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Phase-shift analysis of elastic proton-deuteron scattering cross sections and 3He excited states
l-=-l
Nuclear Physics A221 (1974) 253 -268; @ North-Holland Publishing Co., Amsterdam Not to be
reproduced by photoprint
or microfilm without written permission from the publisher
PHASESHIFT ANALYSIS OF ELASTIC PROTON-DEUTERON SCATTERING CROSS SECTIONS AND 3He EXCITED STATES 5. ARVIEUX + Lawrence Berkeley Laboratory, Berkeley, Calif. 94720 +f Received 27 November
1973
Abstract: A phase-shift analysis of recent experimental data on elastic proton-deuteron scattering and reaction cross sections has been performed for laboratory energies from 1 to 46.3 MeV. Phase shifts up to L = 8 calculated by Faddeev equation techniques have been included in the analysis. A new set of p-d scattering lengths is given which rules out the possibility of a pole term in the effective range expansion of the doublet *S; phase. The Argand plot of the 4P wave is shown to be consistent with a broad resonance in 3He centered around an excitation energy of 14 MeV. This resonance is discussed in the light of preceding experimental and phase-shift analysis results.
1. Introduction
The theoretical treatment of the three-nucleon interaction has recently undergone considerable progress. In the case of elastic nucleon-deuteron scattering, the Faddeev equations have been solved exactly ‘) with a S-wave nucleon-nucleon non-local separable interaction. A treatment using the Pad6 approximants approach has been also used with a central spin-dependent Yukawa interaction “). Both calculations give very similar results and they reproduce fairly well the gross structures of the experimental cross-section data. Recent calculations have been done with separable spin-dependent potentials either in a perturbative treatment including the tensor and the P- and D-nucleon-nucleon interaction 3), or in a more exact treatment including the tensor and the P-force only “). These latter calculations reproduce surprisingly well and without any free parameters, most of the qualitative features of the polarization data including the nucleon polarization and the deuteron vector and tensor analyzing powers “). These calculations indicate how important the effects of some spin-couplings are and the need to take them into account in a serious phase-shift analysis. Such a semi-phenomenological analysis based partly on theoretical predictions is valuable in giving accurate phase shifts which can be then compared to refined theoretical calculations using more realistic N-N potentials. This stochastic process has been the line along which nucleon-nucleon experiments have been analyzed and has proved to be very valuable. There have been many phase-shift analyses of nucleon-deuteron scattering, some of them extending over a large energy region 6-8), some others restricted to a single + Permanent address: Institut des Sciences NuclCaires, BP 257, 38044 Grenoble, France. ++ Work performed
under the auspices of the US Atomic Energy Commission. 253
J. ARVIEUX
254
TABLE 1 Results of the
E
?3
‘93
2P
2.503
-0.044
4l?
2D
4D
“F
QF
-0.0004
0.0022
-0.0036
0.0086
-0.0080
0.0160
2G
Q
1.993
2.837
2.209
-0.112
0.334
2.995
2,689
2.020
-0.117
0.411
0.0116 -0.0176 -___e 0.0231 -0.0387 ~__I~ 0.029 -0.0670
3.998
2.542
1.856
-0.081
0.456
0.025
-0.098
-0.0114
0.0232
-0.116
-0.022
0.0292
0.0042 -0.0082 -___..
0.0324
o.00.51 -0.0102 II___I_
0.029
0.0059 I_____
-0.0115
0.0069 .--
-0.0135
1.0
2.922
0,967 5.002
2.462
1.755
0.893
0.176
0.987
0.999
- 0.040
0.500
0.032
0.955
0.996
0.998
6.007
2.387
1.649
0.523
0.040
0.999
0.912
0.9997
0.995
6.78
0.81 -___ 2.328
1.581
0.001
0.536
0.023
0.76
0.998
0.872
0.985
0.99
ii%+6
1.521
0.066
z-i
0.055
0.997
0.822
0.967
0.983
10.04
0.682 ____2.098
1.409
0.118
0.574
0.090
0.989 1.35
0.749
0.946
0.971
12.18
0.610 ._I_ 1.980
0.227
0.576
0.103
0.55 ____ 1.910
0.985
0.715
0.909
0.959
13.93
1.296
0.273
0.585
0.155
0.973
0.687
0.866
0.95
16.24
0.495 ___1.737
1.247
0.3133
0.574
0.141
0.963
0.682
0.856
0.939
17.12
0.456 -1.686
1.221
0.316
0.577
iGG
0.447
0.971
0.681
0.830
0.936
8.025
-iGG
0.0008 -0.0016 -____ 0.0017 -0.0036 -_____ 0.0030 -0.0060 -____
0.9999 -0.129 0.999 -0.152 0.998 -0.149 0.996 -0.160 0.991 -0.188 0.988 -0.180
-0.041 0.9995 -0.068 0.9989 -0.051 0.997s -0.049 0.9952 -0.046 0.9925 -0.038
0.9998 0.0412 0.9996 0.042
0.0082 -0.0160 ___-
0.9989 0.043
0.0093 --
-0.0177
0.9979 0.040
0.0101 -0.0190 I_______
18.52
19.92
22.0
1.630
1.179
0.340
0.540
0.127
0.428
0.954
0.682
0.842
0.931
1.546
1.155
0,373
0.534
0.419
0.970
0.684
0.836
0.931
1.449
1.133
0.424
0.556
0.178
0.166
0.402
0.959
0.695
0.837
0.92
25.7
1.359
1.109
0.514
0.540
0.223
0.405
0.951
0.710
0.823
0.915
31.0
1.215
1.069
0.537
0.493
OS97
0.415
0.920
0.738
0.849
0.910
1.108
1.017
0.530
0.435
0.220
35.0
0.956 -0.183 0.939 -0.189 0.933 -0.173 0.913 -0.177 0.906 -0.169 0.931 -0.162 0.918 -0.163 0.926 -0.138
0.9901 -0.025 0.9870 -0.043 0.9860 -0.008 0.9840 -0.01’8 0.9823
0.9968 0.039
0.0108 -0.0199 --
0.9954 0.040
0.0111
-0.0202
0.0114 --
-0.0210
0.9948 0.038 0.998 0.040
0.0117 -0.0212 --
0.9928
-0.011
0.034
0.9756 _..~ -0.030
0.9890
0.0129 -0.0226 ~__I 0.9981 0.9961 -0.0136 -0.0230 -0.9973 0.9951
0.054
0.0144 -0.0227
0.971
0.985
0.013
0.040
0.9937 0.9962 -0.041 -0.027
-0.031 0.9788
0.039
0.9910
PHASE-SHIFT
ANALYSIS
25.5
phase-shift analysis
4H
ZH
21
41
2J
4J
2K
4K
x2/p.p.
xz/p.d.f.
Ref.
0.23
0.31
18
1.76
2.09
18
0.96
1.11
18)
>
-0.0002 ___ -0.0004
0.0004 0.0008
-0.0008 ____
0.0016 ___
0.96
1.22
IS
1
-0.0012 ___
0.0023 -
0.65
0.88
18
1
-0.0015 -
0.0030 -
0.62
0.87
18
1
-0.0018 ____
0.0037 ___
-0.0022 -
0.0045 __
1.23
1.73
18
1
-0.0027 _I_
0.0056 -
0.0009 __-
-0.0020 -
0.82
1.30
18
1
-0.0032 ___
0.0064 __
0.0012 -0.0023 ~___
1.54
1.73
19
-0.0035 -
0.0070 __
0.0014 -0.0026 -___
0.73
0.86
-0.0037 ___
0.0075 -
0.0015 -0.0029 -____
1.28
1.62
21
-0.0038 ___
0.0077 __
0.0015 -0.0030 -___
2.45
3.21
21
)
0.0039 ___
0.0079 -
0.0016 -___
-0.0032
1.15
1.49
21
1
-0.0039 -
-
0.0082
0.0017 -0.0033 -___
1.06
1.46
21
1
0.0089 0.9996 ~ 0.0092 __ 0.9994 __ 0.0095 __ 0.9990
0.0018 -___0.9998 0.0020 -p0.9997 ___ 0.0021 ~~ 0.9996 ___ 0.0022 y-___
-0.0042 0.9991 -0.0041 ___ 0.9988 __ -0.0041 ___ 0.9983 --0.0039 ___
0.0096 ~
1
>
20
1
>
-0.0007
0.0015
1.79
2.38
22
0.9999 _I -0.0037 -0.0008
0.0016
3.83
5.94
23
) )
-0.0035
0.9998 -0.0039 -0.0008 0.9997 0.9998 -0.0040 -0.0008
>
0.0017 -
0.0004 _____
-0.0007
0.8
1.12
24
0.0017 __
0.0004 __~
-0.0007
0.78
1.01
22 >
256
J. ARVIEUX TABLE 1
E
40.0
46.3
%
?3
2P
4P
2D
0.437
0.851
0.755
0.816
0.905
1.026
1.092
0.557
0.375
0.215
0.475
0.903
0.773
0.874
0.905
0.888
1.041
0.469
0.360
0.238
0.540
0.757
0.805
0.832
0.905
Column 1: Colums 2-19: Column 20: Column 21: Column 22:
4D
2F
4F
ZG
0.870
0.9688 __ 0.004
0.9830 0.045
0.03 1
-0.030
0.9665
0.971
0.042
0.043
0.9920 -~ 0.027
-0.002
0.9638
0.966
0.9908
-0.096 0.896 -0.090 0.865
0.9929 __-
4G 0.9955
0.9946
0.9934
Energy E of the incident proton in p+d scattering. Phase shifts for I = 0 to I = 8; the first line printed for each energy is the real part 6 in corresponding value is 1. Underlined numbers have been kept fixed during the search. Value of the x2 per point. Value of the x2 per degree of freedom. Reference for the experimental data.
