Nucleon-nucleon phase shift analysisSubmitted on 1 Jan 1987
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Nucleon-nucleon phase shift analysis J. Bystricky, C. Lechanoine-Leluc, F. Lehar
To cite this version: J. Bystricky, C. Lechanoine-Leluc, F. Lehar. Nucleon-nucleon phase shift analysis. Journal de Physique, 1987, 48 (2), pp.199-226. 10.1051/jphys:01987004802019900. jpa-00210432
J. Bystricky (+), C. Lechanoine-Leluc (++) and F. Lehar (+)
(+) DPhPE, CEN-Saclay, 91191 Gif sur Yvette, Cedex, France (++) DPNC, Université de Genève, Geneva, Switzerland
(Requ le 26 mai 1986, accept6 le 2 octobre 1986)
Résumé. 2014 Une analyse en déphasages avec dépendance en énergie est présentée pour les réactions élastiques pp, np et pn entre 10 et 800 MeV. Pour l’analyse de la réaction pp, une solution unique a été trouvée présentant une
dépendance en énergie lisse sauf pour le déphasage 1D2 qui montre un comportement résonnant vers la masse de 2,14 GeV. Pour l’analyse des réactions np et pn très peu de mesures de paramètres complexes de polarisation existent, si bien que la solution trouvée est moins précise surtout au-dessus de 500 MeV. Une analyse à énergie fixe à 1 GeV a aussi été effectuée, pour laquelle une unique solution a été trouvée. Le formalisme, la base de données utilisée et la compatibilité avec d’autres analyses sont discutées.
Abstract. 2014 An energy dependent phase-shift analysis is reported for pp, np and pn elastic scattering between 10 and 800 MeV. For the pp case a unique solution smoothly varying with energy is found. A resonant behaviour is observed only in the 1D2 phase shift near the mass 2.14 GeV. For np and pn scattering only a few complex polarization parameters are available ; therefore the solution is less precise, especially above 500 MeV. A fixed- energy analysis at 1 GeV is also reported providing a unique phase-shift set. The formalism, data base and compatibility with other analyses are discussed.
J. Physique 48 (1987) 199-226 FÉVRIER 1987, 1
Classification
1. Introduction.
Phase shift analyses (PSA) are still the most successfull phenomenological approaches to the nucleon-nucleon amplitude determination in the intermediate energy range. The PSA will provide excellent results for pp scattering up to 1 000 MeV as soon as all SIN, LAMPF and Satume II results from recent experiments will be available. The np solution is not as satisfactory and more data on several complex polarization parameters (e.g. Aookk, Aoonn) are needed. The present phase shift analysis is an updated and
improved version of our previous one [1, 2]. Our approach to the analysis was guided by the following considerations : (i) The amount of experimental data up to 800 MeV is sufficient for an energy dependent PSA. (ii) A fixed-energy PSA is fully justified if a sufficient number of experiments are performed at
practically the same energy. Otherwise, averaging of values measured at different energies introduces sys- tematic uncertainties which cannot be correctly es-
timated. (iii) In an energy dependent analysis, one should be careful to leave enough freedom to the phases in order to describe possible structures.
(iv) Model-dependent input to the analysis should be minimized in order to detect any unpredicted
phenomena. In particular we did not introduce as data the ratio of the real and imaginary parts of the spin- independent forward scattering amplitudes, non the total elastic cross section, calculated by integrating the differential cross sections (included in the analysis). (v) Inelastic total cross sections were used as part of the input. They were obtained by summing the cross sections of all reaction dhannels open in the considered energy interval. Direct bubble chamber data on
Utot (inel) were also introduced. (vi) Certain noncon- troversial kinematical and theoretical results were im-
plemented in the analysis. In particular we enforced the correct threshold behaviour for each partial wave amplitude and used one pion exchange results for
higher order partial waves. Taking these considerations into account, we have
divided the energy range from 10 to 800 MeV in four
overlapping intervals. In each of them our energy expansion of the phase shifts allows two extrema. This leads to four independent analyses for each isospin channel. The isospin I = 1 phase shifts were first determined on the basis of pp data. The data are well distributed between different experimental quantities and the resulting energy dependences of the phase shifts are smooth functions of the energy. The pp phase
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01987004802019900
shifts were then used as fixed input in the np (pn) analysis, where only the isospin I= 0 phase shifts were fitted from np and pn data. Here, the available data are
mostly differential cross-sections. The I= 0 phase shifts are therefore not so well determined, as illus-
trated by their energy dependences. In overlapping energy regions, where 2 analyses give different sol-
utions, either of them can be used to calculate predic- tions, as both of them describe equally well the existing data. This is an analogy to a fixed-energy analysis which results in two different solutions. When results from Satume II will be available, the
pp analysis could be extended above 1 GeV. Meanwhile we give fixed-energy PSA results at 1 GeV, where only one solution remains.
2. Formalism
2.1 AMPLITUDES. - In the analysis we have used the invariant scattering amplitudes a, b, c, d and e. The expression of all observables in terms of these quantities can be found in reference [3]. For each laboratory kinetic energy T and centre-of-mass scattering angle 0, these amplitudes may be written in terms of singlet- triplet amplitudes [3-5] :
The expansion of the singlet-triplet amplitudes in
Legendre polynomial series and partial wave amp- litudes a u, as well as their parametrization by nuclear bar phase shifts BLJ and mixing parameters EJ were
taken from reference [4] :
Up to the one pion production threshold s’s and B’s are real, above the 6’s can be complex with Im 8 , 0. These requirements are introduced to fulfill the unitari- ty condition. In principle a sixth parameter, cp J, should
be introduced as discussed in reference [6]. We have omitted it as our fit to the data is reasonably good with only « five parameter » representation. The six par- ameter approach has been tried and no sizable improve- ment to the fit was observed at 1 GeV [7, Priv. Com.]. Other phase shift analyses in this energy region [7-13] use only five parameters. Denoting Legendre polynomials as P J (cos 0 ) and
their derivatives with respect to cos 0 as P J (cos 0 we write the resulting expansion of the invariant
scattering amplitudes as
where p is the centre-of-mass momentum. The a Lj for J 0 and aJ for J 0 are defined to be zero.
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2.2 ONE PION EXCHANGE CONTRIBUTIONS. - The
expansion of the amplitudes was truncated at the total angular momentum Jmax. The higher angular momen- tum states were replaced, as usual, by the Bom
approximation of the one pion exchange contribution (OPE) [5, 14]. The OPE invariant amplitudes, whose J * Jmax contributions should be subtracted, are :
where
with
Ml, M2 mass of the beam and target nucleon respect- ively,
0 mass of ir 0 u mass of charged pion, E total energy,
f2 pion-nucleon coupling constant = 0.08. The pion-nucleon coupling constant was introduced
as a fixed parameter.
2.3 ELECTROMAGNETIC CONTRIBUTIONS. - The Coulomb amplitudes and Coulomb-nuclear interfer- ences were introduced as in references [4, 5]. Namely, the Coulomb parts of the invariant pp scattering amplitudes are:
where
M = the proton mass.
