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ELASTIC PION SCATTERING AT 50 MeV ON ⁴⁰Ca AND ⁴⁸Ca Rozon, Francis Martin 1985

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ELASTIC PION SCATTERING AT 50 MeV ON  k0Ca AND ^Ca. by FRANCIS MARTIN ROZON B.Eng.(Phys.), The Uni v e r s i t y of Saskatchewan, 1983 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n THE FACULTY OF GRADUATE STUDIES Department of Physics We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA October 1985 © Francis Martin Rozon In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of P H 1 S / C S The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date Qr.{ . Zo , / - i i -Abstract Absolute differential cross-sections have been measured for,elastic ± 12 40 48 12 IT scattering on C, Ca, and Ca using the QQD Spectrometer. The C data are in good agreement with (Sob 84a), indicating that the overall • 40 normalization of the data is good. The TT"1" Ca data does not agree with the previously published data of (Pre 81) but f i t s the potential 48 calculation using the SET E parameters (Car 82) better. Data for Ca and 40 TT Ca have not been previously published. An optical potential model was used to describe the data. The 40 potential parameters were fixed by f i t t i n g to the Ca absolute cross-sections. The ir~ differential cross-section ratios of the measured 48 40 48 pair, ( Ca, Ca), were compared to calculations for which the Ca neutron distribution had been f i t t e d , either by varying the Fermi parameters, or by adding a truncated series of orthogonal polynomials to a starting Fermi form. Two forms of orthogonal polynomials were used; spherical Bessel functions as used in (Gyl 84, Bar 85)), and Laguerre polynomials as used i n (Bar 85). The rms radii differences obtained from the Fermi form f i t t i n g were found not to be independent of the optical potential used and to be sensitive to the inclusion of the ratio data in the diffractive region. Di f f i c u l t i e s were encountered in obtaining reliable results from the orthogonal polynomial f i t s . The rms radii difference produced by the polynomial f i t s were not in agreement with results from the Fermi function f i t . The neutron density distribution difference obtained from the polynomial f i t i s similar in form to the results of (Ray 81), but the distribution peak is shifted toward the nuclear center. The rms radii differences found from the Fermi function and Fourier-Laguerre analysis are; Fermi Fourier-Laguerre A , .222 ± .048 .110 ± .022 (fm) nn' - i v -Table of Contents Abstract i i Table of Contents iv List of Tables v i List of Figures • v i i i Acknowledgements x Chapter I INTRODUCTION 1 1.1 Measuring Nuclear Radii and Density Distributions 5 1.1.1 Charge Probes 6 1.1.2 Hadronic Probes 7 1.1.3 Some Theoretical Approaches 8 Chapter II THE EXPERIMENT 13 2.1 The M13 Beamline 13 2.2 The QQD Spectrometer 16 2.2.1 Detector Equipment 16 2.2.2 Targets 19 2.3 Data Aquisition Electronics 20 Chapter III DATA REDUCTION 25 3.1 Cuts, Coefficients, and Angles 25 3.1.1 Cuts in MOLLI 26 3.1.2 Coefficients 28 Front End Coefficients 28 Spectrometer Coefficients 29 3.1.3 Cuts in QQDANA 32 3.1.4 The Spectrometer and Target Angles 34 3.2 Peak Fitting 35 3.3 Absolute Cross-Sections 39 3.4 Cross-Section Ratios 46 -v-Chapter IV THE THEORY 48 4.1 Scattering Theory 48 4.2 The Klein-Gordon Equation 49 4.3 The Optical Potential 50 Chapter V RESULTS 55 5.1 Absolute Cross-Section Fits 58 5.2 Cross-Section Ratio Fitting 64 5.2.1 Fermi Function Analysis 64 5.2.2 Model Independent Analysis 68 5.2.3 Discussion 70 References 77 - v i -Li 81 of Tables Table 1.1 Some rms charge r a d i i and nuclear radii differences from various experimental and theoretical methods. Errors are indicated where supplied by the references. A l l quantities are in fm 11 Table 2.1 Target mass densities and scattering center densities of the experimental targets 22 Table 3.1 Measured ir~ differential cross-sections for 1 2 C at 49.5 MeV 43 Table 3.2 Measured ir* differantial cross-sections for 4 0Ca at 49.5 MeV 44 Table 3.3 Measured TT* differential cross-sections for 4 8Ca at 49.7 MeV 45 Table 3.5 ir* d i f f e n t i a l cross-section ratios for Calcium 47 Table 4.1 The SET E optical potential parameters (Car 82) 54 Table 5.1 The Fermi distribution parameters used in this analysis. Note that for Krell code f i t t i n g , the charge distribution i s set equal to the proton distribution. MIA refers to the Fermi parameter values used in the Fourier-Laguerre f i t s of section 5.2.2, while FIT B and FIT C refer to the best f i t parameters from ratio f i t t i n g with those potentials to the whole angular range of the data set .56 Table 5.2 The two best potential f i t s to the absolute ' t 0Ca differential cross-section angular distributions. The potential parameters that are not shown in the table remain at SET E values .61 - v i i -Table 5.3 The rms r a d i i differences obtained by f i t t i n g Fermi density forms to the TT- ratio data for the " f u l l " angular distribution and "reduced" set of angles as described i n the text .66 Table 5.4 The results of the Fourier-Laguerre (FL) model independent f i t s to the ratio data using the FIT C potential. The Fermi part of the density i s described by the MIA parameters in Table 5.1 ,71 - v i i i -L i s t of Figures Fig. 1.1 The Tr+p differential cross-sections at a) 160 MeV, and b) 50 MeV (from (Tac 84)) 3 Fig. 1.2 The ir ±p total cross-sections (from (Tac 84)) 4 Fig. 1.3 The density distribution difference produced by the model independent analysis of (Ray 81) (solid line) compared to results of a DME calculation of (Neg 72) (dashed line). The two solid lines represent the error bounds of the obtained density distribution difference 9 Fig. 2.1 The M13 Channel and QQD Spectrometer 14 Fig. 2.2 The QQD Spectrometer 17 Fig. 2.3 Target holder used for the Calcium targets. The nylon thread is shown as a dotted line between holes in the target frame and is anchored to the frame at the outermost holes 21 Fig. 2.4 The experimental electronic logic 24 Fig. 3.1 Typical Time-of-Flight (TOF) histogram for TT~ in the M13 channel 27 Fig. 3.2 Showing a) the position of the ANGL cut, and b) the positions of the DDIF cuts 33 Fig. 3.3 Typical spectra for ir + at 122 degrees for a) CH2, and b) l + 8Ca. ENAV is the average energy calculated from wire chambers 4 and 5 36 Fig. 3.4 Typical spectra for TT+ at 120 degrees for k8Ca 38 - i x -Fig. 5.1 SET E calculations compared to the l f 0Ca data; • this experiment, A from (Pre 81) 59 Fig. 5.2 FIT B (solid line) and FIT C (dashed line) potential f i t s to the h0Ca data 62 Fig. 5.3 FIT B (solid line) and FIT C (dashed line) potential f i t s to the h8Ca data 63 Fig. 5.4 The various Fermi function f i t s to the ratio data; FIT B f u l l set (solid) and reduced set (dash-dot), FIT C f u l l set (long dash) and reduced set (short dash). The f i t s are described in the text 67 Fig. 5.5 The Fourier-Laguerre f i t s to the ratio data; three FL parameters (solid), five FL parameters (dashed) 72 Fig. 5.6 The density distribution difference produced by the Fourier-Lagurre f i t (solid) compered to the proton analysis (dashed) of (Ray 81). The lines indicate the upper and lower error bounds obtained in the analyses 73 — X — Acknowledgements The production of t h i s thesis has been quite an extended e f f o r t . It turned out to be considerably longer than I had o r i g i n a l l y intentioned, but I was unable (unwilling) to s i g n i f i c a n t l y shorten i t , while s t i l l preserving i n t a c t the bulk of what I wished to express ( s t i l l , there has been much l e f t unsaid, including at least f i v e more pages of the possible misspellings of the word c o f f i e n t . . . er, I mean, c o e f f i c i e n t . I also had d i f f i c u l t i e s keeping my sentences short. I would l i k e to thank a l l those who have aided my thesis work i n one way or another. F i r s t l y , I have to thank my supervisor, Dick Johnson, for his e f f o r t s i n t r y i n g to get me to f i n i s h t h i s project, f o r sending me on a s k i trip-conference to Lake Louise, and for i n s t i l l i n g i n me some of the enthusiasm f o r experimental physics that he demonstrates. Dave G i l l has been a constant source of knowledge on the QQD spectrometer and M13 l i n e . Hans Roser was invaluable i n the preparation and running of the experiment i t s e l f , saving me many possible headaches. Bruce Barnett, being the most senior graduate student i n the group, has been greatly appreciated for his knowledge and for h i s sense of humour. His many h e l p f u l suggestions during the ana l y s i s , and guidance i n the use and modification of several lengthy, incompletely documented programs has been invaluable. I would l i k e to thank, en masse, a l l the other graduate students i n the group, including Nigel Hessey, who misspelt my name i n h i s t h e s i s , and a l l others who contributed to the experiment. - x i -TRIUMF also deserves mention for providing the f a c i l i t i e s used in the experiment and analysis, as does NSERC, who funded my work. I would also like to thank Lori Murray. Her extensive patience and excellent backrubs have been greatly appreciated during the course of this work. -1-Chapter I INTRODUCTION The study of the structure of the atomic nucleus has been a topic of much interest for many decades now. In 1911, Rutherford (Rut 11) demonstrated by the scattering of alpha (a) particles that the atomic nucleus had to be smaller than IO - 1 1 cm, several orders of magnitude smaller in size than the atom i t s e l f . The nature of the force that held or bound the nucleus together so tightly remained largely a mystery u n t i l Yukawa (Yuk 35) proposed the existence of a meson which would mediate the nuclear force much in the same way that the photon mediated the electromagnetic interaction. The meson's mass was then estimated to be about 200 times the electron mass, or ~ 100 MeV. The meson was correctly identified in 1947 (Lat 47) as the pion. The pion proved to be an interesting particle, existing in three charge states, one state with a neutral charge and states with plus or minus one electronic unit of charge, a l l with a mass ~ 135 MeV, or about 270 times that of the electron. The pion-nucleon system proved to be a f r u i t f u l experimental ground, yielding several resonances, the most noticeable being the A 3 3 resonance characterized by the quantum numbers of spin 3/2 and isospin 3/2. The A 3 3 resonance i s a wide (width r ~ 100 MeV) resonance which peaks at a laboratory kinetic energy of 195 MeV or a total center-of-mass energy of 1232 MeV. In particle property tables, i t i s denoted by i t s center-of-mass energy as the A(1232) to distinguish i t from other resonances of the same quantum numbers. -2-In the energy region where the A 3 3 resonance dominates, the pion-nucleon system demonstrates a large isospin sensitivity. If one considers the elastic scattering channels only for the moment, then decomposing the pion-proton system (the neutron can be treated in a symmetric manner) into the isospin states gives | TT+ p > - | I = 3/2 > | TT" p > = v H T 7 3 | I = 3/2 > + / T 7 3 | 1 = 1/2 >. If the isospin 1/2 channel is assumed to be negligible at the resonance, then the cross-section ratio (Tr+p/ir~p), which i s proportional to the square of the ratio of the matrix elements describing the elastic processes (or to the fourth power of the ratio of the decomposition coefficients), i s expected to be 9. This treatment i s just i f i e d by examining the 160 MeV cross-sections shown in Fig. 1.1a). If the single charge exchange channel is included, the ratio is expected to drop to 3, as shown in Fig. 1.2. At lower energies ( ~ 50 MeV ) the resonance sensitivity i s much weakened but Fig. 1.1b) demonstrates that there is s t i l l marked isospin preference in the interaction especially at back angles. The pion-nucleus (bound nucleons) interaction i s not the same as the free pion-nucleon interaction. Although one expects there to be relationships between the two processes, the isospin sensitivity may not necessarily carry over into the pion-nucleus system. Several theses by other students i n the PISCAT group have demonstrated that the sensitivity i s indeed s t i l l present in studies of elastic pion scattering (Gyl 84, Bar 85) and inelastic pion scattering (Tac 84, Sob 84a) on nuclei in the s-d shell region. This work attempts to extend these methods into the f 7 / 2 , shell studying the two closed shell calcium -4-0 100 200 300 Tff (MeV) F i g . 