 3L is in centimeters and t is in nanoseconds. Fits were made to each set of three planes in X and Y. In addition, time delays inherent in the electronics require an additional parameter, to, a time offset for each plane. The time offset is defined as t=t'+to, where this quan- tity is substituted into eq. II-S. The results to these fits are given in Table 11-1. Although the coefficients for the X views are different from the Y views by a factor of m 2, expression (II-5) yields R1_values within 0.04 cm of the values for the Y views for the same times. 16 TABLE II-l Time Offsets (to) nanoseconds RDCl RDCZ RDC3 PDC1 9002 PDC3 x View 4.29 7.49 8.62 -9.02 -0.29 -5.73 Y View 11.05 8.51 19.54 g —1.58 5.93 3.32 Coefficients to Eq. II-S Cl . C2 C3x10“ CnxlO6 C5x109 CsxlO12 (cm) (cm/nsec) (cm/nsecz) (cm/nsec3) (cm/nsec“) (cm/nsecsl. RDCX .0475 .00105 .525 -.275 .531 -.537 PDCX .0573 .00144 .484 -.250 .509 -.525 RDCY .0189 .00324 .257 -.153 .378 -.342 PDCY .0280 .00300 .294 -.159 .419 -.382 CHAPTER III DRIFT CHAMBER ALIGNMENT Each plane of the RDC and PDC was constructed with the distance be- tween anode wires held constant within a tolerance of .010 cm and a cum- mulative error in the position of the wires of less than .038 cm. However, in the assembly of the set of 6 planes in the RDC or PDC the registration of one plane relative to another has a precision of only about .1 cm. The separation of the planes within each chamber and the position and orien- tation of the PDC relative to the RDC were measured by optical surveying techniques, yielding a precision of about .2 cm and 2 milliradians, res- pectively. In order to improve the precision of these measurements, the location of each plane was surveyed using the straight tracks recorded with the magnet off. The corrections to the positions in the dimensions parallel to the X and Y planes were determined first. In order to minimize the effect of the uncertainty in the separation of the planes, only tracks with a re- stricted range of slopes, |g§i < .04, [3%4 < .01, were used. A line was fit to the hits in the three RDC planes and projected to the PDC. The drift sign ambiguities were resolved by choosing the fit which gave the smallest differences from both the RDC and PDC hits. Position corrections were found which minimized the differences averaged over the set of tracks. These fits were iterated until there was no change in the corrections. The 2 position of the PDC was found using a 3 point fit to the RDC 17 18 hits, similar to the fit used above,tnrtincluding the whole range of slopes. Again the fits were iterated until the Z corrections remained constant. The 2 position of each end of the PDC was also determined separately to check foria rotation.of'the PDC relative to the RDC. The measured rotation correction was less than 1 milliradian. The Z locations of the individual planes were determined using a straight line fit to all six X planes. Because of the small slope of the tracks in the Y dimension this technique could not be used to de- termine the 2 position of those planes. Instead, the corrections de- termined for the neighboring X planes were added to the previous meas- urements of the Y planes. 0‘ After the Z corrections had been detenmined, the X and Y corrections were again adjusted. This whole procedure. including the drift velocity . parameters described in the preceeding section, was iterated two more times to yield the final corrections, Which are listed in Table III-l. 19 TABLE III-1 Fitted corrections to original optical survey positions plus l980 corrections. X or Y Z Plane corrections cm corrections cm gRDC X1 -.0084 .1056 RDC X2 .0073. .1390 RDC X3 .0066 .1409 PDC X1 .0594 .2580 PDC X2 .0260 .4775 PDC X3 .0172 .4570 RDC Y1 .0036 .2490 RDC Y2 .0009 .2083 RDC Y3 .0101 .2586 PDC Y1 .0358' .2164 PDC Y2 .0353 .2836_ PDC Y3 .0306 .2250, CHAPTER IV SPATIAL RESOLUTION In order to confirm that the coordinate adjustments and velocity parameters improved the accuracy of locating tracks, the differences between the line and the hits to which it was fit were monitored. As the coordinate corrections or velocity are changed, plots of these dif- ferences for each plane show the improvements in accuracy, by centering on 0.0 and/or peaking more sharply. The full widths at half maximum (FWHM) of these distributions are given in Table IV-l, with the posi- tion corrections included, before and after the velocity parameters are applied. ' The six plane fit is susceptible to a number of problems due to the 200 cm track length. A few particles scatter within the spectrometer. These produce a kinked path that the six plane fit treats as one line. A more serious effect results from multiple coulomb scattering of low momentum particles. For example, 50 MeV/c particles have an RMS deflect- ion of 40 milliradians and a RMS deviation of'7 cm from a straight line projected to the PDC. Low momentum particles would also be bent notice- ably by any residual magnetic field. A 30 gauss magnetic field acting on a 50 MeV/c particle would bend the trajectory by 30 milliradians, de- flecting it.3 cm from where it would have passed through the PDC. A high momentum particle would have proportionally smaller effects due to the residual field and even smaller effects from multiple scattering. Fit 20 21 TABLE IV-I FNHM values, before and after velocity corrections, to dis- tributions of the differences between the fitted line and measured coordinates. FNHM, before FNHM, after Plane velocity corrections (cm) velocity corrections (cm) RDC X1 0.105 0.098 RDC X2 0.115 0.098 RDC x3 0.115 0.092 PDC X1 0.155 0.139 PDC X2 0.110 0.078 PDC X3 0.130 0.112 RDC Y1 0.120 0.090 RDC Y2 0.145 0.076 RDC Y3 0.190 0.100 PDC Y1 0.105 0.075 PDC Y2 0.105. 