For angular momenta L higher than Lmax, we have added to the amplitude e the magnetic moment correc- tion [15] given by
where 4 p (= 2.7928456) is the anomalous magnetic moment of the proton. Another form of the electromagnetic contributions
given in reference [16] was also tested, but the resulting phase shifts were not significantly different from the present analysis. The electromagnetic contributions calculated from the one photon exchange were studied in detail in reference [17]. 2.4 ENERGY DEPENDENCE OF PHASE SHIFTS. - The
energy dependence of the phase shifts were fitted by polynomial expansion of the form
in each interval, where To is the central point of the interval and aLJn are variable parameters. In all cases it
turned out that n = 3 was sufficient, in some cases we adopted n = 2, or n = 1. Proper threshold behaviour was assured by multiplying equation (8) by OPE factors, obtained from the appropriately calculated OPE elements of the K-matrix. The higher waves, as mentioned above, were taken to be pure OPE ones.
It is well known that low-L OPE phase shifts do not correspond to the nuclear-bar phase shifts. We there- fore have used for iSo and 3SJ phase shifts the scattering length and effective radius energy dependence. For 61 and 3D1 the energy dependence is properly taken into account by multiplying the polynomials by arctg (T) 3/2 and arctg (T) 5/2, respectively. Above the inelastic threshold, the phase shifts are
allowed to be complex. The imaginary parts of these phase shifts are then written:
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where the threshold energy TLJ is proper to each phase shift, as well as auo, aul and aw which were treated as free parameters. Inelastic unitarity was imposed by constraining the Im 5 LJ to be non-negative. The parametrization equation (9) of the threshold
behaviour of the imaginary parts of the phase shifts is in agreement with general analyticity and unitarity re-
quirement. The inelastic threshold energy T u was left as a free parameter and was fitted in the lowest energy interval in which the corresponding phase shift SLJ receives an inelastic contribution. The same form
equation (9) was used in all intervals, with T u already found in the initial interval.
Continuity and the boundaries of the intervals was not artificially imposed. For pp scattering the energy dependences of the real and imaginary parts of the phase shifts do however turn out to be close to
continuous as a result of the fit. The isospin I = 1 phase shifts found in the pp
analysis were used as a fixed input to the np analysis. The OPE parts of the pp phase shifts were replaced by the np OPE and the multiplying polynomials were changed so that whole np phase shifts reproduce the pp ones (see also Ref. [18]).
3. Data base.
3.1 ANALYSED DATA. - To denote the experimental quantities, we use a four subscript notation [3] : Xsrbt, where the subscripts s, r, b, t refer to the
scattered, recoil, beam and target particle respectively. In the present analysis we have used all the relevant
data available in the compilation of reference [19] as well as recently published data. These latter are listed in table I (pp) and table II (np).
In total 6966 independent pp data points and 6866 np data points were analysed in the energy dependent PSA. For pp scattering 35 % of the data are spin independent measurements, 34 % are polarizations (p = Aoono = Aooon) and 31 % are parameters with two or three subscripts (correlations and Wolfenstein parameters). For np scattering the corresponding per- centages are 81 %, 14 % and 5 %, respectively. A more detailed repartition of the data is shown in table III
(pp) and in table IV (np). For each type of experiment we give the total number of data points as well as their occurrences. For the fixed energy solution 618 indepen- dent data points in the energy range 970-1 040 MeV were used as listed in table III.
3.2 INELASTIC CROSS-SECTIONS. - In the energy re-
gion from 290 to 1000 MeV there ’exist only few independent measurements of the total pp or np inelastic cross sections. On the other hand, about
260 pp and 100 np (pn) measurements of different inelastic channel cross sections are known in this energy range. In order to use this information, we have fitted
the energy dependence of the total cross-section of the reaction j by an expression of the form [75] :
where Pi are polynomials of the energy T and the parameters Toj’ coj, ..., Cnj are fitted for each reaction j. The total inelastic cross-section was then calculated as a sum of all the reaction cross-sections, fitted together with the independent measurements of the atot (inelas- tic). The recent Dubna measurements of otot (np, inel) [58] were also taken into account. The atot (inelastic) were introduced into the PSA in 5 MeV steps with the calculated errors.
Figure 1 compares our fitted pp and np reaction cross-sections to those obtained by Ver West and Arndt [76]. For pp the agreement is excellent up to 750 MeV. From there on, the 2 7T-channel contributions, not
taken into account in reference [76] become significant. For np the disagreement is apparent from 400 MeV on, and becomes more and more pronounced with increas- ing energy. This discrepancy comes mainly from a different treatment of at (np => npir") , which is the most dominant and the least measured reaction. The
bubble-chamber measurements of Qtot (inel) (e.g. [58]), were not taken into account in [76]. Our treat- ment of inelastic pp and np cross-sections will be
discussed in a separate article.
3.3 FITTING PROCEDURE. - All experimental data were fitted according to the standard x2-method, including the error matrix calculation (see e.g. ref. 77) with statistical errors taken from publications. The systematic errors and the discrepancies in the normali- zations between the PSA and the data were taken into account by introducing variable normalization factor multiplying each data set. They were kept free only if their values were different from one by more than their errors. We found that most of the remaining normaliz- ation factors apply to differential cross-sections. Our X 2 sum thus reflects correctly the systematic experimen- tal uncertainties. Experimental points for which the x2-contribution was larger than 10 were omitted from the analysis. These represent about 3 % of pp and 2 % of np data.
Several incompatible sets exist for 4u L (pp ) measurements. This is clearly illustrated e.g; in
figure 12 of reference [25]. Since the SIN 4u L data [25] were obtained simultaneously with .4uL ( pp => 7T+d) measurements which are in good agreement with the values extracted in a separate experiment for
Aookk ( pp =:> dw + ) [78], confidence in the normaliz- ation of the AoL (pp elastic ) results appears justified. We have therefore renormalized all other measure- ments on the SIN data in the 500 MeV interval and used this normalization in the other intervals as fixed
parameters. In the 500 MeV interval the Saclay data
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204
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Table III. - Summary of pp data points between 10 and 800 MeV.
[27] turn out to need no normalization, being in excellent agreement. The TRIUMF data [24] were found too large by 23.2 % and the LAMPF data too small by 1.7 %. In the 260 MeV interval the TRIUMF data at 202.7 and 325.5 MeV have been omitted due to
large X 2 values, showing that a common normalization cannot be applied to the overall set. The apparent disagreement between the RICE-
LAMPF, RICE ZGS data [22] and other Aory measure- ments was found to be due to different treatment of Coulomb-Nuclear interference corrections. No normal- ization of any åUr data set was needed âÎ all, when the two higher energy TRIUMF [24] data points are
omitted (see discussion in Ref. [23]). The Argonne Aoonn measurement at 697.6 MeV [79]
is well fitted with a normalization of 24.9 %. More recent LAMPF [46] and Saclay [43] measurements confirm this normalization factor.
Table IV. - Summary of np/pn data points between 10 and 800 MeV used in PSA.
The np total cross-sections measured at LAMPF [55] and at SIN [56] are nicely fitted without normalization. This shows that above the pion production threshold a good compatibility exists between available forward- scattering data, i.e. differential cross-sections at small angles [63, 80], new total cross-sections [55, 56] and our calculated inelastic cross-sections (see Eq. (10)). Other differential cross-section data are also well fitted with normalization factors close to one, namely about 2 300 data points measured at LAMPF [81]. A disagreement is found in the shape of the angular distributions of Aoono measured at TRIUMF [82] and at LAMPF [83]. In this case, the normalization could not remove the
discrepancies, therefore we have omitted the Aoono data from reference [82]. Several Aoonn ( np ) data points at 665 MeV from LAMPF [84] were also omitted. A detailed discussion of the np data itself would
greatly increase the volume of this PSA article. Such a discussion is being submitted separately to the J.