1.2 The IT* p t o t a l cross-sections (from (Tac 84)) - 5 -nuclei. 1 . 1 M e a s u r i n g N u c l e a r R a d i i a n d D e n s i t y D i s t r i b u t i o n s A property of the nucleus which can be measured with reasonable ease, though not completely unambiguously, i s i t s size. The ambiguity arises from the fact that the nucleus i s not a sharply defined sphere, but has a diffuse edge. As a result of the diffuse edge, the definition of the nuclear radius is somewhat d i f f i c u l t . A common measure of the radius i s the root mean square (rms) radius defined by where p(r) i s the nuclear density. The resulting rms radius i s somewhat model dependent, that i s , the derived rms radius depends to some extent upon the functional form chosen for the density. The earliest definitive size determinations used a particle scattering and examined the cross-sections for deviations from the Rutherford (Coulomb) scattering law. A deviation implied that the probe had reached the interaction range of the strong force. From these early experiments, an empirical nuclear size law was established .1/3 r ~ A nuc where A denotes the atomic number of the nucleus. With the development of various particle accelerators, the use of electron scattering to measure charge distributions to a high degree of accuracy became possible. Several exceptions to the A 1 ^ rule soon appeared. For the two closed shell calcium isotopes, l t 0Ca and i t 8Ca, i t was found (Fro 68) that the rms charge radius of 4 8Ca was actually - 6 -smaller than that of 1 + 0Ca. For a reliable method of comparing various experimental and theoretical results several derived quantities are useful, namely , 2 . 1 / 2 . 2 . 1 / 2 A = < r > - < r > np n p . . 2 . 1 / 2 . 2 . 1 / 2 A t = < r >» ~ < r >., nn n A n A and . 2 . 1 / 2 2 . 1 / 2 A i = < r >» - < r >., PP P A p A' conventionally defined such that A' < A , that i s , the nucleus with less nucleons i s denoted by the primes. For the calcium isotopes under study here, various experiments show that < r 2 > 1 / 2 - < r 2 > 1 / 2 - < r 2 > 1 / 2 n s r p ^ 4 0 ^ r p ^ 4 8 so that essentially the neutron rms radius of 4 8Ca i s the only quantity to be determined. 1.1.1 Charge Probes The lepton-nucleus interaction i s taken to be the well known electromagnetic interaction. As there are then no ambiguities in the results arising from uncertainties in the interaction, one can concentrate upon the nuclear charge distribution parameters by themselves. By about ten years ago or so, the sophistication of electron scattering data had reached the point where the data was of sufficient precision and covering a large enough region of momentum transfer that -7-model independent charge densities became necessary for good data f i t s (for example see (Fri 79)). These model independent densities took the form of sum-of-gaussian (SOG) or Fourier-Bessel (FB) expansions. Another leptonic probe i s the muon. Due to i t s large mass (~200 electron masses), the radius of the muonic orbit lies well inside the electron cloud and i t s wavefunction appreciably overlaps the nucleus. The muonic atom is then quite sensitive to the nuclear charge distribution. The so-called Barrett moments are obtained from the muonic atom data and combining them with the appropriate electron scattering data, one can obtain high precision model independent rms ra d i i (see (Woh 81)). The proton rms r a d i i should be obtainable from the charge radius i n a straightforward manner as the protons carry the bulk charge of the nucleus. However, allowance must be made for the proton and neutron charge form factors as well as a spin-orbit correction (Ber 72) in nuclei where there are unfilled £-shells. The proton distribution can also be unfolded taking these effects into account, but the unfolding procedure can be rather cumbersome. To examine the neutron distribution, leptonic probes are not of much direct use, thus hadronic probes have to be used. 1.1.2 Hadronic Probes Unlike the leptonic probes, hadronic probes are directly sensitive to the neutron distribution through the effects of the strong interaction. However, as there exists no convenient and exact description of the strong interaction as there i s for the electromagnetic probe, an additional source of uncertainty i s introduced into the problem. The pion's capabilities as a probe of the neutron density w i l l be discussed more later in this work. The proton and the a particle have been -8-extensively used as nuclear probes and supply some experimental results for comparison to the pion data. The proton is probably the most common hadronic probe due to i t s availability at high fluxes over wide energy regions. The energy region about 1 GeV kinetic energy has provided some of the best quality proton data for density determination. An optical potential of the Kerman, McManus, and Thaler (Ker 59) type is used in the cross-section calculation and some of the more recent analyses have extended the potential to second order (Ray 81, Cha 78, Var 77). The results of such analyses are usually in good agreement with theoretical models and data from other probes, but the agreement i s not much improved, and often worsened, with the inclusion of these second order terms. The sophistication of these analyses has increased substantially over the past decade to the point where model independent methods are beneficial. The neutron density distribution difference obtained by (Ray 81), for the isptopes l f 8Ca and l t 0Ca, is shown in Fig. 1.3 along with results of a DME calculation (Neg-72) for comparison. The a particle is conceptually a convenient probe as i t has spin and isospin of zero. However, the a is strongly absorbed at a l l energies and thus is sensitive primarily to the nuclear surface. If experimental data i s taken over a sufficiently large angular region, that i s , into the rainbow scattering region, the a can probe the nuclear interior. A good example of this type of experiment and the associated analysis i s given by (Gil 84) and references contained therein. 1.1.3 Some Theoretical Approaches One of the major thrusts of theoretical nuclear structure J I I I I I L 1 i i 1 1 1 1 1 r 0 2 4 6 8 r ( f m ) 1.3 The density distribution difference produced by the model independent analysis of (Ray 81) (solid line) compared to results of a DME calculation of (Neg 72) (dashed l i n e ) . The two solid lines represent the error bounds of the obtained density distribution difference. -10-calculations of the 1960's and early 1970's was the adaptation and development of self-consistent f i e l d (SCF) approaches to nuclear physics. SCF methods were developed in several forms, some of the better models being the density-dependent Hartree-Fock (DDHF) of (Neg 70, Vau 70, Vau 72) or the density-matrix-expansion (DME) of (Neg 72). These theories provided a suitable framework for a variety of calculations but they were not without adjustable parameters. The parameters are f i t to reproduce single particle energies, density distributions, and nuclear matter binding energies. The resulting densities agree reasonably well with selected data, and extracted quantities such as A and A , were for np nn the most part consistent with experimental results, but not for a l l models. In addition to various SCF calculations, the use of the Coulomb energy difference of mirror pairs of nuclei could be used to find a value for ^ . This proved to be an unsatisfactory approach as unreasonable values of A were often required to explain the energy difference (Nolan-Schiffer anomaly (Nol 69)). The r a d i i differences produced in this way were up to an order of magnitude different from experimental values for A > 40. The shell model can also be used to predict the ra d i i differences. However, results tended to overestimate A (Bat 69). More recent np calculations have been done for the sulpher and magnesium isotopes (Gyl 84) which give good agreement but no similar results are available for the calcium isotopes. A summary of some of the results for charge radii and nucleon r a d i i differences obtained by various probes and theoretical approaches i s given in Table 1.1. As the calcium nuclei have been widely studied, there -11-Method Ref. charge rms radius ^Ca charge rms radius " 8Ca A 4 0 np A 4 8 np Ann' e- Fro-68 3.4869 3.4762 v~ Woh-81 3.483(3) 3.482(3) p Ray-81 Ray-79 Var-77 .10(5) -.07 •23(5) .21 •16(5) •13(4) .27 a Gll-84 -.02(4) .17(4) .21(5) DDHF Neg-70 Vau-72 3.49 3.54 -.04 -.05 .23 .18 .31 .18 Jak-77 .14(5) ir-atom Pow-80 .001(34) .207(65) .229(44) Table 1.1 : Some rms charge radii and nuclear r a d i i differences from various experimental and theoretical methods. Errors are indicated where supplied by the references. A l l quantities in fm. -12-1s a good data base f o r comparison to r e s u l t s from t h i s experiment. -13-Chapter I I THE EXPERIMENT This experiment was conducted during the summer of 1984 at the T r i - U n i v e r s i t y Meson F a c i l i t y (TRIUMF). The f a c i l i t y ' s cyclotron has several primary beam l i n e s and can produce intense proton beams i n the energy region of 180 MeV to 520 MeV. The meson h a l l ' s secondary beamlines feed o f f the primary beam l i n e 1A. The pions used i n t h i s experiment are produced at the 1AT1 production target, a 10 mm p y r o l y t i c graphic target, by a 500 MeV unpolarized proton beam, t y p i c a l l y at a current of 130 yA. 2.1 The Ml3 Beamline The pion beam l i n e used i n t h i s experiment i s the M13 (Ora 81) l i n e . The M13 beam l i n e i s capable of handling up to 65 MeV pions, pions of higher energy not being copiously produced at the beam take-off angle of 135 degrees with respect to the primary beam. The beam l i n e produces dispersed f o c i at FI and F2 (see F i g . 2.1) and an achromatic focus at the scat t e r i n g target l o c a t i o n . The beam l i n e transports protons, a's, muons and electrons i n large quantities as well as pions. The heavier p a r t i c l e s are ranged out by a CH 2 absorber placed before the second channel dipole so that they do not make i t to the sc a t t e r i n g target. The separation of the electrons and muons from the pions i s done by t i m e - o f - f l i g h t down the channel. The momentum spread of the incident pion beam i s co n t r o l l e d by mechanical s l i t s placed at F l . For a width of .5% Ap/p (or a 6 mm s l i t width), the TT+ f l u x was - 1.2 (10 6) per second. The F2 p o s i t i o n -14-M13 F i g . 