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What pp parameters need measuring from 200 to 525 MeV? Bugg, D. V. (David Vernon); Oram, C 1975

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TRIUMFWHAT PP PARAMETERS NEED MEASURING FROM 200 TO 525 MeV?D.V. Bugg* and C. Oram*Queen Mary College, London^Present address: TRIUMFMESON FACILITY OF:UNIVERSITY OF ALBERTA SIMON FRASER UNIVERSITY UNIVERSITY OF VICTORIA UNIVERSITY OF BRITISH COLUMBIA TRI-75-^TRI-75-^TRIUMFWHAT PP PARAMETERS NEED MEASURING FROM 200 TO 525 MeV?D.V. Bugg* and C. Oram*Queen Mary College, London"Present address: TRIUMFPostal address:TRIUMFUniversity of British Columbia Vancouver, B.C.Canada V6T 1W5 December 1975I ntroduct ionIt is well known that the pp elastic scattering amplitudes are uniquely determined up to about 600 MeV.1 However, although the phase shift solution is unique, it is not as precise as one might wish, particularly in low partial waves (which contain the essential physics) and at the higher energies. We investigate the improvements possible in the phase shifts from fresh measurements in the range 200 to 520 MeV, readily access­ible at TRIUMF, particularly of Wolfenstein parameters D, R, A, R' and A'. We assume a reasonable goal for fresh measurements to be ±0.8% for da/dft, ±0.01 for P and ±0.025 for other parameters, and seek a combination of measurements which reduces the errors on all observables to these levels.A subsequent report will deal with the less healthy np system.MethodFirst we use all existing data to do single energy phase shift analyses at 200, 320, 380, 425 and 520 MeV. The data set is very close to that of Arndt et at.1 However, we keep, with increased errors, a few data points which they reject. We also add the recent Maryland measurements of da/dfi2 and Geneva data on P at small angles.3 Our phase shift solutions and error matrices closely reproduce the published values of Arndt et at.From these solutions we predict, using the full error matrix, values and errors for all the common experimental observables from 10 to 170 deg (Tables IV). These errors immediately indicate what most needs measuring, and at what level of accuracy. We choose convenient groups of new measurements and find their effects on the error matrix of the phases (Tables V), until the errors on all observables are below the limits specified in the Introduction.The Choice of VariablesPhases up to and including 3Hg are treated as variables; higher partial waves are set to OPE values. At 320 MeV we include inelasticity in 3Pl only. At 380, 425 and 520 MeV we allow inelasticity in ^ 2  only. This is undoubtedly an over-simplification. However, we find that the elastic amplitudes are generally insensitive to small inelasticity (whose effect can be simulated by a small change in phase), and our conclusion is that-  2 -the small inelasticity parameters are better determined by studying the inelastic channels.We do not attempt an energy-dependent analysis, since our feeling is that at this level of accuracy the energy dependence must be flexible enough to accommodate an S-shaped excursion through the chosen energies (due to poles below threshold and inelasticity above 500 MeV). We omit very little data of consequence at intermediate energies: data close toour chosen energies Ej are reliably adjusted to Ej using the energy dependence of the phases.We have investigated the effect of