Physique.
4.1 COMPARISON WITH PREVIOUS ANALYSIS. - The
present phase shift analysis differs from our previous one in the following aspects.
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Fig. 1. - Inelastic pp and np total cross-sections. The full lines are our calculations, the dash-dotted line is that of Ver West and Arndt [76].
1. The data basis has been considerably enlarged in both pp and np scattering. Above 400 MeV the new data are mainly due to Saclay experiments.
2. The fixed energy analysis of pp scattering at
1 GeV has now become unique. The original 9 different possible solutions first collapsed to 4, mainly as a result of the Argonne and Gatchina experiments. The new Saclay data permitted the present unique deter- mination.
3. Correct threshold behaviour were implemented at the elastic and inelastic thresholds (see Sect. 2).
4.2 ENERGY INTERVALS. - The analysis was per- formed in the energy range from 10 to 800 MeV. The entire range was divided into four overlapping inter- vals : 10-220 MeV (« 80 Mev »), 130-450 Mev (« 260 MeV »), 380-610 MeV (« 500 MeV ») and 520-800 MeV (« 670 MeV »). In each of them our energy expansion allows two extrema for every phase shift in order to permit a good fit of possible structures. If the last parameter in equation (8) for a phase shift was found to be less than its error then the parameter was set to zero.
An independent analysis of pp data was performed in each energy interval to find the isospin I =1 phase
shifts. The I =1 phase shifts were used as fixed input to the np analysis. The analyses were performed once more independently in each energy interval defined above.
4.3 RESULTS AND DISCUSSIONS. - Phase shift val- ues found in the energy dependent PSA are given in tables V-IX. The solution at 1 GeV is given in table X. Both real and imaginary phase shifts are calculated in units of degrees. Phase shifts for which the real parts are not mentioned in the tables, were set equal to the OPE (see Eq. (4)) and magnetic moment (see Eq. (6)) contributions. All other phase shifts were fitted. The number of experimental data points used and the X 2-values for each analysis are summarized in table XI. The value in the first column is the energy To in
equation (8). The starting values for the pp scattering analysis were
taken from reference [2]. A unique solution was found in each energy interval. The thresholds of the imaginary parts of all S, P, D, F and G waves were studied. The thresholds of the Im ’So and 3F4 are found to be higher than 800 MeV and the 1m 3P1 is compatible with zero up to 800 MeV. The Im 3Po, also compatible with zero below 800 MeV, is determined with large error and give no improvement of the PSA fit. At 1 GeV only one solution (Tabl. X) remains after
introducing the recent Saclay data [44, 48]. The number of different experimental quantities at 1 GeV is smaller than that necessary for a direct reconstruction of the
scattering matrix. Consequently, values of some phase shifts may change after introducing data recently measured at Satume II.
For the I = 0 analysis, slightly modified pp phase shifts were used as fixed input and only the isospin I = 0 phase shifts were left free. The inelasticities in the S, P, D and F waves were studied. Only the
imaginary parts of 3SJ, lpl, 3D1 and 3D2 should be significantly different from zero, but in the analysis only the two last ones were considered. The energy dependence of the S, P, D, F, G, H and I
phase shifts and mixing parameters together with their « error bars » are shown in figures 2-10 for pp and in figures 11-17 for np and pn scattering. The « corridor of errors » shown in some of figures was calculated as the square root of the corresponding diagonal element of the error matrix. It is much narrower than the error
corresponding to the « confidence level 1 u». The
meaning of the indicated corridor is that it indicates the regions in which further experiments would be particu- larly fruitful.
In the pp scattering, the continuity between solutions in different energy intervals is very good. The point at 1000 MeV is our fixed-energy solution. An interesting behaviour is observed in the real part of 3P2 above 400 MeV. Small discontinuities in pp phase shifts have a strong influence in the np analysis.
208
Table V. - Real parts of pp phase shifts in degrees.
209
degrees.
I --The isospin I = 0 phase shifts are less smooth and *how a need for more experimental data. We mention that no new spin dependent measurements on np scattering in the energy region 70-200 MeV have been published since 1968. Hence many of the I = 0 phase shifts are poorly determined in the « 80 MeV » np interval. Moreover, earlier data are grouped around the energy 140 MeV. We don’t attach a great physical significance to the behaviour of the phase shifts and the corresponding predicted experimental quantities in this energy region. The coupled triplet 3G3 phase shift
(Fig. 15) shows a distinct energy dependent structure,
in complete agreement with the five fixed energy phase shift analyses [8] as well as with our previous PSA [2]. The PSA in the « 500 MeV » interval is strongly constrained by the new Satume II forward differential cross sections at 481 and 582 MeV [63]. The connection with the considerably less known, « 260 MeV » interval was poor. To improve it we imposed a smooth connec- tion for the 3G3 phase shift at 400 MeV. More spin dependent measurements in this region are definitely needed.
In figures 2-17, the real and imaginary parts of our phase shifts are also compared to the results of the most recent PSA, those of Dubois [8] (squares), Higuchi [10] (triangles), Vovchenko [12] (open circles), Grebenyuk [13] (diamonds) and Arndt [7] (dashed-dotted lines). Their phase shifts have been converted into our rep- resentation ; this transformation changes mainly the imaginary parts. Only the phase shifts left free are shown in the figures. The ones imposed from theoretical calculations [8] are omitted. Error corridors of variab- le-energy solution from reference [7] are not available. Other older fixed-energy analyses [5, 9, 11] are not shown, their data base being less complete than ours and the comparison thus less instructive. The dotted lines in figures 2-8 and 11-16 are contributions from OPE (see Eq. (4)) together with electromagnetic corrections (see Eq. (6)). At lower energies, they compare well with the fitted values. A significant comparison between different analyses
can only be done between the present energy dependent analysis, and those of Arndt [7] and of Dubois [8]. This last analysis was carried out at energies where the set of measured experimental quantities approximately cor- responds to a complete set of experiments. Moreover, this analysis is « locally » energy dependent, i.e. in the energy range of fitted data points, and the data sets at different energies are correctly normalized to the
corresponding central values. Results from other fixed energy analyses [10, 12, 13] are very dispersed, and will not be discussed in detail. Our fixed energy solution at 1 GeV will certainly be changed when new Saclay data are included, but is a very good starting point for a variable energy solution between 700 and 1300 MeV. For real as well as for imaginary parts a better
agreement is observed between our solution and
Dubois’s in the entire energy range. As can be seen in
figures 2-8, the agreement between all three analyses is good for I = 1 real phase shifts up to 450 MeV although there are small discrepancies, namely in the Re 3F2, Re 3F3 and Re 3H5 phase shifts. At higher energies, Arndt’s solutions [7] are smoother than ours, however the latter coincides better with the fixed-energy analysis of Dubois [8]. This may be partly due to the fact, that the Arndt’s analyses do not contain completely the data sets measured by the Geneva group [49, 50] and any recent Saclay results [23, 42, 43, 44, 48]. Other reasons could be the different ways in which the inelastic cross sections are taken into account, more or less extensive
211
Table VII. - Real parts of np phase shifts for Isospin = 1 [deg].