2.1 The M13 Channel and QQD Spectrometer -15-contained a p o s i t i o n s e n s i t i v e (1/10 inch wire spacing), f a s t readout wire chamber to monitor the pion path i n the channel and provided a d d i t i o n a l momentum information for the pions. The F2 chamber was only-used for the ir~ data. For the i r + beam se t t i n g , the presence of large fluxes of protons and a's (other p o s i t i v e l y charged heavy ions are possibly also present) i n the pion beam make the F2 counter i n e f f i c i e n t . The F2 chamber allows one to use a wider channel acceptance for increased f l u x . For the tt~ data, a 2% Ap/p (25 mm F l s l i t width) s e t t i n g was used which provided a f l u x of - 1.1 (10 6) ir~ per second. The beam l i n e magnet settings (as well as the spectrometer magnets) are c o n t r o l l e d remotely by REMCON and the f i e l d strengths are monitored by h a l l probes i n the quadrupoles and NMR probes i n the dipoles. The probes are u s e f u l i n reproducing beam tunes and also i n monitoring magnet s t a b i l i t y . The channel sextupoles were not used i n t h i s experiment. The purpose of the sextupoles i s to straighten the f o c a l plane at F2 f o r better momentum d e f i n i t i o n , but t h e i r usefulness has not yet been r e l i a b l y demonstrated. The pion beam f l u x i s monitored by three d i f f e r e n t pairs of p l a s t i c s c i n t i l l a t o r s (NE110) which have been f i t t e d with RC8575R phototubes. The primary beam f l u x monitoring pair are the BM1 and BM2 s c i n t i l l a t o r s . BMl i s a t h i n s c i n t i l l a t o r (.8 mm) and i s placed before the s c a t t e r i n g target, while BM2 i s downstream of the target and i s 6.4 mm t h i c k . The coincidence BM1*BM2 i s a measure of the absolute f l u x . The remaining two pairs of counters are c a l l e d rauon counters as they count muons produced by pions decaying i n the channel. Each pair of counters i s oriented at 7 degrees to the beam l i n e . The coincidences yl«p2 and y3«u4 are then used as r e l a t i v e beam f l u x monitors and are e s p e c i a l l y u s e f u l at very forward -16-angles where the spectrometer blocks the f l u x before i t reaches BM2. The ion chamber and Cerenkov counter around the T l production target can also be used as r e l a t i v e ~ f l u x monitors. 2.2 The QQD Spectrometer The QQD (quadrupole-quadrupole-dipole) spectrometer (Sob 84b) was used to measure the momentum of the scattered pions. The quadrupoles QT1 and QT2 serve to give the spectrometer a large s o l i d angle (~16 msr) and the dipole BT bends the scattered pions 70 degrees h o r i z o n t a l l y to the l e f t f o r eventual dispersion matching. The f o c a l plane of the spectrometer i s beyond the l a s t wire chamber and t i l t e d at 72 degrees to the c e n t r a l ray of the spectrometer. The spectrometer i s shown i n F i g . 2.2. During t h i s experiment only QT2, the v e r t i c a l l y focussing element, was used. When the f i r s t quadrupole, which i s h o r i z o n t a l l y focussing, i s i n use, i t should provide approximately a 5-10% increase i n s o l i d angle but w i l l l i k e l y reduce the r e l i a b i l i t y of the target traceback which i s found using no de l t a dependence. The s o l i d angle improvement has not been very well established however, so that QT1 i s not re g u l a r l y used. 2.2.1 Detector Equipment The pion t r a j e c t o r y i s monitored by four Multi Wire Proportional Counters (MWPCs) or Wire Chambers. Two chambers are positioned before the spectrometer dipole on opposite sides of QT2 at the WCl and WC3 loc a t i o n s . The remaining two chambers are placed a f t e r the dipole. The chambers are constructed with three p a r a l l e l planes of equally spaced wires. The middle or anode plane i s supplied with a p o s i t i v e high voltage - 1 7 -F i g . 2 . 2 The QQD Spectrometer -18-while the outer cathode planes are grounded. The wires of one cathode plane are oriented p a r a l l e l ( h o r i z o n t a l l y ) to the anode wires and provide y - d i r e c t i o n information. The other cathode plane i s oriented perpendicular to the anode wires and provide x - d i r e c t i o n information. The cathode wires connect to printed c i r c u i t delay l i n e s . Both ends of the delay l i n e are timed with t i m e - t o - d i g i t a l convertors (TDC's) and the diff e r e n c e of the times provides a measure of the p a r t i c l e p o s i t i o n i n the chamber. Further d e t a i l s concerning the construction and operation of the counters are provided i n (Tac 84, Hes 85). The method of construction of the wire chambers provides a convenient approach to t h e i r c a l i b r a t i o n . What i s needed i s a conversion factor to convert the TDC differences to a p o s i t i o n i n millimeters and an of f s e t to define the chamber center. That i s , X. = m.• t, + b. I I i I where X, m, t, and b are res p e c t i v e l y the p o s i t i o n , the conversion fa c t o r , the TDC differe n c e value and o f f s e t i n the i 1 " * 1 coordinate. The o f f s e t i s set such that, i n the TDC differe n c e spectrum, the beam spot edges appear at ±X°/2 , X° being the beam size i n that p a r t i c u l a r chamber which i s assumed ( f o r those coordinates that are defined only by one wire chamber segment as i s the case f o r a l l coordinates but WC4X and WC5X) to be symmetric. The conversion factors f o r the y-planes are found by examining the "picket fence" structure that i s obtained i n the y TDC diff e r e n c e data. The y - d i r e c t i o n cathode plane wires are p a r a l l e l to the anode wires. When ionized electrons avalanche around an anode wire, a strong pulse w i l l be created i n that anode wire and through capacitive coupling, i n the nearby cathode wires. The x-plane can see a smooth - 1 9 -spectra of avalanche l o c a t i o n s , but the y-plane sees quantized positions corresponding to the i n d i v i d u a l anode wires. As the anode wires are 2 mm apart, one can obtain from the separation i n the TDC spectra peaks a conversion factor to distance. The two backend chambers are divided h o r i z o n t a l l y into three segments of 203 mm s i z e (Tac 84). A p a r t i c l e passing between the edges of the segments i s l i k e l y to f i r e both segments, thus i f one looks at the positions i n the c e n t r a l segment of the double h i t s with the l e f t or r i g h t segment that occur, one can obtain both the necessary c a l i b r a t i o n factor and o f f s e t for the c e n t r a l segment to put the double h i t s peaks at ± 101.5 mm. The l e f t and r i g h t segments are taken to have the same c a l i b r a t i o n f a c t o r as the middle segment and the o f f s e t i s adjusted to match the c e n t r a l segment edge. The front end chambers are sin g l e segment only, so the chamber edge can not be as conveniently defined. The conversion factor i s then taken equal to the y value. This i s reasonable to assume as the x- and y-planes are constructed i n a symmetric manner and are read by the same TDC. The spectrometer also has three s c i n t i l l a t o r s placed a f t e r the l a s t wire chamber. These s c i n t i l l a t o r s form part of the event d e f i n i t i o n as w i l l be discussed below. The s c i n t i l l a t o r s are large enough to cover most of the back end wire chambers. E l and E2 are 6.4 mm t h i c k and E3 i s 12.8 mm. Three s c i n t i l l a t o r s are used to help reduce the background and random events. 2.2.2 Targets Three targets were used i n t h i s experiment; CH 2, 4 0 C a and l t 8Ca targets. The CH 2 target i s i n the form of a p l a s t i c plate which can be -20-e a s i l y cut to f i t the spectrometer target ladder. The CH 2 target i s large enough such that a l l of the pion beam intercepts the target. The **°Ca target was made of two s e l f supporting plates of m e t a l l i c natural calcium (97% 1 + 0Ca) of s i z e 51 mm by 39 mm held i n a target holder of the design indicated i n F i g . 2.3 by t h i n nylon thread. The target holder i s of dimension such that i t does not intercept any of the beam at the angles measured i n the experiment. The nylon thread i s small enough so that i t s background contribution i s not s i g n i f i c a n t . In t h i s manner the necessity to measure an "empty" target i s eliminated. The l + 8Ca target i s an i s o t o p i c a l l y enriched, s e l f supporting m e t a l l i c plate 31.8 mm by 16.1 mm on loan from Los Alamos and was mounted i n a target holder i n a s i m i l a r manner to the 4 0 C a target. Calcium oxidizes r a p i d l y i n a i r so i t must be keep i n a neutral environment during the experiment so that the target mass does not change during the run. For t h i s purpose, the target chamber was f i l l e d with argon and helium when the targets were being mounted on the target ladder and i t was evacuated during the running. The l | 0Ca target was cleaned i n a neutral atmosphere and weighed before the running period and again a f t e r the run. Due to the expensive nature of the ' t 8Ca target, i t was not cleaned and was heavily contaminated with what i s believed to be p r i m a r i l y oxygen. The 4 8 C a target was also weighed twice. The target mass densi t i e s are given i n Table 2.1 along with the density of s c a t t e r i n g centers. 2.3 Data Acquisition Electronics For t h i s experiment, the e l e c t r o n i c equipment used i s b a s i c a l l y the standard QQD setup as described i n previous group theses (Bar 85, Gyl 84, -21-6" O »- 1 1 •• 0 75" j ( i !<-./5->i — J .3 8 t Fig. 2.3 Target holder used for the Calcium targets. The nylon thread i s shown as a dotted line between holes in the target frame and is anchored to the frame at the outermost holes. -22-Target Nucleus Mass Density ( mg/cm2) N tgt ( cm"2) CH2 1 2C 137 6.87 x 10 2 1 23.0 1.38 x 10 2 2 *°Ca 4°Ca 288 4.33 x 10 2 1 * 8Ca **8Ca 99.9 1.25 x 10 2 1 16 0 12.1 4.55 x 10 2 0 Table 2.1 Target mass densities and scattering center densities of the experimental targets. -23-Sob 84a, Tac 84) with a few minor additions primarily to accomodate the F2 counter and the added moun counters, y3 and y4. The spectrometer data was recorded on magnetic tape for later off-line analysis and was also analyzed to some extent on-line. The TRIUMF standard data acquisition program DA, run on a PDP-11/34 computer using the RSX operating system, collected the spectrometer data which was in the form of TDC, ADC (analog-to-digital convertor) and scaler values plus bit patterns from the C212 and MALU (for the F2 chamber readout) units. The on-line analysis was done with a modified version of MULTI. The program DA responds to LAM's (look-at-me's) generated in a specific CAMAC module. The LAM's are produced by spectrometer events or beam samples. The electronic logic i s shown in Fig. 2.4. A spectrometer event i s defined as the coincidence BM1'E1«E2«E3 with El defining the timing. The large scin t i l l a t o r s E l , E2, and E3 had two or four phototubes each which were mean timed to make the event timing position independent. A l l spectrometer phototube signals were recorded in ADC's for energy information and some were recorded in TDC's for timing information. A useful feature of the logic arrangement i s that the gating used in the electronics serves to remove the necessity of applying dead-time corrections to the data by inhibiting a l l the electronics (including the scalers) while an event i s being processed to tape. The F2X data is used as a software AND only. Data that has no F2X information i s s t i l l recorded on tape, which includes a l l u + and ~ 25% of the ir - data. More detailed descriptions of the electronics can be found in the above mentioned theses. -24-l i n E1R E1L >U" MT l i n E2R E2L MT E3R+ l l n {2= E3R- JMT E3L+ E3L-DISCRIMINATOR AND |MT I MEAN TIMER GG| GATE GENERATOR OR ECL/NIM/ECL CONVERTER Bl B2 F2 SIGNALS BIT 1 GG I s pi L_J X {2 U2 Tl ION Tl £ Tl CAP PROBE MWPC SIGNALS {>-> X CAMAC MALU BIT O CAMAC ADC O CAMAC TDC STOP _ VISUAL AND O CAMAC SCALERS _ CAMAC BIT v PATTERN UNIT I n h i b i t s c a l e r s CAMAC output register (comp busy} F i g . 