varying g2 , the WWHAA coupling constant. At 200 MeV the error on g2 from existing data is ±3-^, and at 320 MeV it is ±3.8. Data at ^25 and 520 MeV give no real determination of g2 . We conclude that g2 is fixed essentially by the lower energy data (particularly around 1 **0 MeV and near threshold), and that it would un­fairly inflate the errors at the higher energies to treat g2 as a variable. Accordingly, most results are presented with g2 fixed at 1^.4. We remark, as is well known, that g2 is essentially determined by data at small angles. Values of D and A' are particularly sensitive to it. Table I shows the accuracy with which g2 is determined by various measurements at 200 MeV.Table II shows the effect on x2 of fixing some or all of the bl­and G-waves at OPE values. Existing data show little deviation from OPE in the H-waves, except perhaps in 3Hit at 200 and 520 MeV. Exchanges of heavier mesons are to be determined from the lower partial waves, particu­larly P, D and F. We maintain that it should be possible subsequently to predict the tiny contributions from heavy boson exchange to G-and H-waves, and hence to reduce the errors on the low partial waves by a bootstrap operation. From an experimental viewpoint one wants to be sure that fresh experiments are more powerful than this type of theoretical constraint.As an example, we show in Table III the reduction in the errors in low partial waves at 320 MeV when G- and H-waves are fixed at OPE. (Small contributions from heavy mesons will not affect the errors significantly.) From this and Table V(b) we can judge which experiments are more powerful than theory, and the conclusion is that theory alone is not enough.-  3 -The Choice of MeasurementsTables IV show the errors on all the common experimental observables from 10 to 170 deg. The situation deteriorates rapidly as the energy rises. Except at 520 MeV, dc/dfi and P are already predicted with an accuracy close to our goal. Tables V display the diagonal elements of the error matrix resulting from existing data W, plus various hypothetical addition. The angular range is arbitrarily broken up into three sections, 0-60 deg, 60-120 deg and 120-180 deg, which require rather different experimental configurations (dictated by the range of the recoil proton). We choose a set of measurements common to all energies and concentrate on Wolfenstein parameters rather than correlation parameters, since we believe they are easier to measure. We assume that one would measure P with an accuracy ±0.017 and Wolfenstein parameters with an accuracy ±0.025, with a common normalization with uncertainty ±0.025; the P measurements serve only to normalize the Wolfenstein parameters. We find that R is generally less useful than D, but the experimental configuration is so similar that we assume both would be measured in practice.We summarize our general observations, and then discuss individual energies.1) Measurements at one particular angle often greatly reduce the errors on other observables at the same angle. However, measure­ments at angles more than 15 deg apart are largely independent, i.e. a measurement at small angles does not usually reduce the errors near 90 deg. An exception is that accurate measurements from 0 to 60 deg generally reduce errors in the range 120 to 180 deg (i.e. transfer parameters) to a low level.2) It is essential to have good (1%) da/dft data over the full angular range, in order to fix the normalization of the large amplitudes.The normalization of existing dcr/dft data is frequently unreliable, and the normalization is therefore fixed by the total cross-section. This is a rather unhappy situation, since no cross-check is avail­able. We suspect that the increase in x2 going from energy- independent analysis to energy-dependent analysis as observed by Arndt et at. is largely due to normalization problems.