212
213
Table VIII. - Real parts of np phase shifts for isospin = 0 [deg].
214
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Table IX. - Imaginary parts of np phase shifts for Isospin 1 and 0. Phase shifts are given in degrees.
use of model dependent input or a different kind of phase shift parametrization. Differences between the present analysis and that of Arndt [7] is more striking in the imaginary parts. A general agreement exists for Im 1D2 only. The dashed lines in figures 9, 10 are
predictions calculated by the Paris group using disper- sion relations [85]. The predictions for Im 1D2 have the right shape, even though the potential was intended only for F and higher waves.
In the I = 0 phase shifts the discrepancies between Dubois, Arndt and our analysis are larger even at lower
energies. The treatment of imaginary parts of phase shifts varies between different authors: in Dubois’
there is no free imaginary phase shift, while in Arndt’s the 1m 3S1’ IPl, 3D1, 3D2 and iF3 are left free, even if several of them are negligible. The dashed line in
figure 17 (Im 3D2 ) is a prediction from the Paris
potential [85]. The last prediction gives non zero
inelastic contributions for other phase shifts, while the np experimental data analysed in our PSA allow in each energy interval only determination of two imaginary
216
Table X. - Real and imaginary phase shifts from fixed energy analysis at 1 GeV.
phase shifts, where all the inelasticities are concen- trated.
The Argand diagrams for the five I = 1 complex phases are given in figures 18a-e. In the energy region studied, only the iD2 partial wave shows, with increas- ing energy, an anticlockwise behaviour reaching a maximum at about 690 MeV. An anticlockwise be-
haviour is also observed in 3P2, 3F3 and IG4 partial waves. The shape of the Argand diagram for 3P2 may be consistent with a possible resonant structure in one of the P waves, suggested by Lomon [86]. The observed structure in Re 3P2 supports this suggestion. In refer-
Fig. 2. - Energy dependences of lSo and 3Po phase shifts. = =- =. Present analysis and our solution at 1 GeV The « corridor of errors » shown in some of figures was calculated as square root of the corresponding element of the error matrix. It is much narrower than the error corresponding to the confidence level 1 Q (see Sect. 4c). Error corridor is shown only if it is large enough to be plotted. -.-.- Analysis of reference [7], 0 Analysis of reference [8], V Analysis of reference [10], 0 Analysis of reference [12], 0 Analysis of reference [13],.... OPE (Eq. 4) and electromagnetic correction (Eq. 6).
Table XI. - Characteristics of the solutions in the four different energy intervals. The values in the first column are energies To from equation (8), the values in the last column correspond to the x2-values per degree offreedom.
217
Fig. 3. - Energy dependences of the phase shifts 3p 1 and 3P2. Symbols have the same meaning as in figure 2.
Fig. 4. - Energy dependence of the phase shift 1D2 and of the mixing parameter s2’ Symbols have the same meaning as in figure 2.
Fig. 5. - Energy dependences of 3F2, 3F3, and 3F4.
Fig. 6. - Energy dependences of the phase shift ’G4 and of the mixing parameter e4.
218
Fig. 7. - Energy dependences of 3H4, 3H5 and 3H6. The 3H5 phase shift at 670 MeV in reference [12] is + 2.45, probably by misprint.
Fig. 8. - Energy dependences of the phase shift 116 and the mixing parameter e6.
ence [87] we have attributed the possible P-wave
structure to the 3Po partial wave, but the recent data determine that Im 3Po is negligible. No conclusion will probably be obtained on the 3F3 structure until more data between 800 and 1000 MeV became available
allowing a care full energy-dependent analysis. A poss- ible resonance-like structure in the lG4 wave may occur at higher energies.
Fig. 9. - Energy dependences of the imaginary parts of the phase shifts 3P2 and lD2. The dashed line is the prediction from the Paris potential [85]. Other symbols have the same meaning as in figure 2. The 1m 3P2 point at 950 MeV from reference [12], at 1 GeV from reference [13] and all points above 830 MeV from reference [10] are off-scale.
Fig. 10. - Energy dependences of the imaginary parts of
3F2, 3F3 and IG4. Dashed lines are the predictions from reference [85]. For other symbols see Figure 2.
219
Fig. 11. - Energy dependence of the phase shift 3S,. -__
present analysis. The « corridor of errors » shown in some of figures was calculated as square root of the corresponding element of the error matrix. It is much narrower than the error corresponding to the confidence level 1 a (see Sect. 4c). Error corridor is shown only if it is large enough to be plotted. -.-.- reference [7], D reference [8].
Fig. 12. - Energy dependences of the phase shift IPl and of the mixing parameter E1. The dotted line is the OPE energy dependence. Other symbols are as in figure 11.
Fig. 13. - Energy dependences of 3DI, 3D2, and 3D3*
Fig. 14. - Energy dependences of 1F3 and E3. Symbols have the same meaning as in figures 11, 12.
220
Fig. 15. - Energy dependences of the phase shifts 3 G3, 3G4 and 3G5. Symbols have the same meaning as in figures 11,12.
Fig. 16. - Energy dependences of lH5, 65 and 31s.
Fig. 17. - Energy dependences of the imaginary parts of the phase shifts 3D1 and 3D2. Dashed lines are the predictions from the Paris potential [85]. For other symbols see figures 11 and 12.
The energy dependences of the ratio of the real to the imaginary part of the spin independent forward amp- litude [ Re ( a + b ) /Im ( a + b ) ] and of the total
cross section differences Ul tot = AoT/2 and - AUL are shown in figures 19 to 24 for pp and np solutions separately. Discontinuities in the np predictions for U 1 tot and - A a, L at the boundaries of each energy interval are more important. Oscillations in the
åu L ( np ) energy dependence at energies between 70-200 MeV are due to the sparseness of spin dependent data points and have no physical meaning (see dis- cussion above). Unlike Arndt [7] and our previous PSA [1], we have not used dispersion relation predic- tions [88] as input to the analysis. Nevertheless, these are in reasonable agreement with our solution as can be seen in figures 19 and 22.
5. Conclusions.
The isospin I =1 phase shifts are fairly well deter- mined. Discrepancies above 450 MeV should disappear as soon as all results from recent experiments at SIN, LAMPF and Satume II are available. The expected Satume II results will even allow an extension of the
energy dependent analysis up to the 1300 MeV region. The number of imaginary phases increases with energy. At 1000 MeV, 8 imaginary phase shifts are necessary as shown in the fixed-energy analysis. The extension of this type of analysis to higher energies would require more experimental data than at lower energies or an independent knowledge of the inelastic channels or another form of theoretical input (see e.g. Ref. [85]).
221
Fig. 18a-e. - Argand’s diagrams. - Present analysis with full circles every 40 MeV.. Our solution at 1 GeV. 201320132013 Analysis of reference [7]. ~ Analysis of reference [8] at 325, 425, 515, 580, 650 and 800 MeV. For each energy only the phase shifts with model-independent imaginary parts are represented.
Fig. 19. - Energy dependence of Re (a + b) / Im ( a + b ) at 6 = 0 for pp scattering. Dispersion relation prediction is shown as dotted line [88].