2.4 The experimental e l e c t r o n i c l o g i c -25-Chapter III DATA REDUCTION In essence, the analysis consists of several stages; determining the number of elast i c a l l y scattered pions from the given target material at each measured angle, applying various correction factors to the total number of pions to obtain the differential cross-section angular distributions, and f i t t i n g an optical model calculation to the angular distributions. The analysis up to the point of obtaining the cross-sections shall be discussed here. The optical model f i t t i n g i s discussed in Chapter 5. 3.1 Cuts, Coefficients, and Angles Two types of event data are written onto the magnetic tape by the on-line computer; spectrometer events defined by the coincidence BM1'E1«E2«E3, and beam sample events defined by BM1«BM2 taken about once per second. The coincidence definition of the spectrometer events reduces the random rate so that the magnetic tapes contain mainly good events and the data taking rate i s reasonable (<100 events per second). The data defined by the BM1*BM2 coincidence is used to determine the beam fraction of pions, muons, and electrons that are detectable in the scin t i l l a t o r s (discriminator thresholds may discard a lot of the electrons). For i r + the pion beam fraction is determined to be 93%, and for TT~ i t is 91%. These fractions are obtained from the time-of-flight spectra (that i s , time-of-flight of the particles down the beam line), an -26-example of which i s shown in Fig. 3.1. 3.1.1 Cuts i n MOLLI The program MOLLI is used to process the data to the point where i t is ready for pion momenta calculations. Various cuts on the data are applied in MOLLI; only spectrometer events are considered, a l l wire chamber TDC's must contain "real" data (that i s , no zeroes or f u l l scales), the sums of the TDC values from each end of the delay line must l i e within specified limits, and in the back two wire chambers, the l e f t and right segments must not f i r e for the same event. These cuts then ensure that reliable position data exists in the x and y coordinates for each wire chamber. The sum cut is useful when a chamber functions poorly with a lot of noise, creating improper position readings or when i t experiences multiple pion hits. WC3 experienced noise problems early i n the run period, but the d i f f i c u l t y was solved by increasing the discriminator thresholds on a l l the wire chamber outputs. The wire chamber cuts provide information on the chamber efficiencies. The efficiency of a single chamber can be written as (using WC5 as an example), = WC1'WC3«WC4'WC5 WC1'WC3»WC4 where WCn indicates a valid f i r i n g i n both the x and y coordinates of t tl of the n chamber. The total wire chamber efficiencies are then the product of the individual efficiencies, and was typically ~ 90%. For the IT" data, MOLLI also removed those events for which F2X did not f i r e or for which multiple hits were registered. The program MOLLI then transfers the data which has passed these - 2 7 -F i g . 3.1 T y p i c a l Time-of-Flight (TOF) histogram for TT" i n the Ml3 channel -28-,cuts to a f i l e readable by the package QQDMP developed by B.M. Barnett (Bar 85). This package allows the determination of the magnetic transfer c o e f f i c i e n t s of the spectrometer for the quantity 6 = (Ap/p)*100% and for peak f i t t i n g of the r e s u l t i n g spectra 3.1.2 Coefficients There are two d i f f e r e n t sets of tra c i n g c o e f f i c i e n t s that are needed i n the a n a l y s i s . For producing pictures of the beam spot on the target, a set of c o e f f i c i e n t s f o r tracing the pion path back through the front two wire chambers to the scattering target i s required. There i s the QT2 magnet i n between WC1 and WC3, so the trace back i s not a simple l i n e a r trace back. However, since the magnetic element i s not a dipole, i t i s assumed that there w i l l not be a large 6 dependence i n the traceback. The other set i s the spectrometer's magnetic transfer c o e f f i c i e n t s . These have very important 6 dependencies as the system these c o e f f i c i e n t s describe contains the spectrometer dipole, BT. Front End Coefficients The target coordinates that are desired are the positions i n x and y (XO and YO) and the angles to the spectrometer's c e n t r a l ray i n the x-and y-planes (THO and PHO). The tracebacks have the form (XO or THO) = a x - X l + a 3«X3 and (YO or PHO) = b ^ Y l + b 3«Y3 . Two s p e c i a l targets constructed of nichrome s l a t s 3 mm wide set about 10 mm apart, one target with h o r i z o n t a l s l a t s , one with v e r t i c a l , are placed at the s c a t t e r i n g target and spectrometer data i s taken. In analyzing -29-this data, one adjusts the coefficients for XO and YO to obtain the correct slat positions. The THO and PHO coefficients can be obtained by forcing the angular positions of the slats as viewed by WC1 to be consistent with the pion trajectory's physical positions at the target and WC1. These coefficients were not the result of a rigorous f i t t i n g procedure, but obtained by hand to the point where they are adequate for use. Substantial effort was not spent on finding the front traceback to a high precision as they are used for target cuts where some small inaccuracy i s tolerable, but not for 6 determination in which good precision i s desired. Spectrometer Coefficients The focal plane of the spectrometer l i e s at an angle of 72 degrees to the central ray of the spectrometer and is beyond the two rear wire chambers. The positions in the back wire chambers by themselves w i l l not give a good indication of the energy spectrum of the scattered pions. To produce a good spectrum, the information of the front two chambers must be used to, in effect, create a software spectrometer. For each back end chamber, the value of 6 can be determined from the x coordinate in the back chamber and a set of front end coordinates. The front end coordinates can be the x and y positions in WC1 and WC3, or i t can be the set of target coordinates described in Raytracing with the target coordinates was used by (Sob 84a). This method has the advantage that i t usually requires a f a i r l y small coefficient set and that a good starting point for the coefficients to f i r s t order in 6 can be taken from the program TRANSPORT. However, higher order terms can be very important in obtaining a good f i n a l resolution. -30-Also, an accurate traceback to determine the target coordinates is very important, especially for the angular terms which are somewhat more d i f f i c u l t to obtain. During the summer of 1984, a d r i f t chamber was tested at the WC1 location replacing the f i r s t two wire chambers. As the dri f t chamber is located before the f i r s t quadrupole, only a simple linear traceback to the target is needed and can provide target coordinates with a high degree of precision, removing some of the disadvantages of the target coordinate approach. The front chamber information can be employed directly to obtain the transfer coefficients in the package QQDMP. The parameterization is of the form (using WC5X as an example), WC5X = A + B*65 + C*6 5 2 with A, B, and C of the form A = (polynomial of order mQ in front-end coordinates) B = (polynomial of order m^  in front-end coordinates) C = (polynomial of order m2 in front-end coordinates) . In practice, the expansion is taken at most to third order, that i s m0=3, m1=2, and m2=l. The parameterization can be meaningfully inverted to obtain 6 5. In a similar manner, 6^  can be obtained. The parameterizations for WC4Y and WC5Y do not yield reliable 6 information as there i s l i t t l e 6 dependence in the y-direction. Two types of data can be used to find the transfer coefficients. A CH2 run at some spectrometer angle where the 2 + (4.44 MeV) state in carbon can be expected to be sizable can be done with the spectrometer and channel settings set to experimental values. In QQDMP, one then defines the elastic and inelastic peak locations and attempts to minimize -31-the peak widths. The two peaks are weighted such that the peak heights are approximately equal so that one region of the energy spectrum is not favored more in the f i t t i n g than other regions. The elastic peak from the hydrogen can also be included in the f i t t i n g , but should be weighted less than the carbon peaks as one expects the hydrogen peak to be broader than the carbon peaks due to kinematics. The method above was used to determine the transfer coefficients for this experiment. Perhaps a better approach i s to take CH2 spectra at some angle with the spectrometer at the experimental f i e l d settings, but then vary the channel settings in a systematic manner to maintain a good channel tune to shift the energy of the elastic carbon peak in the spectrometer. The elastic peaks from different known incident energies are then used in the minimization procedure. One can then be more confident about the constancy of the peak shape over the spectrometer acceptance. This method was employed in the double-charge-exchange (DCX) runs in December, 1984 (Hes 85). To analyze the I T - data, the information from the F2 counter must be used to counteract the degrading of the resolution resulting from the wider Fl s l i t setting (25 mm versus 6 mm for the ir + data). Using the same transfer coefficient set derived above for the T T + data, one examines a scatter plot of F2 position against the calculated 6. A correction of 2% per wire is made to the calculated 6 to remove the F2 dependence in the scatterplot, that i s , 6 = 6, + (F2 - 8)*.2 i = 4 or 5 corr i such that for the center (wire 8) of the F2 counter no correction i s made to the 6 calulated. The same correction i s made for 6^  and 6g. The F2 -32-wire spacing is 1/10 inch, indicating that the channel dispersion at F2 is 2% per inch or 1.25 cm/% which is consistent with (Ora 81). This correction for the F2 position leads to a best resolution obtained of ~ 1.0 MeV for the I T - data as compared to ~ 1.1 MeV for the T T + . 3 . 1 . 