-  k -3) D is sensi tive to tensor parameters £ 2 and ei* and g2 ; it is a vitalparameter over the full angular range 0 to 120 deg.h) R is sensitive to singlet phases; it is less useful than D, andcould be omitted from 60 to 125 deg at b25 MeV.5) Measurements of D and R from 13 to 115 deg reduce errors on most ob­servables to our goal and therefore largely exhaust the necessarymeasurements. Table IV(e) illustrates this at A25 MeV. However, some further measurements are necessary and vary with energy. R' can be measured with a configuration very similar to R, and for this reason we have systematically included R' (33 deg) in Tables V; its weight is not great, and one angle between 0 and 60 deg is enough, except perhaps at 520 MeV where measurements at larger angles would help. More useful, but much more difficult, is A' (longitudinal spin before and after the scatter); it is highly desirable at all energies from 0 to 60 deg. A is a somewhat less satisfactory sub­stitute^.g. Table V(b)).Off-Diagonal Elements of the Error MatrixGenerally these are small and follow the trend of the diagonal ele­ments. However, a striking correlation exists at all energies between 3Pi and 3P2 (Table VI). This is because they are the largest phases and dominate da/dJ2 and CTj at all energies; their similar effect on da/dfi (except in the Coulomb interference region) causes a strong correlation in the phase shift solution. The k2 surface in the two-dimensional sub-space of the 3Pi and 3P2 variables has contours which are narrow ellipses, and the eigenvalues X of the error matrix, given byH r r 2E11 + E22 + E12 + jf[EH "  22are of greater interest than the diagonal elements themselves; they are shown in Table VI. Any measurement which reduces the correlation helps substantially in reducing the larger eigenvalue. In this respect, the otherwise ineffective R f(33 deg) measurement is useful in eliminating a strong correlation between 3Pi, 3P2 and 3H5 , hence its inclusion in the selection of measurements. (However, A r would do the same job better.)-  5 -Cone!usi onsWe discuss our requirements at individual energies.200 MeV: The data are already of such accuracy that the only worth-whileadditions are i) D (13 deg) 'to fix g2 with greater accuracy, ii) D and R close to 90 deg, and iii) A' (13—53 deg).320 MeV: Measurements of D and R from 13 to 115 deg make a valuable re­duction in the present errors. Either A' or A is desirable from 13 to 53 deg, A' being the better.425 MeV: The greatest need is for D and R from 0 to 60 deg. However, toreduce the errors on all observables to the goal given in the Introduction one would like measurements of D from 65 to 115 deg, A f from 0 to 60 deg, and da/dft across the full angular range.520 MeV: The present k2 minimum is very ill defined, and one really needsextensive new measurements. One gets a vast reduction in the errors from measurements of D and R from 0 to 120 deg. New 1% measurements of dcr/dft