Fig. 20. - Energy dependence of 0’1 tot (pp) = - 40’ T/2 ; . Satume II, reference [23]. 0 TRIUMF, reference [24]. 0 RICE-LAMPF, reference [22]. 7 RICE-ZGS, reference [22]. 0 ANL-LAMPF, reference [26]. ð. ANL-ZGS, refer- ence 78/B-248/ in [19]. -, * Present analysis and our solution at 1 GeV. The dashed line and the symbol « + » represent the elastic part of 0’1 tot.
222
Fig. 21. - Energy dependence of - åUL.. SIN, reference
[25]. 2022 Satume II, reference [27]. 0 TRIUMF, reference [24]. 0 LAMPF, reference [26]. 7 ANL-ZGS, reference 78/A-164/ in [19]. -, * Present analysis and our solution at 1 GeV. The dashed line and the symbol « + » represent the elastic part of - åu L.
Fig. 22. - Energy dependence of Re ( a + b ) / Im ( a + b ) at 6 = 0 for np/pn scattering. Dispersion relation prediction from reference [88] is shown as doted line.
The I = 0 phase shifts are not as well determined. There is a general lack of data in the forward hemis- phere and spin dependent parameters are rare
everywhere. Due to the absence of highly polarized neutron beams, Wolfenstein parameters have beeen measured more frequently than spin correlations. How- ever, the analysis will not significantly improve until measurements of spin correlation parameters as well as differences of total cross-sections using polarized beams and targets are done. A crucial point for the determina- tion of the isospin I = 0 imaginary parts of phase shifts is a precise measurement of the total cross-section of the np -+ np1To reaction. It has never been measured with free neutrons.
Fig. 23. - Energy dependence of
olitot(np) =-AaT(nP)/2. Full line present analysis. Dash-dot line is PSA of re-
ference [7].
Fig. 24. - Energy dependence of - 40’ L ( np ) . Full line is
present analysis. Dash-dot line is PSA of reference [7]. Experimental points from reference [59] are calculated from the difference 40’ L ( dp) - 40’ L (pp ) and corrected by Kroll [59]. Oscillations in the 40’ L ( np ) energy dependence at energies between 70-200 MeV are due to the sparseness of spin dependent data points and have no physical meaning (see Sect. 4).
A good way to improve the np analysis would be a simultaneous fit of pp and np data. Such approach can be realized only in the PSA at several fixed energies. On the other hand it is hard to establish an energy dependent structure of phase shifts using only the results of analyses at fixed energies.
223
Acknowledgments.
We are much indebted to P. Wintemitz, A. Gersten, R. Hess, N. Hoshizaki, E. Leader, A. de Lesquen, B. Loiseau, W. Plessas, M. Ross and L. van Rossum for
helpful discussions. We thank also the University of
Geneva group and M. Gargon, M. M. Gazzaly, D. Legrand, L. C. Northcliffe, E. Rossle, J. Saudinos, H. Schmitt, Y. Terrien, S. S. Yamamoto and F. Wellers for providing us with their data before publication. We acknowledge the help of D. Breich in the preparation of the figures for this manuscript.
References
[1] BYSTRICKY, J., LECHANOINE, C. and LEHAR, F., Preprint DPhPE 79-01, Saclay 1979 and Landolt- Bornstein Tables, New Series, Vol. 9a, edited by H. Schopper (Springer Verlag, 1980) and
BYSTRICKY, J., LECHANOINE-LELUC, C. and LEHAR, F., Preprint DPhPE 81-05, Saclay 1981.
[2] BYSTRICKY, J., LECHANOINE-LELUC, C., LEHAR, F., Preprint DPhPE 82-12, Revised February 1984, Saclay 1984.
[3] BYSTRICKY, J., LEHAR, F. and WINTERNITZ, P., J. Physique 39 (1978) 1.
[4] STAPP, H. P., YPSILANTIS, T. J., METROPOLIS, M., Phys. Rev. 105 (1957) 302.
[5] HOSHIZAKI, N., Suppl. Progr. Theor. Phys. 42 (1968) 107.
[6] BRYAN, R., Phys. Rev. C 30 (1984) 305 and refer- ences quoted herein.
[7] ARNDT, R. A., ROPER, L. D., BRYAN, R. A., CLARK, R. B., VERWEST, B.J., SIGNEL, P., Phys. Rev. D 28 (1983) 97 and Private Communi- cation of Arndt R. A.
[8] DUBOIS, R., AXEN, D., KEELER, R., COMYN, M., LUDGATE, G. A., RICHARDSON, J. R., STEWART, N. M., CLOUGH, A. S., BUGG, D. V. and EDGINGTON, J. A., Nucl. Phys. A 377
(1982) 554. [9] HOSHIZAKI, N., Prog. Theor. Phys. 60 (1978) 1796,
and Prog. Theor. Phys. 61 (1979) 129, and HASHIMOTO, K., HIGUSHI, Y. and HOSHIZAKI, N.,
Prog. Theor. Phys. 64 (1980) 1678, and AKEMOTO, M., MATSUDA, M., SUEMITSU, H. and
YONEZAWA, M., Progr. Theor. Phys. 67 (1982) 554.
[10] HIGUCHI, Y., HOSHIZAKI, N., NAKAO, H. and
SUZUKI, K., Contributed paper to « 1983 INS Symposium », Tokyo, 1983.
[11] VOVCHENKO, V. G., GORODNITSKII, G. A., ZHDANOV, A. A., LAZAREV, V. I., MALYUTEN- KOV, E. I., POLYAKOV, V. V., POPOV, V. E., PROKOFEV, A. N., FEDOROV, O. Ya., SHVEDC- HIKOV, A. V. and YAZIKOVA, S. V., Yad. Fiz.
32 (1980) 164, transl. Sov. J. Nucl. Phys. 32 (1) (1980) 85, and
DZHGARKAVA, M.I., KAZARINOV, Yu. M., STRACHOTA, I. and KHAYATOV, M. R., Yad. Fiz. 35 (1982) 65, transl. Sov. J. Nucl. Phys. 35 (1) (1982) 39.
[12] VOVCHENKO, V. G., KAZARINOV, M. Yu., KAZARINOV, Yu. M., PROKOFEV, A. N. and
STRACHOTA, I., Yad. Fiz. 37 (1983) 158, trans. Sov. J. Nucl. Phys. 37 (1) (1983) 90.
[13] GREBENYUK, O. G., KOMAROV, E. N. and
SHKLYAREVSKII, G. M., Yad. Fiz. 39 (1983) 74, transl. Sov. J. Nucl. Phys. 39 (1984) 44.
[14] CZIFFRA, P., MACGREGOR, M. H., MORAVCSIK, M. J. and STAPP, H. P., Phys. Rev. 114 (1959) 880 and
MACGREGOR, M. H., MORAVCSIK, M. J. and STAPP, H. P., Ann. Rev. Nucl. Sci. 10 (1960) 291.
[15] BREIT, G. and RUPPEL, H., Phys. Rev. 127 (1962) 2123.
[16] GERSTEN, A., Nucl. Phys. A 290 (1977) 445, and BYSTRICKY, J., GERSTEN, A., JUNOD, A. and
LEHAR, F., Nucl. Phys. A 285 (1977) 469. [17] LECHANOINE, C., LEHAR, F., PERROT, F. and WIN-
TERNITZ, P., Nuovo Cimento 56A (1980) 201. [18] PLESSAS, W., STREIT, L. and ZINGL, H., Austr.