3 C u t s i n QQDANA QQDANA i s a version of QQDMP which is optimized for analyzing the data, but cannot be used for coefficient determination. Two values of 6 are determined from the transfer coefficients, one value from each back end chamber. Ideally, 6^=65, but in practice this is not true. If a pion decays to a rauon in the spectrometer, the muons path w i l l depart from the pion's trajectory at some angle which is limited by kinematics to <18 degrees in the lab frame (see for example Appendix D in (Tac 84)). If the muon remains in the spectrometer so that 6's are calculated for i t s path, the resulting values of 6 found may be substantially different. If the calculated difference in the 6's was more than 1%, the event was cut (the DDIF cut). Using the value of 6^ , the trajectory to WC5 can be predicted, and a polar angle between the actual trajectory and that predicted can be calculated. If this angle is too large, the event is cut (the ANGL cut). The DDIF and ANGL cuts are overlapping to a large extent, but i t is useful to employ them both. Typical cuts are shown in Fig. 3.2a) and Fig. 3.2b). The cuts efficiencies are calculated from data in the region of the elastic peak. The calcium targets available for this experiment were not large enough to intercept the entire beam. To determine the beam fraction passing through the target, a traceback to the target was necessary. The CH„ target used for reference measurements was large enough to intercept -33-Fig. 3.2 Showing a) the position of the ANGL cut, and b) the positions of the DDIF cuts. -34-the entire beam. Analyzing the CH2 data with the relevant calcium target cuts at each angle, in comparison to a CH2 analysis with no target cuts, established for each angle the beam fraction passing through the target. The assumption i s made that the beam spot does not change substantially between runs at the same angle. 3.1.4 The Spectrometer and Target Angles The spectrometer angle for each measurement i s read from an angle indicator on the supporting track of the spectrometer. The accuracy of this angle depends upon the accuracy with which the track i s laid down before each experimental period, the accuracy to which the pion beam follows the assumed path, and the accuracy to which the original angle markings were established. A method of calibrating any possible angle offset in the reading i s available in the CH2 target data. Due to the small proton mass (relative to the 1 2C nucleus), the energy of the scattered pion w i l l vary greatly; from 50 MeV at 0° to 25.4 MeV at 180°. Spectrometer data was taken at nominal angles of 80° and 90° on the left and right side of the beam. From these spectra, the peak energies of pions elastically scattering off protons are obtained. By adjusting the scattering angle, these peak energies are aligned with the smoothly varying curve of energy versus angle expected from kinematics. By this procedure, i t is found that there i s an angle offset of 2° right necessary to smooth the peak location data. That is to say, an 80° (left) data point is actually at 78° (left) and an 80° (right) at 82° (right). This offset has some effect on the target angle as well. The target is positioned in the transmission mode so that a l l the pions traverse the f u l l target. The nominal target angle i s then given by -35-9 . _ track 9 t g t " — but i s modified by the offset to become 9 6 t g t = 2 + ( _ ) 2 ° for right (left) settings. 3.2 Peak Fitting As mentioned above, QQDANA provides a means of f i t t i n g spectra. These spectra can contain one or more peaks of gaussian or sum-of-gaussian shape (that i s , two gaussians summed to f i t a single peak, one gaussian of large width, one narrow). Parameters describing the peak seperation, width, and position can be fixed or allowed to vary within optional limits. The peak f i t t i n g is crucial for the l t 8Ca data, where the contribution of the 1 60 contaminant must be estimated. The 't0Ca and 1 2C were straightforward to f i t . There was negligible contamination in the l*°Ca target and the nearest inelastic states are 3.8 MeV away. The 1 2C data is from the CH2 target. The proton peak is well separated from the 1 2C ground state at a l l angles measured and the f i r s t excited state in 1 2C i s at 4.44 MeV. The resulting 1 2C and l t 0Ca spectra are very clean (see Fig. 3.3a) and 3.3b)), and the peak areas and locations could be reliably f i t t e d . The 1 + 0Ca fi t t e d mean and width can be used to estimate the l f 8Ca mean and width. This can be done by correcting for the different ionization losses in the two targets for the mean, and by unfolding the estimated difference in target multiple scattering for the width (multiple scattering contribution to the width i s ~ 25% of the ionization losses). The 1 6 0 contamination peak i s taken to have the same width as -36-m r-oo CO CD 1 — i — i — I — i — i — r c o i n CO O CJ c o 0 0 CD ~ i — i — i — I — i — i — r " i — r 2+ -TM-r-1 J L 35 39 43 47 51 55 ENAV (MeV) [RUN720] a) Fig. 3.3 Typical spectra for T T + at 122 degrees for a) CH 2, and b) 1 + 0Ca. ENAV is the average energy calculated from wire chambers 4 and 5. -37-the *°Ca peak and i t s location is estimated from the kinematic energy differences between 1 60 and l t 8Ca. These estimates provide a good starting point for the f i t . It has been assumed that the primary contaminant is ^O. *2C and llfN are other possibilities, however at the most backward angles these contaminants would have been better resolved (1.8 and 1.5 MeV separation with 1 + 8Ca as compared to 1.2 MeV for 1 60 at 130°) than the data and the resulting f i t s indicated. For a sample 't8Ca spectrum see Fig. 3.4. Separate f i t t i n g of the 1 60 and 't8Ca peaks is only possible at ^spec ^ w n e r e t n e P e ak separation and relative peak heights are adequate for reliable f i t s . From the peak areas found for the 1 60, the target density of 1 50 is found to be 4.55 ± .32 (10 2 0) nuclei/cm 2. To calculate this density, reference ir~ - 1 60 cross-sections are required. The ir + reference is taken from (Bar 85) and the ir~ from (Daw 81) There is sufficient inelastic data in the Ca data that one might expect to be able to extract inelastic cross-sections, however several problems with the inelastic data were encountered. In 't0Ca, the 2 + state of interest i s only 170 keV from a possibly strong 3~ state and is thus unresolvable. Some preliminary analysis indicated that the cross-sections obtained are more consistent with the 3 - state than the 2 + state. In 1 + 8Ca, there is a strong 3~ state only 700 keV above the 2 + (3.83 MeV) state and i s again not easily resolvable. The presence of the oxygen contamination makes the f i t t i n g complicated and no consistent cross-sections could be extracted for the inelastic states. Due to these problems, the inelastic data is not presented here. -38-Fig. 3.4 Typical spectra for TT + at 122 degrees for 1 + 8Ca -39-3.3 Absolute Cross-Sections The absolute differential cross-section may be calculated from 27 tgt where N , i s the fi t t e d peak area peak v <j) i s the total beam flux BM1 • BM2 N i s the number of scattering centers i n the target per cm2 tgt J(6) i s the Jacobian converting from the lab to cm. frame and g(8) contains a l l the normalization factors. It can be expressed as r-m C°ai*t*t? 1 1 1 1 M U . ,« s ( e ) An ' WC * 7^ ' 7^e7 Tgt * where 6^  _ i s the target angle corrected for offsets tgt Aft = 16 msr i s the estimated spectrometer solid angle WC i s the total wire chamber efficiency True i s the pion beam fraction Tf-dec = .717 is the fraction of pions traversing the 2.38 m path through the spectrometer without decaying Tgt i s the target cut efficiency MHC i s the multiple hits correction and PS i s the phase-shift normalization. The value taken for the solid angle i s an estimate based on previous group work. The pion decay factor takes as the flight path the nominal central path through the spectrometer to the last wire chamber. The exact central ray i s not known well. However, any uncertainties in -40-the decay factor as well as the solid angle are taken to be absorbed into the overall normalization error of the data (taken as 10%). Differences in these factors between n + and ir~ can be treated in the phase shift normalization factor. The hydrogen cross-sections extracted from the CH2 data can be compared to the output of the d i a l - i n program SAID (Arn 82). SAID + calculates cross-sections from phase shift obtained from TT p experiments. The various experiments are not in excellent agreement, so that the calculated cross-sections cannot be taken as perfect. They are, however, the best available at this time. Results from SAID are shown in Fig. 1.1b). The forward angle hydrogen data were compared to SAID outputs as they are close enough to the carbon peak such that the cuts on the carbon should have the same effect on the hydrogen peak. Care is taken to ensure that there i s no significant carbon inelastic background under the hydrogen peak. The s t a t i s t i c a l errors are at the 5% level for the T T + and about 7% for the ir~ data. It is found that a 15% downward normalization is required for the T T + results to agree with SAID, but that no normalization i s required for the T T ~ . The MHC factor is needed to correct for the occurence of multiple pions per primary beam burst, which becomes more common as the pion beam flux rises. To correct for this, consider a beam burst in which two pions arrive at the target. For the T T + data, the beam monitor BM1'BM2 w i l l register only one pulse. However, as there are two pions, the probability that one w i l l be scattered into the spectrometer is doubled, and hence the effective flux is increased, reducing the cross-section. For T T - , the F2 counter is l i k e l y to register two hits and thus reject the data completely. The effective flux is thus lowered, raising the resulting -41-cross-sections. These effects are s t i l l rather small at the pion rates used in the experiment. Average MHC correction factors are .97 for ir + and 1.02 for T T ~ . The cross-section errors may then be calculated by adding in quadrature a l l terms contributing uncertainty to the cross-section except for those errors which are absorbed into the normalization error. The error can then be written as ^ da ^c.m. ( a¥ )c.m. * W p e a k = [ 1 _ + ( t a n 8 .A6)2 + ( ^  ) 2 + ( *IH£ f + L N v tgt v WC ' y irye J t Acuts 2^ t A N t g t -,2 e Af lux ^2 -.1/2 ^ cuts > ^ N J + I f l U x J J tgt with AG = 1° ( TT/180 radians) Airue irue and A flux = 1% from estimating the beam fraction 2% at the forward angles. flux The angular error i s estimated from the uncertainty in the angle offset and uncertainty in the positioning of the target. The cuts errors are supplied by outputs from f i t t i n g with QQDANA and can be estimated, as can the wire chamber error by AEff r 1-Eff il / 2 1 < y , k , n i ~Yff = L — — J ~ 1/" o r l e s s typically with N being the number of data points in the cut region and Eff the cut efficiency. At forward angles (<60°) the housing of the WC1 starts to intercept the beam before i t can reach BM2, thus BM1•BM2 i s no longer reliable as a -42-measure of the flux. The flux error indicated i s an estimate of the uncertainties of the beam fluxes scaled from the muon counters, and i s taken to be 2%, the approximate st a b i l i t y limit of the muon counters. The density of scattering centers, N , has negligible error for the CH2 and 1 + 0Ca targets as the areas and masses are well known. This i s not the case for the l f 8Ca target where the 1 60 content i s important. However, even for l t 