are also important, and TPP from 30 to 90 deg is highly desirable. For the sake of completeness,Table V(d) lists other measurements which would finally reduce the errors on all observables to the arbitrary level specified in the Introduction; however, it will be better to make fresh judgements after the first round of new measurements.380 MeV: Present data do not give a well-defined k2 minimum. However,much of the data at this energy is of high quality, and the addi­tion of D and R from 0 to 115 deg gives an excellent minimum. Again A' in the range of 0 to 60 deg is desirable. It seems presently that sizable errors in the predictions for C and Aag parameters at small angles can only be eliminated by measuring one of these parameters from 0 to 60 deg, the one with the greatest sensitivity being hzi- However, 380 MeV is unique in this respect, and it may be an artifact which disappears once data define a true k2 minimum.-  6 -AcknowledgementsWe are grateful to K. Shakarchi for assistance in checking the pro­gramme used to calculate observables from pp phase shifts, and for supplying Coulomb phase shifts after inclusion of the Coulomb form factor.References1. R.A. Arndt, R.H. Hackman and L.D. Roper, Phys. Rev. C9_, 555 (197*02. K. Abe, B.A. Barnett, J.H. Goldman, A.T. Laasanen, P.H. Steinberg,G.J. Marmer, D.R. Moffett, E.F. Parker, Phys. Rev. D12, 1 (1975)3. D. Aebischer, B. Favier, G. Greeniaus, R. Hess, A. Junod, C. Lechanoine,J.C. Nicklfes, D. Rapid and D. Werren, Proc. of **th Int. Symposium onPolarization Phenomena in Nuclear Reactions, Zurich (1975)T a b l e  I .  T he  e r r o r  on g2 f r om measurements  a t  D ( 13 deg )  and A ' ( 1 1 . 5  deg)  a t  200 MeV as a f u n c t i o n  o f  t he a c c u r a c y  o f  t he  measurements .Data 6g2Pr esen t  da t a ± 3 - A0<5D (13 deg )  = ±0 . 025 ±1 . 676D( 13  deg )  = ± 0 . 017 ±1 . A7<5D (13 deg )  = ± 0 . 010 ±1 . 326D( 13  deg )  = ±0 . 005 ±1 . 266 A ' ( 1 1 . 5  deg )  = ±0 . 025 ±1 . 77<5A' ( 1 1 . 5  deg)  = ±0 . 017 ±1 . 686 A ' ( 1 1 . 5  deg )  = ±0 . 010 ±1 . 62T a b l e  I I .  Va l ues  o f  x 2 w i t h  v a r i o u s  G-  and H-waves  f i x e d  a t  OPE v a l u e s .V a r i a b l e s 92200Ener gy320(MeV)A25 520A 1 1 G-  and H-waves  f r e e f r e e 50 . 6 117 . 5A1 1 G -  and H-waves  f r e e 1A.A 50 . 9 117 . 9 137 . 6 128 . A3H6 f i x e d  a t  OPE 1 A.  A 52 . 9 122. 2 137 . 6 132. 53H6 and 3H5 f i x e d  a t  OPE 1 A.  A 5A. A 12 A . 6 1A0. 2 13A. 3A l l  H -waves  f i x e d  a t  OPE 1 A.  A 6 1 . 7 125. 2 1A1.5 162. 3 ,  u n s t a b l eG-  and H-waves  f i x e d  a t  OPE 1 A.  A 71 . 8 125 . A 160.1 161. 7T a b l e  I I I .  T he  e f f e c t  on t he  d i a g ona l  e l ement s  of  t he  e r r o r  ma t r i x  ( d e g r e e s 2 ) o f  f i x i n g  H -waves  a t  OPE v a l u e s  a t  320 MeV.  ( g 2 i s  f i x e d  a t  14.1*)V a r i a b l e s 3Po ' So 3Pi 3P2 e2 T a b l D2 T l T 3P'+ ei* Te I. h  T r o 3H6H-waves  f r e e 2 . A8 1. 57 0 . 79 0 . 28 0 . 15 0 . 2 9 0 . 1 8 0 . 3 6 0 . 060 0. 11 0 . 1 A 0 . 0 7  0 . 2 8 0 . 036H-waves  a t  OPE 1. 87 0 . 98 0 . 53 0 . 1 9 0 . 12 0 . 25 0 . 13 0 . 33 0 . 0A0 0 . 05A - 0 . 039  - _G-  6 H -waves  a t  OPE 1. 82 0 . 8 9 0 . A2 0 . 15 0. 11 0 . 23 0 . 13 0 . 33 0 . 039 0. 051 - - -Table IV(a). Predicted errors on common observables at 200 MeV from the phase shiftanalysis of existing data. (g2 fixed at I1*.4; phases varied up to H&)A n g l e  fc.m. dea)6a/o 6P 6D 6R 6A 6 R ' 6 A ' 6 C Kp ng 22 An g g 6C pp 6AXX 6AZZ 6AZXU lllU l uV3 i  10 0. 