Phys. Acta 40 (1974) 272. [19] BYSTRICKY, J. and LEHAR, F., Physics Data Nr 11-1,
Part I and II (1978) Nr 11-2 and 11-3 (1981), edited by H. Behrens and G. Ebel (Fachinfor- mationszentrum Karlsruhe) and
BYSTRICKY, J., CARLSON, P., LECHANOINE, C., LEHAR, F., MONNIG, F., SCHUBERT, K. R., Landolt-Bornstein Tables, New Series Vol. 9a, edited by H. Schopper (Springer-Verlag) 1980.
[20] SHIMIZU, F., KUBOTA, Y., KOISO, H., SAI, K., SAKAMOTO, S. and YAMAMOTO, S. S., Nucl.
Phys. A 386 (1982) 571. [21] DITZLER, W. R., HILL, D., IMAI, K., SHIMIZU, H.,
SPINKA, H., STANEK, R., TOSHIOKA, K., UN- DERWOOD, D., WAGNER, R., YOKOSAWA, A., BURLESON, G. R., COTTINGAME, W. B., GREENE, S. J., JARMER, J. J., JEPPESEN, R. H., Phys. Rev. D 27 (1983) 680.
[22] MADIGAN, W. P., BELL, D. A., BUCHANAN, J. A., CALKIN, M. M., CLEMENT, J. M., COPEL, M., CORCORAN, M. D., JOHNS, K. A., LESIKAR, J. D., MIETTINEN, J. E., MUTCHLER, G. S., NAUDET, C. J., PEPIN, G. P., PHILLIPS, G. C., ROBERTS, J. B., TURPIN, S. E., HUNGERFORD, E. V., MAYES, B. W., HANCOCK, A. D., PINSKI, L. S., SEKHARAN, K. K., HOLLAS, C. L., RILEY, P. J., ALLRED, J. C., BONNER, B. E., CAMERON, P., LINN, T. S., VON WITSCH, W., FURIC, M., VALKOVIC, V., Phys. Rev. D 31 (1985) 966.
[23] PERROT, F., Thesis No 2912, Faculté des Sciences, Orsay, 1984 (preliminary results) and
PERROT, F., AZAIEZ, H., BALL, J., BYSTRICKY, J., CHAUMETTE, P., CHESNY, Ph., DEREGEL, J., FABRE, J., FONTAINE, J. M., GOSSET, J., LEHAR, F., LEO, W. R., DE LESQUEN, A.,
224
NEWSOM, C. R., ONEL, Y., PENZO, A., VAN ROSSUM, L., SIEMIARCZUK, T., VRZAL, J., WHITTEN, C. A. Jr. and YONNET, J., Nucl.
Phys. B 278 (1986) 881.
[24] STANLEY, J. P., STEWART, N. M., BUGG, D. V., EDGINGTON, J. A., STEVENSON, N. R., CLOUGH, A. S., AXEN, D. A., SHYPIT, R., COMYN, M., HEALEY, D. and LUDGATE, G. A., Nucl. Phys. A 403 (1983) 525.
[25] APRILE-GIBONI, E., BYSTRICKY, J., DEREGEL, J., DROMPT, Ph., EISENEGGER, C., FONTAINE, J. M., HEER, E., HESS, R., JACCARD, S., LEHAR, F., LEO, W. R., MANGO, S., MOREN- ZONI, S., ONEL, Y., PERROT, F., RAPIN, D., VRZAL, J. and YONNET, J., Nucl. Phys. A 431 (1984) 637.
[26] AUER, I. P., DITZLER, W. R., HILL, D., IMAI, K., SPINKA, H., STANEK, R., TOSHIOKA, K., UN- DERWOOD, D., WAGNER, R., YOKOSAWA, K., HOFFMAN, E. W., JARMER, J. J., BURLESON, G. R., COTTINGAME, W. B., GREENE, S. J. and STUART, S., Phys. Rev. D 29 (1984) 2435.
[27] BYSTRICKY, J., CHAUMETTE, P., DEREGEL, J., FABRE, J., LEHAR, F., DE LESQUEN, A., VAN ROSSUM, L., FONTAINE, J. M., GOSSET, J., PERROT, F., BALL, J., HASEGAWA, T., NEWS- OM, C. R., YONNET, J., LEO, W. R., ONEL, Y., PENZO, A., AZAIEZ, H., MICHALOWICZ, A., Phys. Lett. 142B (1984) 130.
[28] OTHEWELL, D., WALDEN, P., AULD, E. G., GILES, G., JONES, G., LOLOS, G. J., MCPARLAND, B. J., ZIEGLER, W. and FALK, W., Nucl. Phys. A 412 (1984) 189.
[29] CHATELAIN, P., FAVIER, B., FOROUGHI, F., HOF- TIEZER, J., JACCARD, S., PIFFARETTI, J., WAL- DEN, P. and WEDDIGEN, Ch., J. Phys. G. Nucl. Phys. 8 (1982) 643.
[30] BERDOZ, A., FAVIER, B., FOROUGHI, F., HOF-
TIEZER, J., MUTCHLER, G. S. and WEDDIGEN, Ch., J. Phys. G. Nucl. Phys. 8 (1982) 1363.
[31] PAULETTA, G., ADAMS, G., HAJI-SAIED, S. M., IGO, G. J., MCCLELLAND, J. B., WANG, A. T. M., WHITTEN, C. A. Jr., WRIEKAT, A., GAZZALY, M. M., TANAKA, N., Phys. Rev. C 27 (1983) 282.
[32] BARLETT, M. L., HOFFMANN, G. W., MCGILL, J. A., HOISTAD, B., RAY, L., FERGERSON, R. W., MILNER, E. C., MARSHALL, J. A., AMANN, J. F., BONNER, B. E., MCCLELLAND, J. B., BLANPIED, J. S., ARNDT, R. A., Phys. Rev. C 27 (1983) 682.
[33] GARÇON, M., LEGRAND, D., LOMBARD, R. M., MAYER, B., ROUGER, M., TERRIEN, Y. and NAKACH, A., Nucl. Phys. A 445 (1985) 669.
[34] VELICHKO, G. N., VOROBYOV, A. A., ZALITE, Yu. K., KOROLEV, G. A., MAEV, E. M., TERENTYEV, N. K., KHANZADEEV, A. V. and SHCHEGEL’SKII, A. V., Yad. Fiz. 35 (1982) 1457 transl. Sov. J. Nucl. Phys. 35 (6) (1982) 852.
[35] DOBROVOLSKY, A. V., KHANZADEEV, A. V., KOROLEV, G. A., MAEV, E. M., MEDVEDEV, V. I., SOKOLOV, G. L., TERENTYEV, N. K.,
TERRIEN, Y., VELICHKO, G. N., VOROBYOV, A. A. and ZALITE, Yu. K., Nucl. Phys. B 214 (1983) 1.
[36] SHIMIZU, F., KOISO, H., KUBOTA, I., SAI, F., SAKAMOTO, S. and YAMAMOTO, S. S., Nucl.
Phys. A 389 (1982) 445. [37] BARKER, N. D., COLBY, P. C., HAEBERLI, W. and
SIGNEL, P., Phys. Rev. Lett. 48 (1982) 918 and erratum.