energy “) or a limited domain of energy 1o-12). In a few of them polarization data have been included ‘-11) an d m ’ t wo cases all the couplings have been allowed either below the deuteron break-up energy lo) or at relatively low energy 12). In this last analysis up to 41 parameters have been allowed to vary to analyze data taken from Ed = 6 to 11.5 MeV. When one considers that when going up in energy (and the most interesting region to study 3He excited states is between Ed = 20 and 30 MeV) the number of parameters is even larger, one sees that a good analysis of the cross sections, although it is simplified, is very valuable to give starting values for a more complete analysis including all spin effects. Such a simplified analysis would almost fix the 2S, and 4S, phases and the average of all phases from L = 1 to L,,,, SO reducing the number of free parameters. Moreover, such an analysis permits one to solve problems concerning the effective range expansion of L = 0 phases for which polarizations play an insignificant role. The most extensive analysis of cross sections has been done by Van Oers and Brockmann ‘). It was already semi-phenomenological since partial waves 1 2 2 were calculated in the Born approximation as in the preceding work of Christian and Gammel 6), but it also included inelastic effects. This analysis should be re-done for twomainreasons: (i)thenon-unitaryBornapproximationisnotverygoodforcalculation of low-partial waves and (ii) the inelastic phases in the 2S and 4S states have been assumed to be equal, which is strongly contradicted by results of exact calculations, the absorption being much more important in the doublet than in the quartet state. Finally, new experimental data are now available. We present here a phase-shift analysis of elastic and reaction cross sections between 1 and 46 MeV. The validity of the main predictions of theoretical calculations has been tested and then used whenever possible in order to reduce the number of free parameters. The present analysis will be compared with two recent analyses published at the Los Angeles conference [refs. 13,,,)I.
PHASE-SHIFT
ANALYSIS
257
(continued)
2H 0.9989 ------0.0037 ___ 0.9976 ---__--0.0033 II_ 0.9971 --_---__
4H
*J
41 0.9996
4J
0.9998
2K
4K
x2/p.p.
X2/p.d.f.
Ref.
0.9988
0.9994
0.0097 0.9984
0.0023 -0.0040 -_____0.9993 0.9995
0.0018 0.9999
0.0004 -0.0008 --
1.83
2.75
=)
0.9998
0.0098 0.9980
0.0025 -0.0040 -0.0007 -___0.9991 0.9994 0.9997
0.0018 0.9998
0.0004 -0.0008 __~ 0.9999
1.4
1.95
22)
-0.0008
radians, the second line is the absorption
0.9999
parameter q_ When no absorption
parameter is printed the
2. Formalism
The formula describing the elastic scattering cross section o(6) and the reaction cross section 0, have been given in ref. ‘). In the following the orbital angular momentum is denoted by I; the parameters *S, and ‘6, are respectively the real quartet (S = $) and doublet (S = 4) phases where S is the channel spin; 4qr and ‘qr are the corresponding absorption parameters. The reaction cross section depends only on the inelasticity parameters while the elastic scattering cross section depends on both the real and imaginary phases. The present calculations have been made at the Lawrence Berkeley Laboratory on a CDC 7600 computer. The search routine is VARMIT, a variable metric minimization method based on the routine VARMIN by Davidon 15). The function to be minimized is the x2 given by
where the sum extends over all the elastic 0 ,,,(t?) data at a given energy, do(B) being the experimental uncertainty; Go exp and do, are the experimental reaction cross section and the corresponding uncertainty. The gradient of the x2 with respect to each parameter is calculated analytically which ensures a much faster and safer convergence. The uncertainties in the results of the analysis as given by the covariant matrix, are printed out. 3. The data analyzed No attempt has been made to correct the normalization of the experimental data. ‘When experimental points gave unreasonable x2 values they have been suppressed in the analysis. This occurred for a few energies, especially at very forward angles.
J. ARVIEUX
25s
The data ‘se2 “) are listed in table 1. The listed energy E is the laboratory proton energy in a p + d scattering. The corresponding deuteron lab energy in a d + p scattering is 2E and the c.m. energy is $ E in the non-relativistic approximation. Below E = 1 MeV the old results of Brown 26) are not accurate enough and very different sets of phases give nearly the same x2 value so they have been skipped out in the analysis. Above 1 MeV two sets of recent experimental data are available. One is from Wisconsin 18) and it corresponds to integral energy values from 1 to 10 MeV. The other data set is from Rice 27) from 4.5 to 11.5 MeV for half-integer energy values. Both set have been used in a preliminary analysis 14) and the phases obtained were almost compatible except for the ‘S phase where we have observed a strong oscillation when going from one set of data to the other, the Rice data giving usually higher values for ‘S than the Wisconsin data. Since these latter results have a smaller uncertainty they have been selected for this analysis.
4. Validity of the theoretical calculations There are two calculations based on the solution of the Faddeev equations ‘, “). In both cases the theoretical phase shifts have been made available to us. Two main features predicted by these calculations have been tested extensively. They are the effect of high partial waves and the comportment of absorption parameters. 4.1. EFFECT
OF HIGH PARTIAL
WAVES
In a first step searches have been made up to 15 MeV without including high-order partial waves. For energies as low as 1 MeV it has been found that 1 = 2 phases are “necessary”. We mean by necessary that if the phase of the highest L-value is included in the analysis, the best 11’ value shows a significant improvement over the value TABLE2 Scattering lengths a, effective ranges r and shape parameters P in p-d elastic scattering Quartet
Doublet %-d (fm) Van Oers-Brock-
zPp--d (fm?
+&+a (fm)
1.3 f0.2
%I-+ (fm)
%-ci (fm)
4P*-d (fm3)
0
11.4+r*a -1.2
2.05 10.25
0
0
11.9+o.3 -0.9
1.s9+o.2 -0.05
O
O.OS’+$$
11.88:;:‘:
2.63:z.g;
0
11.00
1.96
mann “) Van Oers-Seagrave b, Present work
2.7310.10
Avishai-Reiner
3.4
(theory) 33) “) Ref. +‘).
“) Ref. 31).
2.2710.12
-0.54&0.1 0
PHASE-SHIFT
ANALYSIS
259
obtained when this phase is not taken into account. F-phases (I = 3) are necessary at 3 MeV to fit the backward peak; G-phases (I = 4) at 8 MeV; H-phases (I = 5) at 12 MeV and I-phases (I = 6) at 15 MeV. This reflects the high importance of exchange effects. In a second step we have used as fixed parameters the phases for high-l values (up to Z = 8 at 46.3 MeV). The two sets of phases as given by Sloan ‘) and Kloet and Tjon (KT) “) d o not differ by more than 10 o/0for Z 2 3, so only Sloan’s values have been used for these phases. The high partial waves used are given in table 2 which shows also what was the maximum Z-value included at each energy. The F-phases have been varied above 5 MeV since this has been shown to give a strong improvement in the x2 value. For phases up to Z = 2 both the Sloan and KT values have been tried. Quartet phases are very similar but the doublet 2S phases can be different by as much as 20”. Our analysis shows that below 5 MeV the search converges towards one unique solution which is somewhat intermediate between the Sloan and KT predictions but at 5 MeV the results of the analysis are much closer to the KT solution. Above 5 MeV the two different sets of starting values lead to two different final solutions with usually the same x2 value, except at 12.18 MeV where KT starting values are again favored. So we think that the KT calculations seem a better approximation to the experimental data and only the corresponding solution will be given. 4.2. ABSORPTION
PARAMETERS
All the calculations based on Faddeev equations show that the absorption in the N-d scattering is much stronger in the doublet than in the quartet state. This assumption has been systematically tested at a few selected energies namely 6,8 and 10 MeV. To increase the sensivity of the search to the absorption parameters, the experimental errors on the reaction cross section have been divided by YEsuch that n2 equals the
I
0
I
I
I
IO
20
30
I
60
50
E (Me’/1
Fig. 1. Experimental reaction cross sections. Black dots are from ref. 16) and open ones from ref. 17). The dotted curve is the interpolated value used when elastic and total reaction cross-section data were taken at different energies. The full curve is the value obtained in our analysis with the absorption parameters of table 1.