011 0 . 0 0 3 0 . 021 0 . 0 1 7 0 . 0 0 8 0 . 0 0 4 0 . 0 4 7 0 . 0 1 0 0 . 0 2 0 0 . 0 1 0 0 . 0 4 6 0 . 0 1 0 0 . 0 4 7 0 . 0 0 620 0 . 0 0 5 0 . 0 0 4 0 . 021 0 . 0 1 3 0 . 0 1 2 0 . 0 0 9 0 . 0 1 9 0 . 0 1 4 0 . 0 3 6 0 . 0 1 8 0 . 0 3 4 0 . 0 1 4 0 . 0 3 8 0 . 0 1 230 0 . 0 0 5 0 . 0 03 0 . 0 1 2 0 . 0 1 0 0 . 0 0 8 0 . 0 1 0 0 . 0 3 6 0 . 0 1 0 0 . 0 1 8 0 . 0 1 7 0 . 0 4 4 0 . 0 1 6 0 . 0 4 5 0 . 0 1 040 0 . 0 0 5 0 . 0 0 3 0. 011 0 . 0 0 9 0 . 0 1 0 0. 011 0 . 0 3 6 0 . 0 0 5 0 . 0 0 8 0 . 0 2 3 0 . 0 4 3 0 . 0 1 7 0 . 0 3 8 0 . 0 1 950 0 . 0 0 6 0 . 0 0 4 0 . 0 1 3 0. 011 0 . 0 1 2 0. 011 0 . 0 2 2 0 . 0 0 4 0 . 0 0 5 0 . 0 2 2 0 . 0 2 9 0 . 0 1 3 0 . 0 2 0 0 . 0 2 060 0 . 0 0 6 0 . 0 0 5 0 . 0 1 3 0 . 0 1 2 0 . 0 1 7 0 . 0 1 2 0 . 0 1 9 0 . 0 0 5 0 . 0 0 4 0 . 0 2 5 0 . 0 2 6 0. 011 0. 011 0 . 0 2 470 0 . 0 0 5 0 . 0 0 5 0. 021 0 . 0 1 9 0 .021 0 . 0 1 8 0 . 0 2 3 0 . 0 1 0 0 . 0 0 7 0 . 0 3 0 0 . 0 3 0 0 . 0 0 9 0 . 0 1 0 0 . 0 3 080 0 . 0 0 6 0 . 0 0 3 0 . 0 3 2 0 . 0 2 7 0 . 0 1 5 0 . 0 2 5 0 . 0 1 5 0 . 0 1 5 0 . 0 1 2 0 . 0 2 3 0 . 021 0 . 0 1 2 0 . 0 1 7 0 . 0 2 290 0 . 0 0 8 0 . 0 0 0 0 . 0 3 8 0 . 031 0 . 0 0 8 0 . 0 2 9 0 . 0 0 8 0 . 0 17 0 . 0 1 4 0 . 0 0 7 0 . 0 0 7 0 . 0 1 4 0 . 0 2 2 0 . 0 0 0100 0 . 0 3 5 0 . 0 2 9 0 . 021 0 . 0 2 8 0 .021110 0 . 0 2 4 0 . 0 2 3 0 . 0 2 5 0 .021 0 . 0 2 6120 0 . 0 1 4 0 . 0 1 6 0 . 0 1 8 0 . 0 1 2 0 . 0 2 0130 0 . 0 1 3 0 . 0 1 8 0 . 0 1 3 0 . 0 1 0 0 . 0 2 0140 0 . 0 1 3 0 . 0 2 4 0. 011 0 . 0 0 7 0 . 0 2 7150 0 . 0 2 0 0 . 021 0. 011 0 . 0 1 6 0 . 0 2 5160 0 .021 0 . 0 1 2 0 . 0 1 5 0 . 0 1 7 0 . 0 1 4170 0 . 0 2 8 0 . 0 0 5 0 . 0 1 0 0 . 0 2 3 0 . 0 0 5T a b l e  I V ( b ) . As I V ( a )  a t  320 MeV.Ang  1 e ( c .m.  deg )6c/a dP 6D 6R 6A 6 R ' 6 A ' 6CKP 6CNN SCKK 6Cpp O) > X X 6AZZ 6AZX10 0 . 0 2 4 0 . 0 0 9 0 . 061 0 . 0 3 8 0 . 0 1 7 0 . 0 1 4 0 . 0 7 9 0 . 0 2 2 0 . 0 3 7 0 . 0 2 4 0 . 0 7 0 0 . 0 2 4 0 . 0 7 3 0 . 0 1 820 0 . 01  1 0 . 0 1  1 0 . 0 5 0 0 . 0 2 6 0 . 0 2 9 0 . 0 2 6 0 . 0 3 3 0 . 0 2 7 0 . 0 5 3 0 . 0 2 8 0 .041 0 . 0 2 2 0 . 0 4 4 0 . 0 2 730 0 . 0 0 9 0 . 0 1 0 0 . 0 3 5 0 . 0 3 7 0 . 0 4 3 0 . 0 3 2 0 . 0 6 2 0 . 0 1 8 0 . 0 3 3 0 . 0 3 5 0 . 0 2 8 0 . 0 3 4 0 . 0 2 6 0 . 0 2 040 0 . 0 0 8 0 . 0 0 9 0 . 041 0 . 0 3 4 0 . 0 5 0 0 . 0 3 5 0 . 0 6 7 0 . 0 3 4 0 . 031 0 . 0 3 3 0 . 0 1 7 0 .041 0 . 0 1 7 0 . 0 2 850 0 . 0 0 8 0 . 0 0 7 0 . 0 4 2 0 . 031 0 . 041 0 . 0 2 9 0 .051 0 . 0 4 4 0 . 0 2 6 0 . 0 1 9 0 . 0 1 8 0 . 0 3 7 0 . 0 4 0 0 . 0 2 860 0 . 0 0 9 0 . 0 0 6 0 . 0 3 5 0 . 0 2 3 0 . 0 4 2 0 . 0 2 9 0 . 0 4 6 0 . 0 3 4 0 . 0 23 0 . 0 1 6 0 . 0 2 7 0 . 0 2 5 0 . 0 4 2 0 . 0 2 270 0 . 0 0 7 0 . 0 0 7 0 . 0 3 5 0 . 0 1 7 0 . 0 4 6 0 . 0 2 8 0 . 0 4 8 0 . 0 3 0 0 . 0 1 9 0 . 0 2 0 0 . 0 2 8 0 .021 0 . 0 3 7 0 . 0 2 580 0 . 0 0 7 0 . 0 0 5 0 . 0 5 5 0 . 0 1 6 0 . 0 2 8 0 . 0 2 0 0 . 0 3 3 0 . 0 4 2 0 . 0 2 2 0 . 0 2 0 0 .021 0 . 0 3 5 0 . 0 4 8 0 . 0 2 090 0 . 0 0 8 0 . 0 0 0 0 . 0 7 8 0 . 0 1 7 0 . 0 2 9 0 . 0 1 6 0 . 031 0 . 0 4 8 0 . 0 2 7 0 . 0 1 4 0 . 0 1 4 0 . 0 4 4 0 . 0 5 6 0 . 0 0 0100 0 . 0 8 3 0 . 0 2 3 0 . 0 5 7 0 . 0 1 6 0 . 0 5 7n o 0 . 0 6 6 0 . 0 3 5 0 . 0 6 4 0 . 0 1 9 0 . 0 6 6120 0 . 0 5 0 0 . 0 4 4 0 . 0 6 0 0 . 0 2 9 0 .061130 0 . 0 5 3 0 . 0 5 7 0 . 0 6 2 0 . 0 4 2 0 . 0 6 7140 0 . 0 4 8 0 .061 0 . 0 6 6 0 . 0 4 5 0 . 0 7 2150 0 . 0 4 2 0 . 0 4 6 0 . 