8Ca this error i s less than 1%. A major contribution to the 1 + 8Ca errors comes from the uncertainty of the 1 60 contribution to the elastic peak area. For the l f 8Ca errors, the fractional error in the peak area i s not ( 1 )1/2 peak but becomes ( ~ ^ ^ ) = t A + ( AA l f i) 2 ] 1 / 2  v N , ' sum 16 J peak where A i s the total peak area including the contamination, and AA, , sum 1 o is the error in the estimated 1 60 area. The resulting differential cross-sections and their errors are shown i n Tables 3.1 to 3.3. The 1 2C results compare very well with the data of (Sob 84a), agreeing to better than 5% for the most part. The T i + - l * 0 C a data do not agree well with the only previous published cross-sections of (Pre 81). Agreement with the unpublished data of (Daw 81) is reasonable, being the best at angles less than 100°. It should be noted that effects due to the f i n i t e spectrometer acceptance and angular width of the pion beam have not been accounted for in finding the differential cross-sections. These effects w i l l be most prominent where the cross-section changes rapidly with angle, that i s , at -43-0 c m ( d e§) da/dfi (mb/sr) 38.6 7.01 + 0.27 48.7 4.20 + 0.15 58.8 2.56 + 0.10 68.9 2.52 + 0.10 78.9 3.64 + 0.14 Tf+ 83.0 3.90 + 0.14 89.0 4.77 + 0.20 93.0 5.01 + 0.14 102.9 5.47 + 0.24 112.9 6.38 + 0.26 122.8 6.18 + 0.29 130.7 5.75 + 0.30 38.6 20.44 + 0.76 48.7 8.63 + 0.36 58.8 3.15 + 0.13 68.9 2.26 + 0.11 78.9 3.27 + 0.15 T T - 83.0 3.90 + 0.17 89.0 5.38 + 0.27 93.0 5.63 + 0.27 102.9 6.56 + 0.28 112.9 6.90 + 0.35 122.8 6.59 + 0.38 130.7 6.13 + 0.38 + Table 3.1 Measured TT differential cross-sections for 1 2C at 49.5 MeV. -44-©cm ( d e8> da/dn (mb/sr) 38.2 37.40 + 1.84 48.2 20.72 + 0.78 58.2 14.99 + 0.45 68.3 14.60 + 0.50 78.3 12.84 + 0.54 T T + 82.3 13.59 + 0.38 88.3 11.82 + 0.51 92.3 10.63 + 0.27 102.3 7.24 + 0.32 112.3 5.04 + 0.23 122.2 3.84 + 0.20 130.2 4.27 + 0.25 38.2 109.5 + 5.4 48.2 38.24 + 1.38 58.2 20.63 + 0.56 68.3 20.68 + 0.83 78.3 16.62 + 0.93 T T - 82.3 16.20 + 0.60 88.3 9.98 + 0.57 92.3 8.29 + 0.35 102.3 2.61 + 0.17 112.3 0.51 + 0.06 122.2 1.49 + 0.13 130.2 2.81 + 0.19 + Table 3.2 Measured TT differential cross-sections for 4°Ca at 49.5 MeV -45-©cm ( d e§) da/dn (mb/sr) 38.1 44.63 + 2.78 48.2 20.97 + 1.37 58.2 10.90 + 0.47 68.2 9.66 + 0.64 82.2 11.44 + 0.47 92.2 8.78 + 0.34 102.2 4.74 + 0.51 112.2 3.15 + 0.35 122.2 2.69 + 0.33 130.2 2.92 + 0.33 38.1 111.7 + 6.7 48.2 45.88 + 2.67 58.2 34.28 + 1.13 68.2 30.03 + 1.71 T T - 82.2 15.92 + 0.70 92.2 5.38 + 0.42 102.2 0.67 + 0.35 112.2 2.48 + 0.37 122.2 7.58 + 0.56 130.2 9.09 + 0.67 + Table 3.3 Measured IT differential cross-sections 48 for Ca at 49.7 MeV -46-very forward angles and in the diffractive region. 3 . 4 Cross-Section Ratios. For obtaining information on the neutron density differences between l t 8Ca and ^ "Ca through the use of fourier-bessel terms in the neutron density, i t is desirable to f i t ratio data. The ratio data w i l l be free of normalization errors and effects due to common errors such as target angle errors and beam flux errors. The absolute t f 8Ca cross-sections have been given above, so that the ratio information is in a sense a restatement of results. However, the data is presented in this manner as the analyses for l t 0Ca and ' t 8Ca are similar, but not identical, the differences being mainly in the effects of different target sizes. The cross-section ratio, R, i s then 48 R = 0 4 0 o with a denoting the differential cross-section. The error in R i s given by ^|=[ ( 4 ^ ) 2 + ( # ) 2 - 2 . ( C . E . ) 2 a a where C.E. i s the sum in quadruture of common errors; target angle errors, pion beam fraction error, and the flux error. The factor of 2 compensates for these errors appearing once each i n the l t 0Ca and 1 | 8Ca calculations. Wire chamber errors are not subtracted as WC1 and WC3 encountered efficiency problems during running and were not always stable (their error contribution is small anyway). The calculated ratios and their errors are given in Table 3.4. -47-©cm ( d eS) . . , ^ 8 a /i+0 a 38.2 1.193 + 0.060 48.2 1.012 + 0.066 58.2 0.727 + 0.033 68.3 0.662 + 0.046 82.3 0.842 + 0.034 92.3 0.826 + 0.029 102.3 0.655 + 0.072 112.3 0.625 + 0.071 122.2 0.701 + 0.085 130.2 0.684 + 0.077 38.2 1.020 + 0.048 48.2 1.120 + 0.067 58.2 1.662 + 0.054 68.3 1.452 + 0.090 T T - 82.3 0.983 + 0.049 92.3 0.649 + 0.053 102.3 0.257 + 0.133 112.3 4.863 + 0.915 122.2 5.087 + 0.518 130.2 3.235 + 0.269 + Table 3.4 ir differential cross-section ratios for Calcium -48-Chapter IV THE THEORY In this chapter, some of the details of the theory used in the description of pion-nucleus scattering w i l l be given. The optical potential and the point in the Klein-Gordon equation at which the potential is inserted to describe the scattering of the pions w i l l be discussed. Application of the theory to the actual data i s l e f t to Chapter 5. 4.1 Scattering Theory The scattering of a particle by a potential V(r) can be described in the Born approximation which gives the differential cross-section of scattering as da c , - 2 where V(q) = < * f| V(r) | > V(q) = / e i q ' r V(r) d r . ty^ and are the f i n a l and incident particle wavefunctions which are taken to be plane waves of momenta k^ and k^ and q = k. - k f is the momentum transfer. For the case of a nucleus, V(r) is the convolution of the interaction of the particle with a point nucleon and the nucleon distribution -49-V(r) = / U(r-r') p(r*) d 3r' . From the properties of a Fourier transform of a convolution F { a(r-r') b(r) } = a(q) b(q) the result is obtained. Thus, both the particle-nucleon interaction and the nucleon distribution are, in general, needed to describe the reaction. 4.2 The Klein-Gordon Equation The equation used to describe the potential scattering of a re l a t i v i s t i c spin zero particle upon a nucleus is the Klein-Gordon equation. The free particle Klein-Gordon equation is given by ( E 2 - p 2 ) = m2 ip where E i s the total particle energy, m i t s mass and p i t s 3-momentum. If one assumes that the nuclear potential V"n can be included along with the static Coulomb potential V"c with the energy E, the equation can then be expressed as (( E - V c - V j 2 - p 2 > = m2 T|> or f E 2 + V 2+ V 2 - 2EV - 2EV - V V - V V - p 2 ) ip = m2 ip . v c n c n c n n c r -1 2 The V term and the Coulomb-nuclear cross terms are usually neglected, n The f i n a l form of the equation is then f E 2 + V 2 - 2EV - 2EV - p 2 1 ip = m2 ip . c c n r ' The term 2EV i s the optical potential to be discussed below. -50-4.3 The Optical Potential A description of the pion-nucleus interaction may be developed in a manner analogous to the interaction of photons with matter. For the photon interaction, the matter through which the photon propagates is modified with respect to the vacuum by its refractive index. In a like manner, the nuclear matter through which the pion propagates is given a complex index of refraction so that i t can both scatter and absorb the incident pions. One purpose of developing a pion-nucleus optical potential is to obtain a semi-phenomenological model, based on theoretical considerations, which can successfully describe data. Ideally, i t should + be able to describe a wide range of TT data with parameters which vary slowly and smoothly with energy with no A dependence over an energy range from negative energies (pionic atoms) to low energies of the order 50 MeV where the A 3 3 is not yet overwhelmingly dominant. To this extent, the SMC potential (Str 79, Str 80, Car 82) discussed below is f a i r l y successful, at least for n + data. It i s not the only possible potential form but incorporates many features which have been demonstrated to be useful in obtaining good data f i t s . For the low energy region, the pion-nucleus scattering amplitude can be written in the form f = b Q + b ^ ' T + ( c Q + c ^ ' t )k«k' - i( d Q + d ^ ' t )a«(kxk') where k, k' are the momenta of the incident and scattered pion, b Q, b x, c Q, C J , d Q, and d x are constants which are related to the s-wave pion-nucleon scattering length and p-wave scattering volume. The constants b Q, c Q, and d Q are referred to as isoscaler while b x, Cj , and -51-are isovector. The operators t, T, and a are respectively the pion isospin, the nucleon isospin, and the nucleon spin. In most studied nuclei, the nucleus tends to be spin saturated (Eis 80) so that the spin term can be ignored. The f i n i t e widths of pionic atom levels imply that absorption of the pion must be included in the potential. The absorption is assumed to predominantly occur on nucleon pairs (absorption on a single nucleon being strongly suppressed by energy conservation). The parameterization of the absorption in the optical potential is then in terms of the square of the nucleon density. It has been noted that to better explain pion data, the potential must be expanded to second order in the multiple scattering series (Eri 66). In second order, the nuclear pair correlations produce two effects. F i r s t , the s-wave parameter b Q is replaced by an effective s-wave scattering length in the nucleus, b Q, given by \ * b 0 - 2? ( b 0 2 + 2 b l 2 ^ where k„ is the Fermi momentum, here taken to be 1.4 fm - 1. Second, the nuclear pair correlations produce an effect analogous to the so-called Lorentz-Lorenz effect which is caused by the scattering of electromagnetic waves in a dense, polarizable medium. This effect introduces a term in the potential referred to as the Lorentz-Lorenz-Ericson-Ericson (LLEE) term and i t s strength in the optical potential is determined by the parameter X. The resulting optical potential including second order effects is then of the form -52-2 a, U - -4TT [ p b(r) + p. B(r) + V2 ( l£l=|)£<£). + (P 2-l>C(r)) opt ^ 2 v 2px 2p2 ; i i 4TTX , . 