J.ARVIEUX
260
number of elastic scattering data points analyzed. At the energies investigated four inelastic parameters are important (I = 0,l). At these energies the assumption that doublet absorption is much more important than the quartet one is strongly supported by the experimental data. It is also found that doublet absorption is much less sensitive to the combined elastic and inelastic data than quartet absorption, so that when both are allowed to vary the doublet absorption parameters remain close to the theoretical values. So they have been kept almost fixed throughout this analysis. The most sensitive absorption parameter is ‘qt which change si~ifi~n~y from the theoretical values between 10 and 20 MeV. The reaction cross sections used in this analysis are from n-d data 16) below 14 MeV and p-d data 17) above 23 MeV. Between these two energies the experimental data have been smoothly interpolated as shown by the dotted line in fig. 1 which shows also the result of our analysis (solid line) and the experimental data. 5. Results The results are summarized in table 1. The real part 2s’161 is given in radians on the same line as the energy E. The second line gives the absorption parameters ” ” ql. When no absorption parameter is printed out 2s “ql = 1. Underlined quantities have been found to vary insignificantly when allowed to vary and have been f. 2. 3.4.5.
01
0.3
IO. 15. 20.
0.5
07
E (MeV) 30.
40.
0.9 k CFm-‘)
Fig. 2. The %Zreal phase shifts in radians as a function of the wave number k in fin-’ the upper scale indicates the eaergy E of the incident proton in the lab system in pi-d
(HOW@Scald;
scattering.The solid curve is a theoretical calculation by Kloet and Tjon =), the dashed line is a calculation by Sloan 1).
1. 2. 3 4.5 I I III
IO. 15. I ,
E (MeV) 30. 40. I I
20. I
E (MeV) 1. 2. 3.4.5
10
15. 20.
30.
40.
.
I
7 ‘11
l
* a
I.1.5 0.1
I
,_:_:: 0.3
0.5
0.7
/
01
0.9
0.3
/
I
I
0.5
0.7
k(fm-1)
Fig. 3. The ?S real phase shifts in radians; the solid line is the theoretical result obtained in an exact three-body calculation by Kloet and Tjon ‘) and Sloan I).
IO. 15. 20 1 1 1
30.
I
0.9
Fig. 4. Same as fig. 2 for the *P (upper graph) and 9 (lower graph) phase shifts.
E (Me!!) 1. 2.3.45 rs1111
,
k (fin-‘)
40. I
E (Me’/! 1.2345
IO. 15 20
30
40
J
-0.d
0.5
0.7
k (fm-‘!
Fig. 5. The 2D, 4D, 2F and 4F phase shifts in radians; the dotted curve is an exact threebody calculation by Sloan ‘>. The predictions by Kloet and Tjon 2, do not differ by more than 10% for these phases and are not shown. The dash-dotted curve is a Born approximation calculation.
0.1
0.3
0.5
0.7
0.9 k (fm-1)
Fig. 6. Absorption parameters zoo. 4~0, 2qr and 4qL. The solid curve is by Kloet and Tjon 2), the dash-dotted curve is the result of a previous phase-shift analysis by Van Oers and Brockmann7).
262
3. ARWEUX
kept fixed to theoretical values during the analysis. According to the modified Levinson theorem 28) the ‘6, and 46, phases start at rc at E = 0 MeV. The results are also displayed in figs. 2 to 6. One can notice in fig. 5 the large difference between the values of 26, obtained in the present analysis when compared with non-unitary Born approximation calculations. Large differences are also observed in fig. 6 for inelastic parameters. In a first analysis all the real phases up to 1 = 3 and imaginary phases up to I = 2 have been allowed to vary. It has been found that some parameters do not change significantly from the theoretical values, especially ‘y10 and *yl. These parameters have been subsequently fixed at their theoretical value, thus reducing the number of free parameters and improving the x2 per degree of freedom. The uncertainties, as calculated by the searching routine VARMIT, are usually very small, of the order of 0.001 radian or less. These are calculated by fixing all the parameters but one and by seeing how the x2 value changes when the free parameter varies. In fact, correlation unoertainties are much more important. Some of these correlations can be an effect of a real physical phenomenon; for example one can compensate for a decrease in 46, by increasing ‘6,) which is an indication of the strong correlation between the ‘P and 4P phases. The indicated errors take into account these different solutions around the selected one given in table 1 and having x2 values differing by usually not more than 20 %. When a solution seems unique a systematic uncertainty of 0.01 radian is applied to all phases. Completely different solutions from the ones listed but with the same 31’value can sometimes also be found. This is the case, for example, with Sloan’s starting values above 10 MeV. These solutions are not discussed on the basis that phase shifts must vary smoothly with energy. 6. Effective range expansions The low energy scattering of two particles may be described by the effective range expansion K = - 1 +&k2+Pk4+
. . .,
a
where a is the scattering length, r the effective range, P the shape parameter and k the relative momentum. For neutral particles K is just K = k ctg 6(k), where 6 is the phase shift, When both particles are charged the usual effective range expansion becomes 2“) K = C; k ctg 6(k)+ ~(~)~~. This expression can be easily reduced to the following directly computable expression for K: jy
zz
C;kctg6(k)+
~[-0.57722+~2~~l~-1(~2-l-r12)-1-lnill. *
PHASE-SHIFT
ANALYSIS
263
where q = e2/nv = 0.15806 E-* and Co2= 2ny/(exp(2ny) - 1). Using this expression, and the values of 6 given in table 1 we have computed the quartet and doublet effective expansion of the ‘S and 4S waves. It is fairly well established that the doublet n-d scattering length shows a remarkable anomaly which would lead to an effective range expansion of the type 34, 35) 2K =
- l/a+$2r,
k2+2Pk4
1 +(k/k,)2
’
Such an expansion has a pole for ki = -k2 in the unphysical region close to the zero energy. Evidence on this phenomenon is based on the triton photo-desintegration [ref. 3“)I, a phase-shift analysis of n-d scattering 31) and a recent zero energy measurement of the n-d scattering lengths 32) giving a very small doublet scattering length 2a,,_d and so a large negative value of 2K at the intercept with the zero energy axis. In the p-d scattering case the situation is not so clear. Existing phase-shift analyses were not accurate enough to yield a conclusion on the existence of such a pole. On a theoretical side, perturbation techniques applied to a dispersion relation calculation of N-d scattering predict a recession of the pole in the ‘K expansion deeper into the unphysical region leading to a near complete extinction of it 33). We have analyzed our results with the expansion (1) using the four following combinations of free parameters: ‘a and 2r; ‘a, ‘r and 2P; 2a, 2r and 2ko ; 2a 2r, 2P and 2ko. The best fit to the overall 20 data points between 1 and 46.3 MeV is ‘obtained with the following parameters, 2a = 2.73*0.1 fm,
‘r. = 2.27f0.12
2P = O.OS;$~ fm3,
fin,
with a x2 of 0.87 per degree of freedom. Different pole terms have been tried with starting values at -0.1 MeV and - 1.0 MeV; in all cases the pole is rejected deeper in the negative energy region (E 2 -22 MeV). The best fit (full line) is shown in fig. 7 together with the best straight fit
0
IO
20
30
40
E (Me’/)
Fig. 7. Effective range expansion of 260 and 460.
264
J. ARVIEUX
(dotted line) for E = 1 to 18.52 MeV with the parameters ‘u = 2.73 fm, ‘r = 2.29 fm and ‘P = 0. Our results are compared in table 2 with the results of a preceding phase-shift analysis ‘) and a p-d elastic scattering calculation using dispersion relations 33). It can be noted that our value of Zap-d is much closer to the theoretical one than the old one. Our result for 4ap_ d, given in table 2 agrees with preceding results but the effective range parameter is significantly larger than preceding estimate. Moreover we have been able to get a fairly accurate value of the shape parameter 4Pp_d which is of a negative sign as is clear in fig. 7. 7. Excited states of the 3He system The excited states of the three-nucleon system are like the Loch Ness monsters. Many people claim to have seen some but when a careful scientific expedition is designed to track them, then they vanish out. Kim et al. 36) pretended to have seen many of them but they have been later infirmed. There has been a subsequent great number of experiments investigating two-body, three-body, and even four-body reactions [see ref. 3‘) for a summary of the experimental situation in 19711. Since 1971 the attention has been mainly put on photo-disintegration of 3He and p + d capture reactions [see ref. 38) for a summary of the situation in 19731 again with conflicting results. On the experimental point of view the sole indications of possible experimental 3He* broad resonances which have not been subsequently confirmed are the 3H(p, n)3He* and 3He(p, n)3p reactions 3“) at 30 and 50 MeV and the p + d + p + d” excitation function 40, 41). These experiments are ambiguous, the first one because it implies the subtraction of a four-body phase space and the others because they depend on the width of the integration performed over the p-n final state. The p + d + p + da2experiments are nevertheless remarkable in the sense that it is a unique case of positive agreement between experimentalists although the interpretations differ, Niiler et al. 40) emphasizing “resonance” effects when Van der Weerd et al. 41) speak of “threshold” effects. On the theoretical point of view it has been pointed out that resonant two-body forces could give excited levels of the compound three-nucleon system 42). The most detailed calculation showing possible three-nucleon broad resonances makes use of the dispersion relation 43) and predict a T = 3 4P state at 15- 5i MeV and ‘P at 18 - 8i MeV. More exact calculations based on Faddeev equations agree with a possible 4P resonance. Such resonances must appear clearly in an Argand-plot representation. If we write the amplitude T = (rje’““-1)/2i, the Argand plot is the graph giving Im T as a function of Re T. In the case of a nonresonant elastic scattering (q = 1) the Argand plot describes a circle of radius 3 centered on (Re T = 0; Im T = 3). If y changes smoothly in a non-resonant way when the energy increases the circle becomes a spiraloid of which the radius decreases
PHASE-SHIFT
ANALYSIS
265
smoothly. Now if there is a sharp resonance and if the non-resonant phase shift remains constant the real phase-shift changes abruptly by J+n and the Argand plot across the resonance describes a circle of radius 3~ in a counter-clockwise direction [see for example ref. ““)I. When the resonance is broad the preceding pattern is smeared out and the circle is not fully described, The ‘5 and 4S Argand plots move smoothly clockwise indicating no resonance effect, The 2P plot is more complex but there is no indication of a resonance in this energy region, but the 4P phase shift shows clearly a resonance structure centered around E = 13.0+0.8 MeV or EsHe*= 14.2 MeV-tO.5 MeV with a width of about 10 MeV, as shown in fig. 8.