0 6 0 0 . 0 3 2 0 . 0 5 6160 0 . 0 5 5 0 . 0 2 2 0 . 0 5 5 0 . 0 2 3 0 . 0 2 8170 0 . 0 6 4 0 . 0 1 2 0 . 0 4 3 0 . 0 4 4 0 . 0 0 9Table IV(c). As IV(a) at 425 MeV.Ang l e  ( c .m.  deq6o/a <5P 6D <5R 6A 6R ' 6 A ' 6CKP 6cNN 6CKK 6Cpp 6AXX 6AZZ 6AZX10 0 . 024 0. 011 0 . 072 0. 031 0 . 030 0 . 021 0 . 117 0 . 029 0 . 057 0 . 029 0 . 106 0 . 030 0. 1 10 0. 02120 0 . 014 0 . 014 0 . 087 0 . 022 0 . 0 50 0 . 042 0. 061 0 . 037 0. 071 0 . 023 0 . 057 0 . 025 0 . 062 0 . 03330 0 . 017 0 . 012 0 . 066 0 . 028 0 . 057 0 . 049 0 . 056 0 . 028 0 . 040 0 . 038 0 . 046 0 . 036 0 . 039 0 . 03440 0 . 015 0 . 010 0 . 059 0 . 026 0 . 057 0 . 043 0 . 060 0 . 030 0 . 017 0 . 037 0 . 037 0 . 027 0 . 017 0 . 04250 0 . 013 0 . 008 0 . 048 0. 021 0 . 038 0 . 027 0 . 048 0 . 034 0 . 016 0. 021 0 . 022 0 . 023 0 . 023 0 . 03360 0 . 014 0 . 006 0. 021 0 . 012 0 . 014 0. 011 0 . 019 0. 021 0 . 015 0 . 014 0 . 013 0 . 017 0 . 023 0 . 01570 0 . 013 0 . 005 0 . 018 0 . 013 0. 011 0 . 013 0 . 017 0 . 022 0. 011 0 . 013 0 . 012 0 . 017 0 . 025 0 . 01480 0 . 015 0 . 004 0 . 032 0 . 019 0 . 016 0 . 018 0 . 027 0 . 050 0 . 018 0. 011 0 . 012 0 . 047 0 . 053 0. 01190 0 . 0 1 9 0 . 000 0 . 036 0 . 019 0 . 025 0 . 018 0 . 028 0 . 062 0 . 023 0. 011 0. 011 0 . 062 0 . 066 0 . 000100 0 . 033 0 . 017 0. 031 0 . 014 0 . 027110 0 . 022 0 . 013 0 . 023 0 . 012 0 . 020120 0 . 026 0 . 016 0 . 019 0 . 015 0 . 018' 30 0 . 049 0 . 027 0. 041 0 . 022 0 . 040140 0 . 057 0 . 033 0 . 057 0 . 033 0 . 053150 0 . 055 0 . 02 8 0 . 056 0 . 035 0 . 042160 0 . 047 0 . 017 0 . 052 0. 021 0 . 020170 0 . 048 0 . 010 0 . 052 0 . 035 0 . 009. . . . . .  .T a b l e  I V ( d ) .  P r e d i c t e d  e r r o r s  on common o b s e r v a b l e s  a t  520 MeV f r om the  phase s h i f t  a n a l y s i s  o f  e x i s t i n g  d a t a .  ( g 2 f i x e d  a t  14 . 4 ;  phases  v a r i e d  up t o  3Hg)  T hese  e r r o r s  a lmos t  c e r t a i n l y  a r e  an u nd e r e s t i ma t e ,  s i n c e  x 2 = ' 2 8  f o r  72 deg o f  f r eedom in t he  phase s h i f t  f i t .Ang 1 e (c.m.  deg)Sa /a 6P 6D 6R 6A 6R ' 6 A ' <$CKp nf 22 6CKK 6Cpp 6AXX 6AZZ 6AZX10 0. 101 0 . 019 0 . 068 0. 121 0 . 077 0 . 073 0 . 165 0 . 143 0 . 1 9 3 0 . 267 0 . 209 0 . 290 0 . 200 0 . 1 2 320 0 . 052 0. 031 0 . 069 0 . 066 0 . 097 0 . 105 0 . 046 0 . 179 0. 066 0 . 078 0 . 226 0 . 117 0 . 182 0 . 1 8 930 0 . 036 0 . 027 0 . 098 0 . 060 0 . 084 0. 091 0 . 037 0. 131 0. 094 0 . 083 0. 141 0 . 070 0 . 125 0 . 14240 0 . 015 0 . 013 0 . 103 0 . 094 0 . 1 0 3 0 . 095 0 . 046 0 . 107 0. 098 0 . 100 0 . 089 0 . 085 0 . 1 2 5 0 . 09850 0 . 012 0 . 009 0 . 116 0 . 096 0 . 099 0. 101 0 . 057 0 . 118 0 . 1 2 7 0 . 104 0. 091 0 . 082 0 . 163 0 . 09060 0. 011 0 . 006 0 . 117 0 . 088 0 . 073 0 . 083 0 . 050 0. 121 0.  124 0 . 072 0 . 095 0 . 083 0 . 184 0 . 05570 0. 011 0 . 004 0 . 106 0 . 087 0 . 078 0 . 078 0 . 070 0.  140 0. 111 0 . 046 0. 091 0 . 117 0 . 1 9 2 0 . 02780 0. 011 0 . 003 0 . 092 0 . 100 0 . 086 0 . 079 0 . 096 0 . 196 0 . 130 0 . 039 0 . 100 0 . 162 0 . 258 0 . 01790 0 . 019 0 . 000 0 . 069 0 . 096 0 . 097 0 . 085 0 . 1 0 9 0 . 2 3 1 0 . 149 0 . 074 0 . 074 0 . 185 0. 301 0 . 000100 0 . 058 0. 071 0 . 104 0 . 082 0 . 099110 0 . 055 0 . 049 0. 101 0. 061 0 . 083120 0 . 048 0 . 073 0 . 104 0 . 043 0 . 100130 0 . 052 0 . 1 0 9 0 . 094 0 . 035 0 . 122140 0 . 084 0 . 1 1 7 0 . 093 0 . 050 0 . 120150 0 . 109 0 . 163 0 . 1 2 3 0 . 053 0. 141160 0 . 