1 + L(r) with b(r) = b 0 p(r) - e w bj 6p(r) c(r) = c Q p(r) - 6p(r) B(r) = BQ p 2(r) - Bx p(r) 6p(r) C(r) = C Q p 2(r) - Cx p(r) 6p(r) L(r) = p j " 1 c(r) + p 2 _ 1 C(r) where e = ± 1 i s the pion charge , and and P 2 1 + ^ m a) 1 + --mA 1+ £ 1 + 2mA are kinematic factors resulting from the transformation from the pion-nucleon to the pion-nucleus center-of-mass frame. The density terms are and p(r) = p n(r) + p (r) 5p(r) = p n(r) - p (r) with p^, Pp, and p normalized to N, Z, and A respectively. The nucleon mass i s m (~ 931 MeV) and u> is the total center-of-mass energy. -53-The terms in V 2p 2 and V2p are a consequence (Thi 76) of the transformation of the factor in the p-wave term of the scattering amplitude from the pion-nucleon to the pion-nucleus center-of-mass system. It is referred to as the angle transformation. B Q and C Q are the isoscaler absorption parameters. The isovector absorption factors, Bj and , are usually taken to be zero due to insufficient data to establish a reliable value. The parameter set that i s used as a starting point in the f i t t i n g of the calcium data (described in Chapter 5) i s the parameter "SET E" of (Car 82). This parameter set was obtained as follows; Re B and Re C were taken from theoretical values, Im b and Im c from pion-nucleon phase-shift values modified by the effect of the Pauli principle, Im B and Im C from f i t s to absortion cross-sections, and Re b and Re c were adjusted to f i t existing T T + elastic scattering data. No T T - elastic scattering data was used in the f i t t i n g procedure for SET E as there were only limited data published at energies < 50 MeV at the time. It i s therefore reasonable to expect then that SET E may not adequately describe T T - scattering data. This has been shown to be true, at least for nuclei larger than carbon, in recent work (Tac 84, Sob 84, Gyl 84). The poor T T - description is also plainly evident in the data set of this experiment. The T T + 4 0 C a data of (Pre 81) was part of the data set used to obtain SET E. There is a discrepancy between the SET E f i t and that data set as can be noted from (Car 82). The SET E parameters are shown in Table 4.1 -54-SMC Optical Potential parameters coefficients (Re + i Im) b 0 (fm) -.061 .006 b x (fm) -.13 -.002 c 0 (fm 3) .70 .028 c x (fm3) .46 .013 B 0 (fm1*) -.02 .11 C 0 (fm6) .36 .54 X 1.4 Table 4.1: The SET E optical potential parameters (Car 82) -55-Chapter V RESULTS The optical potential used to describe pion scattering in this thesis was discussed in Chapter IV. In addition to the optical potential, the nucleon and charge densities are required for the cross-section calculation to proceed. The form of density taken i s the three parameter Fermi (3PF) form given by ( \ - 1 + w(r/c) 2 P U ; " ° 0 1 + exp((r-c)/t) where c is the half density radius , 4»ln(3)«t i s the skin thickness, and w is the "wine-bottle" parameter. For l + 0Ca and l | 8Ca, the charge distribution parameters are taken from (Sin 73) and (Fro 68). The 4 8Ca charge parameters have been adjusted slightly to reproduce the rms radius given i n (Woh 81). By adjusting the c parameter of the distribution, the expected proton r a d i i are reproduced. The neutron distribution of 4 0Ca is taken equal to the proton distribution, which i s consistent with a and proton experimental data. The 3PF parameters used in the analysis are shown in Table 5.1. The computer code BRENDA was used to determine the optical potential through f i t t i n g the absolute 4 0Ca data. BRENDA i s a modified form of DWPI (Eis 76) that contains the SMC potential, and can handle a variety of density forms. The ratio f i t t i n g was done using the code of Kr e l l and Thomas (Kre 68, Tho 81) in the form used in (Bar 85, Gyl 84) (referred to as the Krell code). The absolute cross-sections produced by these two codes agree to better than 2% over most of the angular range of -56-Nucleus Parameter Distribution Charge Proton Neutron c (fm) 3.768 3.629 3.629 *°Ca t (fm) .5865 .5865 .5865 w -.161 -.161 -.161 FIT B FIT C MIA 4 8Ca c (fm) t (fm) 3.754 .526 3.642 .526 3.867 .552 4.018 .529 3.642 .526 w -.03 -.03 -.03 -.03 -.03 Table 5.1 The Fermi distribution parameters used in this analysis. Note that for Krell code f i t t i n g , the charge distribution i s set equal to the proton distribution. MIA refers to the Fermi parameter values used in the Fourier-Laguerre f i t s of section 5.2.2, while FIT B and FIT C refer to the best f i t parameters from ratio f i t t i n g with those potentials to the whole angular range of the data set. -57-concern in this experiment. One of the main differences between the codes is the form taken for the charge density. In BRENDA, the charge parameters are separate from the proton parameters. In the K r e l l code, the charge distribution i s set equal to the proton distribution for a Fermi density form. Differences between the codes should cancel to f i r s t order in the ratio calculation. The method of solution employed by the optical codes is to numerically integrate the radial Klein-Gordon equation incorporating the nuclear and Coulomb potentials for each potential wave out to some predetermined radius. The resulting wavefunctions are then matched to the asymptotic Coulomb wavefunctions at that radius. In this manner, the phase shifts for each partial wave are determined, and thus the differential cross-sections. In the thesis of (Gyl 84), some detail was given to considering the sensitivity of the ratios to the optical potential. Ratios in the diftractive region were shown to be very sensitive to the potential, and for that reason, data was taken only to ~ 105° for the sulpher and magnesium experiments. For the calcium isotopes, the diffractive minima occur at even smaller angles than the sulpher isotopes, so the angular range of the data would have to be greatly limited (maximum angle <95°) to reproduce the sulpher analysis methods. However, data points were taken to 130° as the inelastic data was originally also of interest. Thus, there may not be sufficient points at <95° for an extensive and reliable model independent analysis. One should note that the errors in the back angle ratio data tend to be large due to the contribution of the 1 6 0 contamination. The optical potential sensitivity in the diffractive region should be somewhat reduced by these large errors so that i t may be -58-possible to obtain consistency with the inclusion of the back angle data in the analysis. 5 . 1 A b s o l u t e C r o s s - S e c t i o n F i t s As discussed in section 4.3, the SET E optical parameters are not expected to f i t the tr - data very well. Also, the Tr +-' t 0Ca data set of (Pre 81) i s not f i t well by the SET E calculation. The SET E calcution i s shown in Fig. 5.1 and agrees well with the present T T + data except at backward angles. There appears to be a normalization difference between this data set and the results of (Pre 81). As the author believes his data to be normalized correctly, then i t must be concluded that the data of (Pre 81) i s i n error. The energies used in these calculations are 49.5 MeV for l t 0Ca and 49.7 MeV for **8Ca. These energies correspond to the energy at the center of the targets assuming a 50 MeV incident beam energy. If the incident energy is not exactly 50 MeV, then the f i t to the optical potential should compensate by adjusting the potential appropriately to mimic the energy effects. From Fig. 5.1, i t i s seen that SET E does not describe the I T - data. This disagreement is consistent with trends observed in (Gyl 84,Sob 84a, Tac 84). To better reproduce the data, the parameters of the optical potential are adjusted through the minimization of the x 2/v defined by where the sum i s over n data points and v i s the degree of freedom. The X2/v of the SET E f i t to the ir~ and ir + combined is 61. In order to constrain the potential better, the f i t t i n g i s done to T T - and T V + Fig. 5.1 SET E calculations compared to the 4 0 C a data; • this experiment, • from (Pre 81) -60-simultaneously. Different combinations of parameters were allowed to vary from SET E in attempting to f i t the data. It became clear that to obtain a good f i t , the LLEE parameter, X, had to be one of the parameters varied. One of the effects of f i t t i n g X ( i t increases in the f i t s ) is to "soften" the edge of the potential so as to f i l l in the diffraction minima. The two best f i t s obtained are shown in Table 5.2, which have three parameters different from SET E. If more than three parameters are free, i t becomes increasingly d i f f i c u l t to find a unique minimum in the parameter space due to correlations in the parameters. The results of FIT B and FIT C are shown in Fig. 5.2 for l t 0Ca. Each f i t has certain points of weakness. FIT C f a i l s to f i t the Coulomb-nuclear interference region in the T T + data, while FIT B does not reproduce the diftractive minimum as effectively. The large value of X for FIT C is somewhat distasteful (Jen 85) as i t is a large departure from the SET E value. This, as well as the slightly better x 2/v tends to favor FIT B as the better potential The absolute cross-section data for 1*8Ca was not extensively analyzed for determining rms radii differences. This i s l e f t to the ratio analysis. To demonstrate that the potentials generated in f i t t i n g the 1 + 0Ca data reasonably describe the data, the angular distributions generated are shown in Fig. 5.3. The density parameters used are given in Table 5.1 and the resulting x 2/v are in Table 5.2. The 68° T T + 4 8Ca point looks a bit questionable although no analysis error has been uncovered to date. As the T T + l t 8Ca data i s not extensively analyzed here, there i s no cause for concern on this one data point. - 6 1 -Parameter FIT B FIT C Re b Q - . 0 7 1 4 Im CQ . 1 0 0 3 . 0 6 3 5 Re CQ 1 . 5 5 2 X 1 . 8 3 6 2 . 6 5 6 X 2 ^°Ca 7.2 "•°Ca 7.9 V **8Ca 7.6 k8Ca 6.8 Table 5.2 The two best potential f i t s to the absolute l f 0Ca differential cross-section angular distributions. The potential parameters that are not shown in the table remain at SET E values. Fig. 5.2 FIT B (solid line) and FIT C (dashed line) potential f i t s to the t*°Ca data. Fig. 5.3 FIT B (solid line) and FIT C (dashed line) potential f i t s to the t t 8Ca data. -64-5 . 2 C r o s s - S e c t i o n R a t i o , F i t t i n g The advantage of f i t t i n g ratio data versus absolute cross-sections is that many of the systematic errors inherent in the absolute cross-sections w i l l cancel out or at least be substantially reduced in importance. Essentially, only s t a t i s t i c a l errors in the peak areas contribute to the uncertainty in the ratio. Thus, the determination of A . and the density distribution difference, Ap , should be obtained nn' J ' n' more reliably than by f i t t i n g to the absolute cross-sections. The ratio data is f i t t e d by varying the shape of the l t 8Ca neutron distribution, either by changing the parameters of the Fermi distribution (section 5.2.1) or by adding a series of orthogonal polynomials to a starting neutron density to obtain some degree of model independency (section 5.2.2) . 5 . 2 . 1 F e r m i F u n c t i o n A n a l y s i s In the calculation of the rms radius, p(r) is weighted by a factor of r1* (versus a factor of r 2 for the integrated charge). The density near the surface of the nucleus should then be well determined as most of the contribution to the rms radius comes from this part of the nucleus. It i s reasonable to expect that i f the Fermi density parameters f i t the cross-section ratios, the resulting rms radii difference should be valid. This statement is supported by the results of (Gyl 84). The rms radius of the larger isotope obtained from this type of f i t w i l l be less reliable than the rms r a d i i difference, being quite dependent upon the parameters assumed for the starting densities not being f i t t e d . The sensitivity of the extracted rms r a d i i difference to the inclusion of the data in the diffractive region can be checked by f i t t i n g -65-only those data points at less than 95°. The same potentials that are used to f i t the whole angular range, that i s , FIT B and FIT C, are used to f i t the reduced data set. The results of f i t t i n g to the whole angular range as well as the reduced range are given in Table 5.3. The results are not s t a t i s t i c a l l y independent (they a l l use data from the same data set), so that an overall average should not be found by treating them as s t a t i s t i c a l l y independent. Instead, to reasonably represent the results, taking an unweighted average of the rms radii differences produced by the various f i t s with an error large enough to encompass a l l the values produced, we have A , = .222 ± .048 fm. nn The error quoted here does not include an estimate of an error contributed due to absolute normalization uncertainties as was done i n (Gyl 84) where an estimated contribution of ±.013 fm was taken. The results in Table 5.3 show that there i s