Fig. 8. Argand plot of the 4P phase. The solid line is the result of the phase-shift analysis, while the dotted line is the circle of radius 4 centered on (Re 2’ = 0, Im T = 4). The dash-dotted circle indicates a broad resonance centered around 13 MeV and with a width of about 10 MeV.
These results are almost in agreement with a recent phase-shift analysis by a Groningen group 13). The main difference between this latter and the present analysis is that we have analyzed both the elastic and the total reaction cross section, that the analysis has been extended downwards (to 1 MeV compared to 2.995 MeV) and upwards (to 46.3 MeV compared to 30.4 MeV). Also experimental data at 6.78 MeV and 12.18 MeV have been included in the present work. When comparing the method of analysis we have tried in the present work to tighten the number of free parameters to ensure more reliable results. In particular no free scaling factor has been used. Despite these differences there is good agreement between this analysis and the Groningen one, except for the 2S phase which is rather insensitive to the elastic scattering cross section, making the disagreement not sigrmicant. At the Los Angeles conference we have presented prelimina~ results of another phase-shift analysis 14) pointing out the possible existence of a ‘P state. This analysis was also based on Faddeev-type calculations but integrated to first-order only (poleterm 45). From E = 3 to 8 “first order” phase shifts are systematically higher in absolute value than “exact” phase shifts by 20 %, whatever the phase shift or the energy. Another important feature of this analysis was that all absorption parameters
266
J. ARVIEUX
have been kept fixed at Sloan’s values. We have since investigated carefully the region between E = 10 and 20 MeV and reached the following conclusions: (i) If all inelastic parameters are kept fixed one can find two solutions for the ‘P and 4P couple of parameters, the one with the best x2 value having a large positive ‘P phase and a 4P phase smaller than theoretical predictions. This solution which will be called “irregular” has been presented in ref. 14). The other solution follows approximately the patterns of the theoretical calculations and will be called the “regular” one. (ii) If real doublet and quartet phases and only doublet ‘qO and ‘yl absorption parameters are varied in the analysis one finds again a ‘P resonating phase reaching 0.7 radian between 15 and 20 MeV in connection with a decrease of 4P down to 0.44 radian. The absorption parameter ‘qO first decreases to a value of 0.6 at 15 MeV but then increases again and reaches 1 at 20 MeV and stays at this value for higher energies. The ‘ql absorption parameter decreases sharply to values 0.2-0.3 between 15 and 20 MeV and then increases again. These features are very different from the ones observed in the solution discussed in the present paper although this “irregular” solution has x2 values of the same order or even smaller than the “regular” one. It is only when the four absorption parameters for I = 0 and 1 are varied at the same time that the regular solution seems favored. This is a reflection of the existing strong coupling between the different waves. In fact there is no reason why the channel spin S might be conserved and if there is any resonant wave of spin J, its effect must appear in both ‘ZJ and “IJ states. Our conclusion concerning the excited states of 3He* is the following: there is no possibility of explaining the elastic p-d scattering cross section between 10 and 20 MeV without allowing either the ‘P or the 4P wave to induce a broad resonance. This is an indication that there is probably a broad J” = &- or 3- state in the 3He* system at around 14 MeV excitation energy. A spin $- is ruled out by the fact that one can observe such a resonance on the ‘P phase shift. Besides this resonance, our results are consistent with a strong 2P-4P coupling (spin mixing). 8. Conclusion
Our results confirmthelackof sensitivity of the cross sections alone to the doublet phase shifts and emphasize the need for a complete phase-shift analysis including all non-central effects. Such an analysis has been done at low energy I’) but to solve the 3He* problem it needs to be done at higher energy. The formalism is available and there are many new interesting results concerning spin-dependent experimental data; in particular spin correlation coefficients are now available. It remains that such an analysis represents a large computational effort and we hope that the present analysis, by giving a good set of starting values for the phase-shifts, will be a first step. The development of “exact” theoretical calculations makes such an analysis even more valuable.
PHASE-SHIFT
ANALYSIS
267
The author is indebted to Dr. A. E. Conzett for having permitted him to spend a fruitful year in Berkeley and to use the remarkable computational facilities of this laboratory. References 1) I. H. Sloan, Phys. Rev. 165 (1968) 1587 2) W. M. Kloet and J. A. Tjon, Phys. Lett. 37B (1971) 460 3) S. C. Pieper, Phys. Rev. C6 (1972) 1157 4) P. Doleschall, Phys. Lett. 4OB (1972) 443 5) A. Fiore, J. Arvieux, Nguyen Van Sen, G. Perrin, F. Merchez, J. C. Gondrand, C. Perrin, J. L. Durand and R. Darves-Blanc, Phys. Rev. C8 (1973) 2019 6) R. S. Christian and J. L. Gammel, Phys. Rev. 91 (1953) 100 7) W. T. H. van Oers and K. W. Brockmann, Nucl. Phys. A92 (1967) 561 8) J. Arvieux, Nucl. Phys. A102 (1967) 513 9) R. Griitzschel, B. Ktihn, H. Kumpf, K. Moller and J. Miissner, Nucl. Phys. Al74 (1971) 301 10) W. Trlchslin and L. Brown, Nucl. Phys. A90 (1967) 593 11) R. Briining, B. Zeitnitz and J. Arvieux, Proc. Third Int. Symp. on polarization phenomena in nuclear reactions, Madison 1970, ed. H. H. Barschall and W. Haeberli, p. 445 12) P. A. Schmelzbach, W. Grtiebler, R. E. White, V. Kiinig, R. Risler and P. Marmier, Nucl. Phys. Al97 (1972) 273 13) J. W. Smits, L. P. Kok and R. A. Mallliet, Proc. Int. Conf. on few particle problems in the nuclear interaction, Los Angeles 1972, ed. I. Slaus, S. A. Moszkowski, R. P. Haddock and W. T. H. van Oers, p. 547 14) J. Arvieux, A. Fiore, J. Jungerman and Nguyen Van Sen, Proc. Int. Conf. on few particle problems in the nuclear interaction. Los Angeles 1972, ed. I. Slaus, S. A. Moszkowski, R. P. Haddock and W. T. H. van Oers, p. 519 15) W. C. Davidon, Argonne National Laboratory Report ANL 5990 (1959) 16) H. C. Catron, M. D. Goldberg, R. W. Hill, J. M. Le Blanc, J. P. Stoering and M. A. Williamson, Phys. Rev. 123 (1961) 213 17) R. F. Carlson, Y. Slaus, S. Y. Tin, W. T. H. van Oers, P. Doherty and D. J. Margaziotis, Proc. Int. Conf. on few particle problems in the nuclear interaction, Los Angeles 1972, ed. I. Slaus, S. A. Moszkowski, R. P. Haddock and W. T. H. van Oers, p. 475 18) D. C. Kocher and T. B. Clegg, Nucl. Phys. Al32 (1969) 455 19) W. T. H. van Oers and K. W. Brockmann, Nucl. Phys. 21 (1960) 189 20) S. Kikuchi et al., J. Phys. Sot. Japan 15 (1960) 9 21) T. A. Cahill, J. Greenwood, H. Willmes and D. J. Shadoan, Phys. Rev. C4 (1971) 1499 22) S. N. Bunker, J. M. Cameron, R. F. Carlson, J. R. Richardson, P. Tomas, W. T. H. van Oers and J. W. Verba, Nucl. Phys. All3 (1968) 461 23) G. E. Bixby and R. Smythe, Phys. Rev. 166 (1968) 946 24) C. C. Kim, S. M. Bunch, D. W. Devins and H. H. Foster, Nucl. Phys. 58 (1964) 32 25) J. H. Williams and M. K. Brussels, Phys. Rev. 11 (1958) 136 26) R. J. S. Brown, G. D. Freier, H. D. Holmgren, W. R. Stratton and J. L. Yarnell, Phys. Rev. 88 (1952) 253 27) A. S’Wilson, M. C. Taylor, J. C. Legg and G. C. Phillips, Nucl. Phys. A130 (1969) 624 28) I. H. Sloan, Phvs. Lett. 34B (1971) 243 29) J. D. Jackson and J. M. Blatt, Rev. Mod. Phys. 22 (1950) 77 30) L. M. Delves, Phys. Rev. 118 (1960)1380 31) W. T. H. van Oers and J. D. Seagrave, Phys. Lett. 24B (1967) 562 32) W. Dilg, L. Koester and W. Nistler, Phys. Lett. 36B (1971) 1973 33) Y. Avishai and A. S. Rinat (Reiner), Phys. Lett. 36B (1971) 161 34) A. C. Phillips and G. Barton, Phys. Lett. 28B (1969) 378 35) A. S. Reiner, Phys. Lett. 28B (1969) 387 36) C. C. Kim, S. M. Bunch, D. W. Devins and H. H. Forster, Phys. Lett. 22 (1966) 314