095 0 . 219 0 . 1 9 2 0. 111 0 . 1 7 2170 0 . 089 0 . 1 5 2 0 . 174 0 . 197 0 . 128Table IV(e). Predicted errors on common observables at 1(25 MeV, after the addition ofP D R (13,33,53 deg) and R'(33 deg) to existing data. Otherwise, as Table IV(c).Ang l e  (c.m. deg)6a/a 6P SD 6R 6A 6R ' 6 A ' SCKP 6CNN nf mm SC pp 6AXX SAZZ SAZX10 0 . 020 0 . 008 0 . 020 0 . 014 0 . 0 1 1( 0 . 013 0.01(3 0. 011 0.0l ( 2 0 . 023 0 . 055 0 . 023 0 . 0560 . 00820 0 . 013 0 . 009 0. 021 0 . 015 0 . 018 0 . 017 0 . 022 0. 01i ( 0 . 033 0 . 018 0.01(10 . 016 0. 0l ( 2 0 . 0 1 1(30 0 . 015 0 . 008 0 . 019 0 . 015 0 . 02b 0 . 019 0 . 035 0 . 012 0 . 018 0 . 020 0. 031 0 . 0190 . 0 26 0 . 0171(0 0 . 0 1 1( 0.008 0. 021 0 . 0 1 1( 0. 031 0. 021 0 . 039 0 . 015 0 . 013 0 . 019 0 . 022 0 . 0 16 0 . 013 0. 02150 0 . 013 0 . 006 0 . 019 0 . 013 0 . 025 0 . 017 0 . 028 0 . 015 0. 011 0 . 012 0 . 015 0 . 0130 . 01b 0 . 01560 0 . 013 0 . 005 0 . 012 0 . 009 0. 011 0 . 009 0 . 012 0 . 010 0 . 010 0. 008 0. 011 0 . 008 0 . 01b 0 . 00770 0. 011 0 . 005 0 . 0 15 0 . 010 0 . 010 0 . 0 10 0 . 012 0 . 017 0 . 010 0 . 009 0 . 010 0 . 013 0. 021 0 . 00980 0 . 015 0. 00i ( 0 . 020 0.01b 0. 011 0 . 013 0 . 015 0 . 032 0 . 017 0 . 010 0 . 010 0 . 030 0 . 037 0 . 00790 0 . 0 19 0 . 000 0 . 023 0 . 0 1 1( 0 . 016 0 . 013 0 . 017 0 . 039 0. 021 0. 011 0. 011 0 . 0370.0l ( l ( 0 . 000100 0 . 025 0 . 012 0 . 022 0. 011 0 . 020110 0 . 020 0. 011 0 . 018 0 . 009 0 . 016120 0 . 017 0 . 012 0 . 016 0 . 010 0 . 0 15130 0. 02l ( 0 . 0 1 9 0 . 0 30 0 . 013 0 . 029H O 0 . 023 0 . 025 0 . 036 0 . 015 0 . 036150 0 . 023 0 . 023 0 . 030 0 . 017 0 . 02 8160 0 . 029 0 . 0 1 A 0 . 03b 0 . 015 0 . 015170 0.01(0 0 . 008 0 . 0 b 1 0 . 023 0 . 007Table V(a). Diagonal elements of the error matrix (degrees2) at 200 MeV from existing data, W, and various>t/><><D ^ 3p° 1$0 3pl 3p2 e2 3F2 X02 3F3 3Fl4 e„ 3H4 1g4 3h5 3h6     0.264 0.272 0.042 0.028 0.016 0.059 0.078 0.031 0.027 0.007 0.050 0.010 0.035 0.022W + PDR (13,33,53 deg) 0.21.5 0.249 0.035 0.025 0.014 0.056 0.064 0.027 0.025 0.005 0.038 0.009 0.026 0.018Ditto + R' (33 deg) 0.244 0.249 0.031* 0.024 0.014 0.056 0.061 0.026 0.025 0.005 0.037 0.009 0.026 0.018Ditto + PDR (65-115 deg, 10 deg steps) 0.087 0.088 0.025 0.020 0.008 0.032 0.018 0.023 0.019 0.003 0.028 0.004 0.020 0.014Ditto + Ar (13,33»53 deg)_______________________________0.072 0.066 0.022 0.018 0.006 0.029 0.011 0.015 0.018 0.002 0.015 0.004 0.011 0.008W + PDR(65-115 deg, 10 deg steps) O.O96 0.110 0.029 0.022 0.009 0.036 0.021. 0.027 0.021 0.004 0.035 0.005 0.025 0.017Ditto + A'd3,33,53 deg) 0.077 0.070 0.023 0.019 0.006 0.030 0.012 0.016 0.019 0.003 0.017 O.0Q4 0.011 0.009  3p0 Is0 3pl 3p2 3p2 % 3F3 3F4 3RM ESND UIG URF^   2.1*7 1.62 0.81 O.29 0.15 0.29 0.19 0.36 0.061 0.11 0.14 0.073 0.28 0.035w + PDR(13,33,53 deg) 1.86 0.52 0.54 0.18 0.11 0.25 0.10 0.33 0.036 0.021. 0.058 0.034 0.18 0.024Ditto + R'(33 deg) I.74 0.49 0.44 0.13 0.092 0.25 0.091 0.31 0.032 0.024 0.057 0.033 0.14 0.021Ditto + PDR(65~115 deg, 10 deg steps) = X 0.21 0.26 0.22 0.10 Q.Ql.2 0.053 0.061 0.055 0.020 0.011 0.032 0.027 0.057 0.014Ditto + A'(13,33,53 deg) 0.21 0.23 0.084 0.060 0.022 0.043 0.050 0.037 0.019 0.009 0.024 0.023 0.023 0.013Ditto + A(13,33,53 deg) 0.20 0.022 0.065 0.047 0.021 0.040 0.048 0.033 0.018 0.009 0.024 0.022 0.022 0^013X + A(13,33,53 deg)_______________________________________°-21 °-27 0.11 0.058 0.029 0.044 0.060 0.040 0.019 