[38] BERDOZ, A., FAVIER, B., FOROUGHI, F. and WED- DIGEN, C., J. Phys. G. Nucl. Phys. 9 (1983) L261.
[39] MCNAUGHTON, M. W. and CHAMBERLIN, E. P., Phys. Rev. C 24 (1981) 1778.
[40] MCNAUGHTON, M. W., CHAMBERLIN, E. P., JAR- MER, J. J., KING, N. S. P., WILLARD, H. B. and WINKELMAN, E., Phys. Rev. C 25 (1982) 2107.
[41] HOLLAS, C. L., CREMANS, D. J., MCNAUGHTON, K. H., RILEY, P. J., RODEBAUGH, R. F., SHEN- WU XU, BONNER, B. E., MCNAUGHTON, M. W., OHNUMA, H., VAN DYCK, O. B., SUN TSU-HSUN, TURPIN, S., AAS, B. and WESTON, G. S., Phys. Rev. C 30 (1984) 1251.
[42] BYSTRICKY, J., CHAUMETTE, P., DEREGEL, J., FABRE, J., LEHAR, F., DE LESQUEN, A., VAN ROSSUM, L., FONTAINE, J. M., GOSSET, J., PERROT, F., WHITTEN, C. A. Jr., BALL, J., CHESNY, P., NEWSOM, C. R., YONNET, J., NIINIKOSKI, T., RIEUBLAND, M., MICHALOWICZ, A., DALLA TORRE-COLAUTTI, S., Nucl. Phys. B 262 (1985) 727.
[43] GARÇON, M., Thesis, Faculté des Sciences, Orsay, 1985 and Private Communication of Garçon M., Legrand D. and Saudinos J. (to be published).
[44] BYSTRICKY, J., CHAUMETTE, P., DEREGEL, J., FABRE, J., LEHAR, F., DE LESQUEN, A., PETIT, F., VAN ROSSUM, L., FONTAINE, J. M., PERROT, F., BALL, J., MICHALOWICZ, A., ONEL, Y., PENZO, A., Nucl. Phys. B 262 (1985) 715.
[45] BHATIA, T. S., GLASS, G., HIEBERT, J. C., KENEFICK, R. A., NORTHCLIFFE, L. C., TIP-
PENS, W. B., BOISSEVAIN, J. G. J., JARMER, J. J., SIMMONS, J. E., TRIPARD, G. E., FITZ-
GERALD, D., HOLT, J. and MOKHTARI, A., Phys. Rev. Lett. 49 (1982) 1135.
[46] PAULETTA, G., Private Communication Feb. 1985 and GAZZALY, M. M., Private Communication, Nov. 1985.
[47] AUER, I. P., COLTON, E., DITZLER, W. R., HILL, D., MILLER, R., SPINKA, H., THEODOSIOU, G., TAVERNIER, J.-J., TAMURA, N., TOSHIOKA, K., UNDERWOOD, D., WAGNER, R., YOKOSAWA, A., KROLL, P. and JAVCH, W., Phys. Rev. Lett. 51 (1983) 1411.
[48] BYSTRICKY, J., CHAUMETTE, P., DEREGEL, J., FABRE, J., LEHAR, F., DE LESQUEN, A., VAN ROSSUM, L., FONTAINE, J. M., PERROT, F., BALL, J., HASEGAWA, T., NEWSOM, C. R., PENZO, A., AZAIEZ, H., MICHALOWICZ, A., Nucl. Phys. B 258 (1985) 483.
[49] APRILE, E., HAUSAMMANN, R., HEER, E., HESS, R., LECHANOINE-LELUC, C., LEO, W. R., MORENZONI, S., ONEL, Y., RAPIN, D., MANGO, S., Phys. Rev. D 27 (1983) 21 and ;
225
APRILE, E., HAUSAMMANN, R., HEER, E., HESS, R., LECHANOINE-LELUC, C., LEO, W. R., MOREN- ZONI, S., ONEL, Y., RAPIN, D., MANGO, S., Phys. Rev. D 27 (1983) 2600.
[50] LECHANOINE-LELUC, C., Invited talk to the 5th Int. Symposium on High Energy Spin Physics, Brookhaven 1982, AIP Conf. Proc. 95, Particles Fields Subseries 28 (1982) 193, Editor G. Bunce.
[51] HOLLAS, C. L., CREMANS, D. J., RANSOME, R. D., RILEY, P. J., BONNER, B. E., MCNAUGHTON, M. W. and WOOD, S., Phys. Lett. 143B (1984) 343.
[52] BARLETT, M. L., HOFFMANN, G. W., MCGILL, J. A., FERGERSON, R. W., MILNER, E. C., MARSHALL, J. A., AMANN, J. P., BONNER, B. E. and MCCLELLAND, J. B., Phys. Rev. C 30 (1984) 279.
[53] MCNAUGHTON, M. W., BONNER, B. E., HOFFMAN, E. W., VAN DYCK, O. B., HOLLAS, C. L., RILEY, P. J., MCNAUGHTON, K. H., IMAI, K., TOSHIOKA, K., ROBERTS, J., TURPIN, S. E., AAS, B. and RAHBAR, A., Phys. Rev. C 26
(1982) 249. [54] MCNAUGHTON, M. W., BONNER, B. E., CORNELIUS,
W. D., HOFFMAN, E. W., VAN DYCK, O. B., YORK, R. L., RANSOME, R. D., HOLLAS, C. L., RILEY, P. J., TOSHIOKA, K., Phys. Rev. C 25 (1982) 1967.
[55] LISOWSKI, P. W., SHAMU, R. E., AUCHAMPAUGH, G. F., KING, N. S. P., MOORE, M. S., MORGAN, G. L. and SINGLETON, T. S., Phys. Rev. Lett. 49 (1982) 225, numerical values taken from Private Communication of P. W. Lisowski, Oct. 24, 1984.
[56] GRUNDIES, V., FRANZ, J., ROSSLE, E., SCHMITT, H., Phys. Lett. 158B (1985) 15.
[57] KEELER, R. K., DUBOIS, R., AULS, E. G., AXEN, D. A., COMYN, M., LUDGATE, G., ROBERTSON, L. P., RICHARDSON, J. R., BUGG, D. V., ED- GINGTON, J. A., GIBSON, W. R., CLOUGH, A. S., STEWART, N. M. and DIETERLE, B., Nucl. Phys. A 377 (1982) 529.
[58] ABDIVALIEV, A., BESLIU, C., COTOROBAI, F., GRUIA, S., IERUSALIMOV, A. P., MOROZ, V. I., NIKITIN, A. V., TROYAN, Yu. A., Preprint JINR D1-81-756, Dubna 1981.
[59] AUER, I. P., DITZLER, W. R., HILL, D., SPINKA, H., TAMURA, N., THEODOSIOU, G., TOSHIOKA, K., UNDERWOOD, D., WAGNER, R. and
YOKOSAWA, A., Phys. Rev. Lett. 46 (1981) 1177 and Kroll P. , Private Communication of
Sept.1982. [60] BOL, A., DUPONT, C., LELEUX, P., LIPNIK, P.,
MACQ, P., NINANE, A., Preprint Dep. of Phys. Université Catholique de Louvain, 1983.