268
J. ARVIEUX
37) K. H. Bray, S. N. Bunker, M. Jain, K. S. Jayaraman, G. A. Moss, W. T. H. van Oers, D. 0. Wells and Y. I. Wu, Phys. Rev. C3 (1971) 1771 38) G. Paic, Proc. Int. Conf. on few particle problems in the nuclear interaction, Los Angeles, 1972, ed. I. Slaus, S. A. Moszkowski, R. P. Haddock and W. T. H. van Oers, p. 539 39) L. E. Williams, C. J. Batty, B. E. Bonner, C. Tschalar, H. C. Benohr and A. S. Clough, Phys. Rev. Lett. 23 (1969) 1181 40) A. Niiler, W. von Witsch, G. C. Phillips, C. Joseph and V. Valkoviti, Phys. Rev. Cl (1970) 1342 41) J. C. van der Weerd, T. R. Canada, C. L. Fink and B. L. Cohen, Phys. Rev., C3 (1971) 66 42) V. Efimov, Phys. Lett. 33B (1970) 563 43) W. Ebenhoh, A. S. Rinat (Reiner) and Y. Avishai, Phys. Lett. 29B (1969) 638 44) R. D. Tripp, Baryon resonances, in rendiconti della Scuola Internazionale, E. Fermi, Varenna 1966 (Academic Press, New York) p. 70 45) M. Durand, Thesis, University of Grenoble 1972, unpublished
Nuclear Physics A221 (1974) 253 -268; @ North-Holland Publishing Co., Amsterdam Not to be
reproduced by photoprint
or microfilm without written permission from the publisher
PHASESHIFT ANALYSIS OF ELASTIC PROTON-DEUTERON SCATTERING CROSS SECTIONS AND 3He EXCITED STATES 5. ARVIEUX + Lawrence Berkeley Laboratory, Berkeley, Calif. 94720 +f Received 27 November
1973
Abstract: A phase-shift analysis of recent experimental data on elastic proton-deuteron scattering and reaction cross sections has been performed for laboratory energies from 1 to 46.3 MeV. Phase shifts up to L = 8 calculated by Faddeev equation techniques have been included in the analysis. A new set of p-d scattering lengths is given which rules out the possibility of a pole term in the effective range expansion of the doublet *S; phase. The Argand plot of the 4P wave is shown to be consistent with a broad resonance in 3He centered around an excitation energy of 14 MeV. This resonance is discussed in the light of preceding experimental and phase-shift analysis results.
1. Introduction
The theoretical treatment of the three-nucleon interaction has recently undergone considerable progress. In the case of elastic nucleon-deuteron scattering, the Faddeev equations have been solved exactly ‘) with a S-wave nucleon-nucleon non-local separable interaction. A treatment using the Pad6 approximants approach has been also used with a central spin-dependent Yukawa interaction “). Both calculations give very similar results and they reproduce fairly well the gross structures of the experimental cross-section data. Recent calculations have been done with separable spin-dependent potentials either in a perturbative treatment including the tensor and the P- and D-nucleon-nucleon interaction 3), or in a more exact treatment including the tensor and the P-force only “). These latter calculations reproduce surprisingly well and without any free parameters, most of the qualitative features of the polarization data including the nucleon polarization and the deuteron vector and tensor analyzing powers “). These calculations indicate how important the effects of some spin-couplings are and the need to take them into account in a serious phase-shift analysis. Such a semi-phenomenological analysis based partly on theoretical predictions is valuable in giving accurate phase shifts which can be then compared to refined theoretical calculations using more realistic N-N potentials. This stochastic process has been the line along which nucleon-nucleon experiments have been analyzed and has proved to be very valuable. There have been many phase-shift analyses of nucleon-deuteron scattering, some of them extending over a large energy region 6-8), some others restricted to a single + Permanent address: Institut des Sciences NuclCaires, BP 257, 38044 Grenoble, France. ++ Work performed
under the auspices of the US Atomic Energy Commission. 253
J. ARVIEUX
254
TABLE 1 Results of the
E
?3
‘93
2P
2.503
-0.044
4l?
2D
4D
“F
QF
-0.0004
0.0022
-0.0036
0.0086
-0.0080
0.0160
2G
Q
1.993
2.837
2.209
-0.112
0.334
2.995
2,689
2.020
-0.117
0.411
0.0116 -0.0176 -___e 0.0231 -0.0387 ~__I~ 0.029 -0.0670
3.998
2.542
1.856
-0.081
0.456
0.025
-0.098
-0.0114
0.0232
-0.116
-0.022
0.0292
0.0042 -0.0082 -___..
0.0324
o.00.51 -0.0102 II___I_
0.029
0.0059 I_____
-0.0115
0.0069 .--
-0.0135
1.0
2.922
0,967 5.002
2.462
1.755
0.893
0.176
0.987
0.999
- 0.040
0.500
0.032
0.955
0.996
0.998
6.007
2.387
1.649
0.523
0.040
0.999
0.912
0.9997
0.995
6.78
0.81 -___ 2.328
1.581
0.001
0.536
0.023
0.76
0.998
0.872
0.985
0.99
ii%+6
1.521
0.066
z-i
0.055
0.997
0.822
0.967
0.983
10.04
0.682 ____2.098
1.409
0.118
0.574
0.090
0.989 1.35
0.749
0.946
0.971
12.18
0.610 ._I_ 1.980
0.227
0.576
0.103
0.55 ____ 1.910
0.985
0.715
0.909
0.959
13.93
1.296
0.273
0.585
0.155
0.973
0.687
0.866
0.95
16.24
0.495 ___1.737
1.247
0.3133
0.574
0.141
0.963
0.682
0.856
0.939
17.12
0.456 -1.686
1.221
0.316
0.577
iGG
0.447
0.971
0.681
0.830
0.936
8.025
-iGG
0.0008 -0.0016 -____ 0.0017 -0.0036 -_____ 0.0030 -0.0060 -____
0.9999 -0.129 0.999 -0.152 0.998 -0.149 0.996 -0.160 0.991 -0.188 0.988 -0.180
-0.041 0.9995 -0.068 0.9989 -0.051 0.997s -0.049 0.9952 -0.046 0.9925 -0.038
0.9998 0.0412 0.9996 0.042
0.0082 -0.0160 ___-
0.9989 0.043
0.0093 --
-0.0177
0.9979 0.040
0.0101 -0.0190 I_______
18.52
19.92
22.0
1.630
1.179
0.340
0.540
0.127
0.428
0.954
0.682
0.842
0.931
1.546
1.155
0,373
0.534
0.419
0.970
0.684
0.836
0.931
1.449
1.133
0.424
0.556
0.178
0.166
0.402
0.959
0.695
0.837
0.92
25.7
1.359
1.109
0.514
0.540
0.223
0.405
0.951
0.710
0.823
0.915
31.0
1.215
1.069
0.537
0.493
OS97
0.415
0.920
0.738
0.849
0.910
1.108
1.017
0.530
0.435
0.220
35.0
0.956 -0.183 0.939 -0.189 0.933 -0.173 0.913 -0.177 0.906 -0.169 0.931 -0.162 0.918 -0.163 0.926 -0.138
0.9901 -0.025 0.9870 -0.043 0.9860 -0.008 0.9840 -0.01’8 0.9823
0.9968 0.039
0.0108 -0.0199 --
0.9954 0.040
0.0111
-0.0202
0.0114 --
-0.0210
0.9948 0.038 0.998 0.040
0.0117 -0.0212 --
0.9928
-0.011
0.034
0.9756 _..~ -0.030
0.9890
0.0129 -0.0226 ~__I 0.9981 0.9961 -0.0136 -0.0230 -0.9973 0.9951
0.054
0.0144 -0.0227
0.971
0.985
0.013
0.040
0.9937 0.9962 -0.041 -0.027
-0.031 0.9788
0.039
0.9910
PHASE-SHIFT
ANALYSIS
25.5
phase-shift analysis
4H
ZH
21
41
2J
4J
2K
4K
x2/p.p.
xz/p.d.f.