0.011 0.030 0.024 0.036 0.014W + PDR(65-115 deg, 10 deg steps) 0.50 0.61 0.42 0.21 0.074 0.11 0.13 0.067 0.029 0.029 0.045 0.043 0.089 0.021Table V(c). As V(a) at *(25 MeV.Data 3po ls0 3pl 3p2 F2 3F2 ]D2 3F3 3F4 e4 3Hi, 1gA 3h5 3h6W 2.13 1.16 1.16 0.57 0.55 0.27 0.25 0.16 0.050 0.19 0.13 0.048 0.19 0-071W + PDR(13,33,53 deg) 0.76 0.43 0.55 0.30 0.16 0.11 0.11 0.069 0.043 0.035 0.057 0.036 0.14 0.034 Ditto + R' (33 deg) 0.64 0.37 0.30 0.15 0.13 0.093 0.11 0.065 0.034 0.035 0.057 0.036 0.11 0.030Ditto + PDR(65-115 deg, 10 deg steps) 0.40 0.28 0.25 0.13 0.090 0.075 0.096 0.057 0.031 0.021 0.045 0.032 0.069 0.027Ditto + A'(13,33,53 deg) 0.36 0.27 0.19 0.11 0.064 0.061 0.081 0.046 0.028 0.016 0.036 0.032 0.030 0.024Ditto + da/dn(10-90 deg, 5 deg steps) 0.29 0.23 0.11 0.0750.056 0.033 0.056 0.023 0.011 0.014 0.029 0.013 0.023 0.014Data 3p0 1 So 3Pi 3P2 F2 3F2 xD2 3F3 3F4 e4 3H4 % 3H5 3H6w 19.2 35.8 8.77 1.53 9.09 2.12 23.6 3.55 0.54 2.00 1.10 4.49 1-36 0.40W + PDR(13,33,53 deg) 6-33 5.50 2.88 0.62 1.32 0.42 4.60 0.47 0.37 0.56 0.35 1.21 0.31 0.17Ditto + R'(33deg) 5-87 5-33 1.94 0.52 1.32 0.40 3-10 0.45 0.16 0.45 0.25 0.99 0.31 0.16Ditto + PPR(65-115 deg, 10 deg steps) = X 0.87 1■04 0.50 0.33 0-65 0-16 2.29 0-31 0-12 0.095 0-15 0-23 0-15 0.11s u tD (13 33,53 deg) _ 0.71 0.98 0.39 0.19 0.51 0.12 1.56 0.24 0.086 0.089 0.097 0.20 0.10 0.072 X + da/dft(10-90 deg, 5 deg steps) = Y 0.62 0.78 0.28 0.17 0.47 0.072 1.31 0.21 0.035 0.075 0.059 0.17 0.066 0.028Ditto + CNN(30-90 deg, 10 deg steps) = Z 0.52 0.46 0.27 0.11 0-17 0.069 0.44 0.19 0-025 0.045 0-041 0.087 0.059 0.027Y + a(65-115 deg, 10 deg steps) 0.58 0.75 0.25 0.10 0.22 0.053 0.58 0.16 0.028 0.058 0.045 0.14 0.051 0.024z + ckp(90 deg) 0.51 0.46 0.26 0.10 0.13 0.063 0.39 0.19 0.025 0.040 0.041 0.087 0.058 0.027Ditto + r(160 deg) 0.48 0.44 0.24 0.096 0.068 0.060 0.17 0.14 0.022 0.024 0.041 0.042 0.013 0.025Table V(d). As V(a) at 520 MeV.BEST-PRlNTER CO LTD VANCOUVER. B CTable V(e). Diagonal elements of the error matrix (degrees2) at 380 MeV from existing data, W, and various additions to it. In the additions <5P = ±0.017 and other errors = ±0.025. 92 is fixed at 14.4.The minimum addition to existing data which gives a x2 minimum is shown in the first line of the table.Data 3p0 ls„ 3P! 3p2 r2 3Fz l|j2 3^ 3^ F|f SHl( lGl| 3Hs 3HgW + PDR(13,33,53 deg) 1.89 1.28 0.93 0.47 0.22 0.48 0.085 0.48 0.14 0.068 0.22 0.043 0.28 0.17Ditto + R'(33 deg) 1.88 0.75 0.79 O.38 0.22 0.40 0.063 0.47 0.11 0.057 0.22 0.039 0.19 0.14Ditto + PDR(65-115 deg, 10 deg steps) 0.63 0.46 0.35 0.20 0.056 0.11 0.052 0.10 0.040 0.022 0.092 0.026 0.046 0.056Ditto + A'(13,33,53 deg) 0.44 0.37 0.23 0.125 0.046 0.082 0.043 0.089 0.035 0.016 0.081 0.023 0.036 0.049Ditto + AZZ(13,33,53 deg 0.29 0.32 0.084 0.061 0.035 0.076 0.033 0.049 0.034 0.013 0.044 0.015 0.024 0.027Table VI. Diagonalization of the sub-error matrix in the 3px-3P2 space. E33 andare the diagonal elements of the error matrix for 3Pl antj 3f>2> respectively, and E34 is the correlation term between them. Then Aa and A[j are the eigenvalues of the error matrix, ignoring correlation with other phases.Units are degrees2.633 61+1, 631, Ag A^200 0.04214 0.02828 0.02784 0.0639 0.0064320 0.84226 0.28903 0.40825 1.0588 0.0725425 1.16435 0.56648 0.68271 1.6107 0.1201520 8.77246 1.53218 1.08396 8.9313 1.3733380 0.92729 0.46921 0.58576 1.3272 0.0693

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