0.077 PDC Y3 0.095 0.018 22 quality cuts are made to minimize these effects when the data are used for alignment and velocity fits. The fact that these problems are sym- metric (assuming equal numbers of positive and negative charged particles) also helps to average out these effects. Because of these effects, thedistributions 'of differences are related to the resolution of the chambers in a complicated way. The value of the resolution can be more easily determined from fits to the 3 planes within each chamber. The three plane fit will avoid the problems incurred due to the long path through the spectrometer. Multiple coulomb scattering contributes less than .005 cm in this case. However, this fit may choose a set of drift sign ambiguities that give a wrong solution. Since the best fit is chosen, the false solution will have . smaller residuals than the correct solution. This method therefore gives a low estimate for the resolution. A three plane fit in which the ambiguities were resolved by the six plane fit would avoid the problem of false solutions because of the extra constraints provided by the other chamber. If the track changed direc- tion in the magnet, these 3 plane fits probably would have a worse set of residuals than the true line. This method therefore gives a high estimate of the true resolution. The relationship between the resolution and the distribution of re& siduals is best seen if a set of three equally spaced planes is used. The RDC and PDC planes are close enough to equal spacing to make this a good approximation. A chi-squared minimization is used to find the slope and intercept of the best line 1 ,2 x2 =. 1 "Mm . (IV-1) 23 where e is the resolution assumed to be the same for all three planes. If the center plane is at z = 0 and the outer two planes are atz = :20 the minimization gives b = x3 ' x1 20 + + and a = x1 g2 x3 , for the slope and intercept. These in turn give 2 3 (X1 + X3 - 2X2)2 662 The average chi-squared for one degree of freedom is ($3) = 1 = »/6€2 = <§€> /562 . where U = X1 + X3 ' 2X2 X The residual for the second plane is defined as R= "1*”‘2i')<3.,x =X1+X3-2X2 _ 3 2 3 on: If the mean of R is zero, then the variance of R is given by «2 = 0 42> From (IV-5), ‘ 52 = (P2 /5 = 982/5 If the residuals have a normal distribution, _ 2 2 P(R) = Ae R ’2“ then the half maximum is given by - 2 2 - P(R%) = Ae R8/2° = A/2: Ae 1" 2 , resulting in Rg=taV2ln2 (IV-2) (IV-3) (IV-4) (IV-5) (IV-6) (IV-7) (IV-8) (IV-9) (IV-10) (IV-11) (IV-12) 24 The full width at the half maximum is ZRg. or FWHM = 20 V2 ln 2 = 0V8 ln 2 . (IV-13) The resolution, therefore, is from (IV-9) FWHM = 1.5 = .52 FNHM. IV-14 6 Y min- - ( ) In Table IV-2 the FNHM of the second plane and the resolution calcu- lated from it are given for each chamber, using both the high and low method of estimating the residuals. 25 TABLE IV-2 Upper and lower limit estimates of the RDC and PDC resolution. I UPPER L‘MI' L NE T L: FWHM (cm? s (Lcm) FWHM cm cm RDCX .0920 .0488 .0823 .0428 PDCX .0800 .0416 .0773 .0402 RDCY .0690 .0359 .0690 .0359 PDCY .0800 .0416 .0743 .0387 CHAPTER V SUMMARY AND CONCLUSIONS The goal of this study was to improve the spatial resolution of the chambers and therefore the momentum resolution of the spectrometer. The average of the high and low estimates for both chambers is 0.0433 cm in the X coordinate. For the Y coordinate the average is 0.0380 cm. These values are close to the design values of 0.03 cm for these chambers.(5) It should be noted that the alignment and velocity parameters were fit to the magnet-off data only. Variations in the gas mixture are expected to be as large as i 5%, which could contribute additional errors of as much as 0.03 cm to other data runs.(2) The difference in the X and Y resolutions may be due to a number of effects. The larger range of angles of the X tracks could make the X residuals larger. However, when the X angles were restricted to a range similar to that in Y, the X residuals remained larger than those in Y. Cumulative errors in the positions of the anode wires within the planes are larger for the larger X dimension. Andther difference between the two coordinate views is that there are more loose and missing wires in the X planes. In addition, there is some evidence that the planes are not truly planar, but have warped slightly along the X dimension. In conclusion, the Monte Carlo study has helped to define the im- portant parameters in the performance of these drift chambers, while the alignment and velocity fits have significantly improved the resolution of the spectrometer. ' 26 LIST OF REFERENCES LIST OF REFERENCES 0. Lowenstein gt al, A High Resolution Magnetic Spectrometer for Antiproton Phy51cs at Rest and Low Energies, IV European Antiproton Symposium, Strasbourg, France, June 25-30 (1978) 669. F. Sauli, Principles of Operation of Multiwire Proportional and Drift Chambers, CERN 77-09 (1977). W. Davies-White gt_gl, NIM 169.(l979) 227. B. Jean-Marie, NIM 1§g_(1979) 213. D. Lowenstein et al, Search for Y Transitions in fip Annihilations at Rest and Low EfiE'TTes. ASS Proposal for BroOkhaven National Laboratory, Marcfi 31 (I977). 27 IllINN“"WIN||||llllHiiHlllWllHlH 8 3 2 9 7 7 1 3 0 3 9 “2 1 3 Illililillili