Ref.
0.23
0.31
18
1.76
2.09
18
0.96
1.11
18)
>
-0.0002 ___ -0.0004
0.0004 0.0008
-0.0008 ____
0.0016 ___
0.96
1.22
IS
1
-0.0012 ___
0.0023 -
0.65
0.88
18
1
-0.0015 -
0.0030 -
0.62
0.87
18
1
-0.0018 ____
0.0037 ___
-0.0022 -
0.0045 __
1.23
1.73
18
1
-0.0027 _I_
0.0056 -
0.0009 __-
-0.0020 -
0.82
1.30
18
1
-0.0032 ___
0.0064 __
0.0012 -0.0023 ~___
1.54
1.73
19
-0.0035 -
0.0070 __
0.0014 -0.0026 -___
0.73
0.86
-0.0037 ___
0.0075 -
0.0015 -0.0029 -____
1.28
1.62
21
-0.0038 ___
0.0077 __
0.0015 -0.0030 -___
2.45
3.21
21
)
0.0039 ___
0.0079 -
0.0016 -___
-0.0032
1.15
1.49
21
1
-0.0039 -
-
0.0082
0.0017 -0.0033 -___
1.06
1.46
21
1
0.0089 0.9996 ~ 0.0092 __ 0.9994 __ 0.0095 __ 0.9990
0.0018 -___0.9998 0.0020 -p0.9997 ___ 0.0021 ~~ 0.9996 ___ 0.0022 y-___
-0.0042 0.9991 -0.0041 ___ 0.9988 __ -0.0041 ___ 0.9983 --0.0039 ___
0.0096 ~
1
>
20
1
>
-0.0007
0.0015
1.79
2.38
22
0.9999 _I -0.0037 -0.0008
0.0016
3.83
5.94
23
) )
-0.0035
0.9998 -0.0039 -0.0008 0.9997 0.9998 -0.0040 -0.0008
>
0.0017 -
0.0004 _____
-0.0007
0.8
1.12
24
0.0017 __
0.0004 __~
-0.0007
0.78
1.01
22 >
256
J. ARVIEUX TABLE 1
E
40.0
46.3
%
?3
2P
4P
2D
0.437
0.851
0.755
0.816
0.905
1.026
1.092
0.557
0.375
0.215
0.475
0.903
0.773
0.874
0.905
0.888
1.041
0.469
0.360
0.238
0.540
0.757
0.805
0.832
0.905
Column 1: Colums 2-19: Column 20: Column 21: Column 22:
4D
2F
4F
ZG
0.870
0.9688 __ 0.004
0.9830 0.045
0.03 1
-0.030
0.9665
0.971
0.042
0.043
0.9920 -~ 0.027
-0.002
0.9638
0.966
0.9908
-0.096 0.896 -0.090 0.865
0.9929 __-
4G 0.9955
0.9946
0.9934
Energy E of the incident proton in p+d scattering. Phase shifts for I = 0 to I = 8; the first line printed for each energy is the real part 6 in corresponding value is 1. Underlined numbers have been kept fixed during the search. Value of the x2 per point. Value of the x2 per degree of freedom. Reference for the experimental data.
energy “) or a limited domain of energy 1o-12). In a few of them polarization data have been included ‘-11) an d m ’ t wo cases all the couplings have been allowed either below the deuteron break-up energy lo) or at relatively low energy 12). In this last analysis up to 41 parameters have been allowed to vary to analyze data taken from Ed = 6 to 11.5 MeV. When one considers that when going up in energy (and the most interesting region to study 3He excited states is between Ed = 20 and 30 MeV) the number of parameters is even larger, one sees that a good analysis of the cross sections, although it is simplified, is very valuable to give starting values for a more complete analysis including all spin effects. Such a simplified analysis would almost fix the 2S, and 4S, phases and the average of all phases from L = 1 to L,,,, SO reducing the number of free parameters. Moreover, such an analysis permits one to solve problems concerning the effective range expansion of L = 0 phases for which polarizations play an insignificant role. The most extensive analysis of cross sections has been done by Van Oers and Brockmann ‘). It was already semi-phenomenological since partial waves 1 2 2 were calculated in the Born approximation as in the preceding work of Christian and Gammel 6), but it also included inelastic effects. This analysis should be re-done for twomainreasons: (i)thenon-unitaryBornapproximationisnotverygoodforcalculation of low-partial waves and (ii) the inelastic phases in the 2S and 4S states have been assumed to be equal, which is strongly contradicted by results of exact calculations, the absorption being much more important in the doublet than in the quartet state. Finally, new experimental data are now available. We present here a phase-shift analysis of elastic and reaction cross sections between 1 and 46 MeV. The validity of the main predictions of theoretical calculations has been tested and then used whenever possible in order to reduce the number of free parameters. The present analysis will be compared with two recent analyses published at the Los Angeles conference [refs. 13,,,)I.
PHASE-SHIFT
ANALYSIS
257
(continued)
2H 0.9989 ------0.0037 ___ 0.9976 ---__--0.0033 II_ 0.9971 --_---__
4H
*J
41 0.9996
4J
0.9998
2K
4K
x2/p.p.
X2/p.d.f.
Ref.
0.9988
0.9994
0.0097 0.9984
0.0023 -0.0040 -_____0.9993 0.9995
0.0018 0.9999
0.0004 -0.0008 --
1.83
2.75
=)
0.9998
0.0098 0.9980
0.0025 -0.0040 -0.0007 -___0.9991 0.9994 0.9997
0.0018 0.9998
0.0004 -0.0008 __~ 0.9999
1.4
1.95
22)
-0.0008
radians, the second line is the absorption
0.9999
parameter q_ When no absorption
parameter is printed the
2. Formalism
The formula describing the elastic scattering cross section o(6) and the reaction cross section 0, have been given in ref. ‘). In the following the orbital angular momentum is denoted by I; the parameters *S, and ‘6, are respectively the real quartet (S = $) and doublet (S = 4) phases where S is the channel spin; 4qr and ‘qr are the corresponding absorption parameters. The reaction cross section depends only on the inelasticity parameters while the elastic scattering cross section depends on both the real and imaginary phases. The present calculations have been made at the Lawrence Berkeley Laboratory on a CDC 7600 computer. The search routine is VARMIT, a variable metric minimization method based on the routine VARMIN by Davidon 15). The function to be minimized is the x2 given by
where the sum extends over all the elastic 0 ,,,(t?) data at a given energy, do(B) being the experimental uncertainty; Go exp and do, are the experimental reaction cross section and the corresponding uncertainty. The gradient of the x2 with respect to each parameter is calculated analytically which ensures a much faster and safer convergence. The uncertainties in the results of the analysis as given by the covariant matrix, are printed out. 3. The data analyzed No attempt has been made to correct the normalization of the experimental data. ‘When experimental points gave unreasonable x2 values they have been suppressed in the analysis. This occurred for a few energies, especially at very forward angles.
J. ARVIEUX
25s
The data ‘se2 “) are listed in table 1. The listed energy E is the laboratory proton energy in a p + d scattering. The corresponding deuteron lab energy in a d + p scattering is 2E and the c.m. energy is $ E in the non-relativistic approximation. Below E = 1 MeV the old results of Brown 26) are not accurate enough and very different sets of phases give nearly the same x2 value so they have been skipped out in the analysis. Above 1 MeV two sets of recent experimental data are available. One is from Wisconsin 18) and it corresponds to integral energy values from 1 to 10 MeV. The other data set is from Rice 27) from 4.5 to 11.5 MeV for half-integer energy values. Both set have been used in a preliminary analysis 14) and the phases obtained were almost compatible except for the ‘S phase where we have observed a strong oscillation when going from one set of data to the other, the Rice data giving usually higher values for ‘S than the Wisconsin data. Since these latter results have a smaller uncertainty they have been selected for this analysis.
4. Validity of the theoretical calculations There are two calculations based on the solution of the Faddeev equations ‘, “). In both cases the theoretical phase shifts have been made available to us. Two main features predicted by these calculations have been tested extensively. They are the effect of high partial waves and the comportment of absorption parameters. 4.1. EFFECT
OF HIGH PARTIAL
WAVES
In a first step searches have been made up to 15 MeV without including high-order partial waves. For energies as low as 1 MeV it has been found that 1 = 2 phases are “necessary”. We mean by necessary that if the phase of the highest L-value is included in the analysis, the best 11’ value shows a significant improvement over the value TABLE2 Scattering lengths a, effective ranges r and shape parameters P in p-d elastic scattering Quartet
Doublet %-d (fm) Van Oers-Brock-
zPp--d (fm?
+&+a (fm)
1.3 f0.2
%I-+ (fm)
%-ci (fm)
4P*-d (fm3)
0
11.4+r*a -1.2
2.05 10.25
0
0
11.9+o.3 -0.9
1.s9+o.2 -0.05
O
O.OS’+$$
11.88:;:‘:
2.63:z.g;
0
11.00
1.96
mann “) Van Oers-Seagrave b, Present work
2.7310.10
Avishai-Reiner
3.4
(theory) 33) “) Ref. +‘).