enough small angle data to obtain a r a d i i difference with errors comparable to the results from the f u l l data set for the Fermi density form. The ratios produced by these f i t s are shown in Fig 5.4. Fitting only those data at < 95° produces values of A t that are about 40 mfm less than the results from f i t t i n g the whole angular range for the same potential, and there i s a 55 mfm radii difference between the results from the two potentials. The differences between ra d i i obtained from the reduced data set and the whole data set are somewhat larger than one standard deviation as are the differences in ra d i i from the two potentials. This implies that dependence upon the optical potential has not been removed effectively enough and that the sensitivity of the diffractive region i s s t i l l large in spite of the -66-X2 Density Potential Data Set A (fm) Parameters nn' V FIT B f u l l .217 ± .024 1.75 a reduced .174 ± .028 1.37 b FIT C f u l l .269 ± .038 3.28 c reduced .230 ± .038 2.14 d Table 5.3 The rms ra d i i differences obtained by f i t t i n g Fermi density forms to the TT " ratio data for the " f u l l " angular distribution and "reduced" set of angles as described in the text. The density parameters are: a: c = 3.867 fm, t = .552 fm n n b: c = 3.976 fm, t = .492 fm c: c = 4.018 fm, t = .529 fm n n d: c = 4.124 fm, t = .468 fm -67-20 60 g 100 140 c m . Fig. 5.4 The various Fermi function f i t s to the ratio data; FIT B f u l l set (solid) and reduced set (dash-dot), FIT C f u l l set (long dash) and reduced set (short dash). The f i t s are described in the text. -68-substantial errors in the back angle data. It is important to note that model dependent f i t s tend to underestimate errors. Realistic errors would improve agreement somewhat, but lik e l y , the errors would not increase enough to markedly reduce the poor agreement. Possible reasons for this lack of agreement w i l l be discussed in section 5.3. 5 . 2 . 2 M o d e l I n d e p e n d e n t A n a l y s i s The 3PF density form used in section 5.2.1 can give accurate descriptions of the rms radii difference that should not have too substantial of an optical model dependence. However, the density distribution difference obtained from these f i t s would be very dependent upon the exact form taken for the density. The neutron density of l t 8Ca can be described in a model independent form by adding a series of orthogonal polynomials to a starting density that approximates the neutron density. A Fourier-Bessel expansion can be used as the orthogonal series such that the 4 8Ca neutron density, ^ p (r), can be written as " 8 P n ( ' ) = * Vr) + I °n S l n ( r A n ^  = " Vr) + P F B ( r ) where _ nir n " R~ c with R cbeing the cutoff radius beyond which the Fourier-Bessel contribution is set to zero, and the a are the f i t t e d coefficients. n I + 8Pp i s normalized to the number of neutrons in l + 8Ca. i s then constrained to have no net contribution to the neutron number. The model independent analysis used the codes implemented by Gyles and Barnett for their thesis work. Further details concerning the analysis techniques can be found in those theses ((Gyl 84, Bar 85)). -69-The starting density for 4 8 p „ was taken to be either the best f i t r parameters in Table 5.3 or i t was set such that (c,t,w) n = (c,t,w) p. If the procedure i s truly model independent enough, the starting density used should not affect the f i n a l result. If too many fourier terms are f i t to the ratio data, the resulting density distribution difference can oscillate rapidly with radius and tends to have very large errors as a result of correlations in the fourier parameters. Due to the limited momentum transfer available in the data, such oscillations can not be r e a l i s t i c a l l y resolved by the pion. If the fourier series i s truncated too early, then the polynomials do not have enough freedom to f i t any arbitrary density form. That i s , a short series i s not completely model independent. It was found to be impossible to obtain results from the Fourier Bessel approach which had at the same time reasonable errors and no fine radial structure in the density distribution difference when f i t t i n g the entire angular range of the data. If only the\reduced angular data set was f i t t e d , the resulting errors were always very large, indicating that there is insufficient low angle data to f i t in this manner. It is possible that, for this data set, the Fourier-Bessel expansion can not attain enough modelling freedom to f i t the neutron density. A different set of orthogonal polynomials may be more suited to this data set, a possibility being the Fourier-Laguerre (FL) expansion described in Appendix VIII of (Bar 85). Fitting with this form of expansion encountered many of the same problems as the Fourier Bessel f i t t i n g , however two adequate f i t s were obtained using the FIT C potential. Fitting with the FIT B potential tended to produce large density variations near the center of the nucleus and the results were -70-discarded. The results of these f i t s are summarized in Table 5.4 and the ratio f i t s are shown in Fig. 5.5. The neutron radii difference extracted from these two f i t s agree with each other considering the size of the error in the second f i t . Taking the result of the f i t with three Fourier-Laguerre parameters (the second f i t agrees within i t s errors), we have A , = .110 ± .022 fm nn for the model independent analysis. This is not in agreement with the Fermi function f i t . The density distribution difference obtained is shown in Fig. 5.6. The density distribution difference is displayed in the form 4Tr«r 2»Ap (r) so that the plot indicates the amount of extra neutron n density as a function of radius. The shape of the density difference obtained is similar to the results of (Ray 81) except that the maximum in the difference distribution in shifted more to the nuclear interior. This shift i s reflected in the smaller rms ra d i i difference obtained in this model independent analysis than that obtained by (Ray 81) (see Table 1.1). 5.3 Discussion From the discussion of the density analyses above, several points can be made about that calcium data: a) the neutron radii differences obtained have been shown to not be free of model dependency in the description of the neutron density, b) the sensitivity of the ratio data in the diffractive region to the optical parameters cannot be ignored, and c) the model independence that was demonstrated for analysis of data for the sulpher and magnesium isotopes (Gyl 84) has not carried over into -71-Number X 2 Potential of FL A (fm) terms nn' V FIT C 3 .110 ± .022 1.68 5 .081 ± .054 2.38 Table 5.4 The results of the Fourier Laguerre (FL) model independent f i t s to the ratio data using the FIT C potential. The Fermi part of the density i s described by the MIA parameters in Table 5.1. -72-20 60 a 100 140 c m . Fig. 5.5 The Fourier-Laguerre f i t s to the ratio data; three FL parameters (solid), five FL parameters (dashed). I I I I I I L n — i — i — i — i — i — i i r 0 2 4 6 8 r ( fm) 5.6 The density distribution difference produced by the Fourier-Laguerre f i t (solid) compared to the proton analysis (dashed) of (Ray 81). The lines indicate the upper and lower error bounds obtained in the analysis. -74-the calcium data. The difference in atomic number, A, between the two isotopes studied in this experiment i s larger than in the isotope (and isotone) pairs previously measured by group members using similar analysis techniques. It is possible that the potential fitted to describe the l t 0Ca angular distribution does not adequately describe the 't8Ca data. That i s , even though the x 2/v for the l t 8Ca f i t s are comparable to the f i t s to the l t 0Ca data, the potential may not follow trends developing in the l t 8Ca angular distribution shape well enough, so that any short comings in the potential are forcibly absorbed into the neutron density of ' t 8Ca. This might distort the shape of the resulting fi t t e d neutron distribution. The d i f f i c u l t i e s created by taking two nuclei so far apart in A could be reduced by doing an experiment including ^Ca and then analyze the ratios I + l ta/ l t 0a, 4 8a/ 1 + l ta, and h 8 a / t * ° a to try to obtain consistency .To reduce the ratio's sensitivity to the optical potential in the diffractive region, more data would be needed below the minima than was taken for this data set. The d i f f i c u l t i e s in f i t t i n g the optical potential to the data could also indicate that the form of the potential i t s e l f is not adequate. With a reasonable amount of T T - data now available, i t is possible that a new global optical potential parameter set with the inclusion of terms that had up to this point been ignored (Bj_, C^, v c 2 » a n d ^ c v n terms) could be generated to f i t both the T T + and T T - data. Agreement between overlapping data sets from different labs, or sometimes even from the same lab, is not always good (for example, this T T + l t 0Ca data and that of (Pre 81)), so that some choice of relative normalization would have to be made, otherwise attempts to improve the optical potential would be limited. -75-The consistency of the analysis method used for the TT" ratio data could be checked by analyzing the i r + ratios. The ratio difference obtained should be consistent with values from the precision electron and muon experiments. This has not yet been done for this data, but the model independent methods have been applied to i r + scattering (see especially (Bar 85)) where consistency has been demonstrated. The results obtained for the r a d i i difference, A ,, are ' nn' ' A . = .222 ±.048 fm nn' for the Fermi function f i t , and A , = .110 ±.022 fm nn' for the Fourier-Laguerre f i t t i n g . The results do not agree well. Some possible causes for this discrepancy have been suggested above. For incompressible nuclear matter, the radius should follow an A 1 / 3 trend so that the rms ra d i i difference would be This model gives A , = .227 fm nn' which agrees well with the Fermi function f i t result. The A . from the Fermi f i t i s more more consistent with the most nn' recent model independent analyses of proton (Ray 81) and a (Gil 84) scattering than i s the result from the Fourier-Laguerre f i t t i n g (see Table 1.1). However, as the analysis of this calcium data set has not been demonstrated to be free of dependence upon the density form chosen for the l f 8Ca neutron distribution, nor to be free of the diffractive -76-sensitivity in the optical potential, a conclusive comparison is not yet warranted. Further analysis may solve some of these d i f f i c u l t i e s . However, i t appears, at present, that the calcium isotopes (and other nuclei in the f 7 / 2 shell) are just slightly too large in atomic number for elastic pion scattering at 50 MeV to be analyzed i n a model independent fashion through the methods developed and used in (Gyl 84, Bar 85). REFERENCES Arn 82 R.A. Arndt and L.D. Roper, SAID: Scattering Analysis and Interactive Dialin Program, Center for Analysis of Particle Scattering, Virginia Polytechnic Institute and State University Internal Report CAPS-80-3 (1982). Bar 85 B.M. Barnett, Ph. D. Thesis, Univ. of British Columbia, unpublished, (1985). Bat 69 C.J. Batty and G.W. 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