[61] LELEUX, P., Private Communication of Dec. 1983. [62] NORTHCLIFFE, C. L., Private Communication 1985. [63] WELLERS, F., Thesis, Orsay, January 15, 1986. [64] TORNOW, W., LISOWSKI, P. W., BYRD, R. C. and
WALTER, R. L., Nucl. Phys. A 340 (1980) 34. [65] BROCK, J. E., CHRISHOLM, A., DUDER, J. C., GAR-
RETT and POLETTI, J. L., Nucl. Phys. A 361 (1981) 368.
[66] WILCZYNSKI, J., HANSMEYER, J., BRADY, F. P., DOLL, P., HEERINGA, W., HIEBERT, J. C., KLAGES, H. O. and PLISCHKE, P., Nucl. Phys. A 425 (1984) 458.
[67] SROMICKI, J., Private Communication, August 1, 1985.
[68] KOROLEV, G. A., KHANZADEEV, A. V., PETROV, G. E., SPIRIDENKOV, E. M., VOROBYOV, A. A., TERRIEN, Y., LUGOL, J. C., SAUDINOS, J., SILVERMAN, B. H., WELLERS, F., Phys. Lett. 165B (1985) 262.
[69] SAKUDA, M., ISAGAWA, S., ISHIMOTO, S., KABE, S., MASAIKE, A., MORIMOTO, K., OGAWA, K., SUETAKE, M., TAKASAKI, F., WATASE, Y., KIM, N., KOBAYASHI, S., MURAKAMI, A., DE LESQUEN, A., NAKAJIMA, K., NAKADA, S., WADA, T., YAMAUCHI, I., Phys. Rev. D 25
(1982) 227. [70] BYSTRICKY, J., DEREGEL, J., LEHAR, F., DE LES-
QUEN, A., VAN ROSSUM, L., FONTAINE, J. M., PERROT, F., ARVIEUX, J., HASEGAWA, T., NEWSOM, C. R., ONEL, Y., PENZO, A., AZAIEZ, H., MICHALOWICZ, A., RAYMOND, C., Nucl. Phys. A 444 (1985) 597.
[71] BHATIA, T. S., GLASS, G., HIEBERT, J. C., NORTH- CLIFFE, L. C., TIPPENS, W. B., BONNER, B. E., SIMMONS, J. E., HOLLAS, C. L., NEWSOM, C. R., RILEY, P. J. and RANSOME, R. D., Phys. Rev. Lett. 48 (1982) 227.
[72] SCHROEBERL, M., KUIPER, H., SCHMELZER, R., MERTENS, G. and TORNOW, W., J. Phys. G. Nucl. Phys. 10 (1984) 247.
[73] BAGATURIYA, Yu. Sh., KAZARINOV, Yu. M., KAZARINOV, M. Yu., LIBURG, M. Yu., MATAFONOV, V. N., MACHARASHVILLI, G. G., POTASHNIKOVA, I. K., STRACHOTA, J., STRACHOTOVA, M., USOV, Yu. A., KHACHATUROV, B. A. and KHAYATOV, M. R., Yad. Fiz. 33 (1981) 1237, transl. Sov. J. Nucl.
Phys. 33 (1981) 659. [74] RANSOME, R. D., HOLLAS, C. L., RILEY, P. J.,
BONNER, B. E., GIBBS, W. R., MCNAUGHTON, M. W., SIMMONS, J. E., BHATIA, T. S., GLASS, G., HIEBERT, J. C., NORTHCLIFFE, L. C. and TIPPENS, W. B., Phys. Rev. Lett. 48 (1982) 781.
[75] LEHAR, F., SIEMIARCZUK, T., WINTERNITZ, P. and BYSTRICKY, J., Preprint DPhPE 79-28, Sac-
lay 1979. [76] VERWEST, B. J. and ARNDT, R. A., Phys. Rev. C 25
(1982) 1979. [77] ARNDT, R. A. and MACGREGOR, M. H., Methods
Comp. Phys. 6 (1966) 263. [78] APRILE, E., HAUSAMMANN, R., HEER, E., HESS,
R., LECHANOINE-LELUC, C., LEO, W. R., MORENZONI, S., ONEL, Y., RAPIN, D., EATON, G. H., JACCARD, S. and MANGO, S., Nucl.
Phys. A 415 (1984) 365. [79] BELL, D. A., BUCHANAN, J. A., CALKIN, M. M.,
CLEMENT, J. M., DRAGOSET, W. H., FURIC, M., JOHNS, K. A., LESIKAR, J. D., MIETTINEN, H. E., MULERA, T. A., MUTCHLER, G. S., PHIL- LIPS, G. C., ROBERTS, J. B. and TURPIN, S. E., Phys. Lett. 94B (1980) 310.
226
[80] CARLINI, R., DIETERLE, B., DONAHUE, J., LEAVITT, C., RUPP, T., THOMAS, W., WOLFE, D., AUERBACH, L. B., HIGHLAND, V. L., JOHN- SON, K. F., MCFARLANE, W. K., PRATT, J. and BENTLEY, R., Phys. Rev. Lett. 41 (1978) 1341.
[81] BONNER, B. E., SIMMONS, J. E., GRASS, G., JAIN, M., HOLLAS, C. L., NEWSOM, C. R. and RILEY, P. J., Phys. Rev. Lett. 41 (1978) 1200.
[82] CLOUGH, A. S., GIBSON, W. R., AXEN, D., DUBOIS, R., FELAWKA, L., KEELER, R., LUDGATE, G. A., ORAM, C. J., AMSLER, C., BUGG, D. V., EDGINGTON, J. A., ROBERTSON, L. P., STEWART, N. M., BEVERIDGE, J. and RICHARD- SON, J. R., Phys. Rev. C 21 (1980) 988.
[83] NEWSOM, C. R. : Free neutron-proton analysing power at medium energies, PhD thesis, Univer- sity of Texas at Austin, 1980, and Private communications of Newson C. R. and Simmons, J. E.
[84] BHATIA, T. S., GLASS, G., HIEBERT, J. C., NORTH- CLIFFE, L. C., TIPPENS, W. B., BONNER, B. E., SIMMONS, J. E., HOLLAS, C. L., NEWSOM,
C. R., RANSOME, R. D., RILEY, P. J. 5th Int. Symposium on Polarization Phenomena in Nuc- lear Physics, AIP Proceedings (1981) 123 and
BHATIA, T. S., Private Communication (see /B-293/ in ref. [19]).
[85] CÔTÉ, J., LACOMBE, M., LOISEAU, B. and COTTING- HAM, W. N., Nucl. Phys. A 379 (1982) 349, and
LACOMBE, M., LOISEAU, B., RICHARD, J. M., VINH MAU, R., CÔTÉ, J., PIRÈS, P. and DE TOURREIL, R., Phys. Rev. C 21 (1980) 861 and
LOISEAU, B., Private Communication.
[86] LOMON, E. L., Aussois Meeting, March 1980 and Private Communication (1984).
[87] BYSTRICKY, J., DEREGEL, J., LEHAR, F., FONTAINE, J. M., PERROT, F., LECHANOINE-LELUC, C., LEO, W. R., ONEL, Y., NEWSOM, C. R., YON- NET, J., PENZO, A., WINTERNITZ, P., Nuovo Cimento 82A (1984) 385.