“) Ref. 31).
2.2710.12
-0.54&0.1 0
PHASE-SHIFT
ANALYSIS
259
obtained when this phase is not taken into account. F-phases (I = 3) are necessary at 3 MeV to fit the backward peak; G-phases (I = 4) at 8 MeV; H-phases (I = 5) at 12 MeV and I-phases (I = 6) at 15 MeV. This reflects the high importance of exchange effects. In a second step we have used as fixed parameters the phases for high-l values (up to Z = 8 at 46.3 MeV). The two sets of phases as given by Sloan ‘) and Kloet and Tjon (KT) “) d o not differ by more than 10 o/0for Z 2 3, so only Sloan’s values have been used for these phases. The high partial waves used are given in table 2 which shows also what was the maximum Z-value included at each energy. The F-phases have been varied above 5 MeV since this has been shown to give a strong improvement in the x2 value. For phases up to Z = 2 both the Sloan and KT values have been tried. Quartet phases are very similar but the doublet 2S phases can be different by as much as 20”. Our analysis shows that below 5 MeV the search converges towards one unique solution which is somewhat intermediate between the Sloan and KT predictions but at 5 MeV the results of the analysis are much closer to the KT solution. Above 5 MeV the two different sets of starting values lead to two different final solutions with usually the same x2 value, except at 12.18 MeV where KT starting values are again favored. So we think that the KT calculations seem a better approximation to the experimental data and only the corresponding solution will be given. 4.2. ABSORPTION
PARAMETERS
All the calculations based on Faddeev equations show that the absorption in the N-d scattering is much stronger in the doublet than in the quartet state. This assumption has been systematically tested at a few selected energies namely 6,8 and 10 MeV. To increase the sensivity of the search to the absorption parameters, the experimental errors on the reaction cross section have been divided by YEsuch that n2 equals the
I
0
I
I
I
IO
20
30
I
60
50
E (Me’/1
Fig. 1. Experimental reaction cross sections. Black dots are from ref. 16) and open ones from ref. 17). The dotted curve is the interpolated value used when elastic and total reaction cross-section data were taken at different energies. The full curve is the value obtained in our analysis with the absorption parameters of table 1.
J.ARVIEUX
260
number of elastic scattering data points analyzed. At the energies investigated four inelastic parameters are important (I = 0,l). At these energies the assumption that doublet absorption is much more important than the quartet one is strongly supported by the experimental data. It is also found that doublet absorption is much less sensitive to the combined elastic and inelastic data than quartet absorption, so that when both are allowed to vary the doublet absorption parameters remain close to the theoretical values. So they have been kept almost fixed throughout this analysis. The most sensitive absorption parameter is ‘qt which change si~ifi~n~y from the theoretical values between 10 and 20 MeV. The reaction cross sections used in this analysis are from n-d data 16) below 14 MeV and p-d data 17) above 23 MeV. Between these two energies the experimental data have been smoothly interpolated as shown by the dotted line in fig. 1 which shows also the result of our analysis (solid line) and the experimental data. 5. Results The results are summarized in table 1. The real part 2s’161 is given in radians on the same line as the energy E. The second line gives the absorption parameters ” ” ql. When no absorption parameter is printed out 2s “ql = 1. Underlined quantities have been found to vary insignificantly when allowed to vary and have been f. 2. 3.4.5.
01
0.3
IO. 15. 20.
0.5
07
E (MeV) 30.
40.
0.9 k CFm-‘)
Fig. 2. The %Zreal phase shifts in radians as a function of the wave number k in fin-’ the upper scale indicates the eaergy E of the incident proton in the lab system in pi-d
(HOW@Scald;
scattering.The solid curve is a theoretical calculation by Kloet and Tjon =), the dashed line is a calculation by Sloan 1).
1. 2. 3 4.5 I I III
IO. 15. I ,
E (MeV) 30. 40. I I
20. I
E (MeV) 1. 2. 3.4.5
10
15. 20.
30.
40.
.
I
7 ‘11
l
* a
I.1.5 0.1
I
,_:_:: 0.3
0.5
0.7
/
01
0.9
0.3
/
I
I
0.5
0.7
k(fm-1)
Fig. 3. The ?S real phase shifts in radians; the solid line is the theoretical result obtained in an exact three-body calculation by Kloet and Tjon ‘) and Sloan I).
IO. 15. 20 1 1 1
30.
I
0.9
Fig. 4. Same as fig. 2 for the *P (upper graph) and 9 (lower graph) phase shifts.
E (Me!!) 1. 2.3.45 rs1111
,
k (fin-‘)
40. I
E (Me’/! 1.2345
IO. 15 20
30
40
J
-0.d
0.5
0.7
k (fm-‘!
Fig. 5. The 2D, 4D, 2F and 4F phase shifts in radians; the dotted curve is an exact threebody calculation by Sloan ‘>. The predictions by Kloet and Tjon 2, do not differ by more than 10% for these phases and are not shown. The dash-dotted curve is a Born approximation calculation.
0.1
0.3
0.5
0.7
0.9 k (fm-1)
Fig. 6. Absorption parameters zoo. 4~0, 2qr and 4qL. The solid curve is by Kloet and Tjon 2), the dash-dotted curve is the result of a previous phase-shift analysis by Van Oers and Brockmann7).
262
3. ARWEUX
kept fixed to theoretical values during the analysis. According to the modified Levinson theorem 28) the ‘6, and 46, phases start at rc at E = 0 MeV. The results are also displayed in figs. 2 to 6. One can notice in fig. 5 the large difference between the values of 26, obtained in the present analysis when compared with non-unitary Born approximation calculations. Large differences are also observed in fig. 6 for inelastic parameters. In a first analysis all the real phases up to 1 = 3 and imaginary phases up to I = 2 have been allowed to vary. It has been found that some parameters do not change significantly from the theoretical values, especially ‘y10 and *yl. These parameters have been subsequently fixed at their theoretical value, thus reducing the number of free parameters and improving the x2 per degree of freedom. The uncertainties, as calculated by the searching routine VARMIT, are usually very small, of the order of 0.001 radian or less. These are calculated by fixing all the parameters but one and by seeing how the x2 value changes when the free parameter varies. In fact, correlation unoertainties are much more important. Some of these correlations can be an effect of a real physical phenomenon; for example one can compensate for a decrease in 46, by increasing ‘6,) which is an indication of the strong correlation between the ‘P and 4P phases. The indicated errors take into account these different solutions around the selected one given in table 1 and having x2 values differing by usually not more than 20 %. When a solution seems unique a systematic uncertainty of 0.01 radian is applied to all phases. Completely different solutions from the ones listed but with the same 31’value can sometimes also be found. This is the case, for example, with Sloan’s starting values above 10 MeV. These solutions are not discussed on the basis that phase shifts must vary smoothly with energy. 6. Effective range expansions The low energy scattering of two particles may be described by the effective range expansion K = - 1 +&k2+Pk4+
. . .,
a
where a is the scattering length, r the effective range, P the shape parameter and k the relative momentum. For neutral particles K is just K = k ctg 6(k), where 6 is the phase shift, When both particles are charged the usual effective range expansion becomes 2“) K = C; k ctg 6(k)+ ~(~)~~. This expression can be easily reduced to the following directly computable expression for K: jy
zz
C;kctg6(k)+
~[-0.57722+~2~~l~-1(~2-l-r12)-1-lnill. *
PHASE-SHIFT
ANALYSIS
263
where q = e2/nv = 0.15806 E-* and Co2= 2ny/(exp(2ny) - 1). Using this expression, and the values of 6 given in table 1 we have computed the quartet and doublet effective expansion of the ‘S and 4S waves. It is fairly well established that the doublet n-d scattering length shows a remarkable anomaly which would lead to an effective range expansion of the type 34, 35) 2K =
- l/a+$2r,
k2+2Pk4
1 +(k/k,)2
’
Such an expansion has a pole for ki = -k2 in the unphysical region close to the zero energy. Evidence on this phenomenon is based on the triton photo-desintegration [ref. 3“)I, a phase-shift analysis of n-d scattering 31) and a recent zero energy measurement of the n-d scattering lengths 32) giving a very small doublet scattering length 2a,,_d and so a large negative value of 2K at the intercept with the zero energy axis. In the p-d scattering case the situation is not so clear. Existing phase-shift analyses were not accurate enough to yield a conclusion on the existence of such a pole. On a theoretical side, perturbation techniques applied to a dispersion relation calculation of N-d scattering predict a recession of the pole in the ‘K expansion deeper into the unphysical region leading to a near complete extinction of it 33). We have analyzed our results with the expansion (1) using the four following combinations of free parameters: ‘a and 2r; ‘a, ‘r and 2P; 2a, 2r and 2ko ; 2a 2r, 2P and 2ko. The best fit to the overall 20 data points between 1 and 46.3 MeV is ‘obtained with the following parameters, 2a = 2.73*0.1 fm,
‘r. = 2.27f0.12
2P = O.OS;$~ fm3,
fin,
with a x2 of 0.87 per degree of freedom. Different pole terms have been tried with starting values at -0.1 MeV and - 1.0 MeV; in all cases the pole is rejected deeper in the negative energy region (E 2 -22 MeV). The best fit (full line) is shown in fig. 7 together with the best straight fit
0
IO
20
30
40
E (Me’/)
Fig. 7. Effective range expansion of 260 and 460.
264
J. ARVIEUX
(dotted line) for E = 1 to 18.52 MeV with the parameters ‘u = 2.73 fm, ‘r = 2.29 fm and ‘P = 0. Our results are compared in table 2 with the results of a preceding phase-shift analysis ‘) and a p-d elastic scattering calculation using dispersion relations 33). It can be noted that our value of Zap-d is much closer to the theoretical one than the old one. Our result for 4ap_ d, given in table 2 agrees with preceding results but the effective range parameter is significantly larger than preceding estimate. Moreover we have been able to get a fairly accurate value of the shape parameter 4Pp_d which is of a negative sign as is clear in fig. 7. 7. Excited states of the 3He system The excited states of the three-nucleon system are like the Loch Ness monsters. Many people claim to have seen some but when a careful scientific expedition is designed to track them, then they vanish out. Kim et al. 36) pretended to have seen many of them but they have been later infirmed. There has been a subsequent great number of experiments investigating two-body, three-body, and even four-body reactions [see ref. 3‘) for a summary of the experimental situation in 19711. Since 1971 the attention has been mainly put on photo-disintegration of 3He and p + d capture reactions [see ref. 38) for a summary of the situation in 19731 again with conflicting results. On the experimental point of view the sole indications of possible experimental 3He* broad resonances which have not been subsequently confirmed are the 3H(p, n)3He* and 3He(p, n)3p reactions 3“) at 30 and 50 MeV and the p + d + p + d” excitation function 40, 41). These experiments are ambiguous, the first one because it implies the subtraction of a four-body phase space and the others because they depend on the width of the integration performed over the p-n final state. The p + d + p + da2experiments are nevertheless remarkable in the sense that it is a unique case of positive agreement between experimentalists although the interpretations differ, Niiler et al. 40) emphasizing “resonance” effects when Van der Weerd et al. 41) speak of “threshold” effects. On the theoretical point of view it has been pointed out that resonant two-body forces could give excited levels of the compound three-nucleon system 42). The most detailed calculation showing possible three-nucleon broad resonances makes use of the dispersion relation 43) and predict a T = 3 4P state at 15- 5i MeV and ‘P at 18 - 8i MeV. More exact calculations based on Faddeev equations agree with a possible 4P resonance. Such resonances must appear clearly in an Argand-plot representation. If we write the amplitude T = (rje’““-1)/2i, the Argand plot is the graph giving Im T as a function of Re T. In the case of a nonresonant elastic scattering (q = 1) the Argand plot describes a circle of radius 3 centered on (Re T = 0; Im T = 3). If y changes smoothly in a non-resonant way when the energy increases the circle becomes a spiraloid of which the radius decreases
PHASE-SHIFT
ANALYSIS
265
smoothly. Now if there is a sharp resonance and if the non-resonant phase shift remains constant the real phase-shift changes abruptly by J+n and the Argand plot across the resonance describes a circle of radius 3~ in a counter-clockwise direction [see for example ref. ““)I. When the resonance is broad the preceding pattern is smeared out and the circle is not fully described, The ‘5 and 4S Argand plots move smoothly clockwise indicating no resonance effect, The 2P plot is more complex but there is no indication of a resonance in this energy region, but the 4P phase shift shows clearly a resonance structure centered around E = 13.0+0.8 MeV or EsHe*= 14.2 MeV-tO.5 MeV with a width of about 10 MeV, as shown in fig. 8.
Fig. 8. Argand plot of the 4P phase. The solid line is the result of the phase-shift analysis, while the dotted line is the circle of radius 4 centered on (Re 2’ = 0, Im T = 4). The dash-dotted circle indicates a broad resonance centered around 13 MeV and with a width of about 10 MeV.
These results are almost in agreement with a recent phase-shift analysis by a Groningen group 13). The main difference between this latter and the present analysis is that we have analyzed both the elastic and the total reaction cross section, that the analysis has been extended downwards (to 1 MeV compared to 2.995 MeV) and upwards (to 46.3 MeV compared to 30.4 MeV). Also experimental data at 6.78 MeV and 12.18 MeV have been included in the present work. When comparing the method of analysis we have tried in the present work to tighten the number of free parameters to ensure more reliable results. In particular no free scaling factor has been used. Despite these differences there is good agreement between this analysis and the Groningen one, except for the 2S phase which is rather insensitive to the elastic scattering cross section, making the disagreement not sigrmicant. At the Los Angeles conference we have presented prelimina~ results of another phase-shift analysis 14) pointing out the possible existence of a ‘P state. This analysis was also based on Faddeev-type calculations but integrated to first-order only (poleterm 45). From E = 3 to 8 “first order” phase shifts are systematically higher in absolute value than “exact” phase shifts by 20 %, whatever the phase shift or the energy. Another important feature of this analysis was that all absorption parameters
266
J. ARVIEUX
have been kept fixed at Sloan’s values. We have since investigated carefully the region between E = 10 and 20 MeV and reached the following conclusions: (i) If all inelastic parameters are kept fixed one can find two solutions for the ‘P and 4P couple of parameters, the one with the best x2 value having a large positive ‘P phase and a 4P phase smaller than theoretical predictions. This solution which will be called “irregular” has been presented in ref. 14). The other solution follows approximately the patterns of the theoretical calculations and will be called the “regular” one. (ii) If real doublet and quartet phases and only doublet ‘qO and ‘yl absorption parameters are varied in the analysis one finds again a ‘P resonating phase reaching 0.7 radian between 15 and 20 MeV in connection with a decrease of 4P down to 0.44 radian. The absorption parameter ‘qO first decreases to a value of 0.6 at 15 MeV but then increases again and reaches 1 at 20 MeV and stays at this value for higher energies. The ‘ql absorption parameter decreases sharply to values 0.2-0.3 between 15 and 20 MeV and then increases again. These features are very different from the ones observed in the solution discussed in the present paper although this “irregular” solution has x2 values of the same order or even smaller than the “regular” one. It is only when the four absorption parameters for I = 0 and 1 are varied at the same time that the regular solution seems favored. This is a reflection of the existing strong coupling between the different waves. In fact there is no reason why the channel spin S might be conserved and if there is any resonant wave of spin J, its effect must appear in both ‘ZJ and “IJ states. Our conclusion concerning the excited states of 3He* is the following: there is no possibility of explaining the elastic p-d scattering cross section between 10 and 20 MeV without allowing either the ‘P or the 4P wave to induce a broad resonance. This is an indication that there is probably a broad J” = &- or 3- state in the 3He* system at around 14 MeV excitation energy. A spin $- is ruled out by the fact that one can observe such a resonance on the ‘P phase shift. Besides this resonance, our results are consistent with a strong 2P-4P coupling (spin mixing). 8. Conclusion
Our results confirmthelackof sensitivity of the cross sections alone to the doublet phase shifts and emphasize the need for a complete phase-shift analysis including all non-central effects. Such an analysis has been done at low energy I’) but to solve the 3He* problem it needs to be done at higher energy. The formalism is available and there are many new interesting results concerning spin-dependent experimental data; in particular spin correlation coefficients are now available. It remains that such an analysis represents a large computational effort and we hope that the present analysis, by giving a good set of starting values for the phase-shifts, will be a first step. The development of “exact” theoretical calculations makes such an analysis even more valuable.
PHASE-SHIFT
ANALYSIS
267
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37) K. H. Bray, S. N. Bunker, M. Jain, K. S. Jayaraman, G. A. Moss, W. T. H. van Oers, D. 0. Wells and Y. I. Wu, Phys. Rev. C3 (1971) 1771 38) G. Paic, Proc. Int. Conf. on few particle problems in the nuclear interaction, Los Angeles, 1972, ed. I. Slaus, S. A. Moszkowski, R. P. Haddock and W. T. H. van Oers, p. 539 39) L. E. Williams, C. J. Batty, B. E. Bonner, C. Tschalar, H. C. Benohr and A. S. Clough, Phys. Rev. Lett. 23 (1969) 1181 40) A. Niiler, W. von Witsch, G. C. Phillips, C. Joseph and V. Valkoviti, Phys. Rev. Cl (1970) 1342 41) J. C. van der Weerd, T. R. Canada, C. L. Fink and B. L. Cohen, Phys. Rev., C3 (1971) 66 42) V. Efimov, Phys. Lett. 33B (1970) 563 43) W. Ebenhoh, A. S. Rinat (Reiner) and Y. Avishai, Phys. Lett. 29B (1969) 638 44) R. D. Tripp, Baryon resonances, in rendiconti della Scuola Internazionale, E. Fermi, Varenna 1966 (Academic Press, New York) p. 70 45) M. Durand, Thesis, University of Grenoble 1972, unpublished