- Email: [email protected]
Microscopic calculation of four-nucleon scattering observables in dd→dd and dd→p3H
NUCLEAR PHYSICS A ELSEVIER
Nuclear Physics A631 (1998) 675c~79c
Microscopic Calculation of Four-Nucleon Scattering Observables in dd--+dd and dd----)p3H A.C. Fonseca Centro Fisica Nuclear da U.L., Av. Prof. Gama Pinto, 2, 1699 Lisboa, Portugal
The four-body equations of Alt, Grassberger and Sandhas are solved for a system of four nucleons, using realistic NN interactions. The results of the calculations are compared with data for the reactions and d d ~ dd and d d ~ p3H. Preliminary calculations indicate that the nucleonnucleon p-waves have a strong effect on 4N observables.
Unlike the three-nucleon (3N) scattering problem where the qualitative features of the observables seem to be well-described even by relatively simple models of the nucleon-nucleon (NN) interaction, the four-nucleon (4N) system has remained a challenge for many years. Our aim is to make a serious attempt to overcome this barrier and provide a framework where one can reach an understanding of the data presently available in terms of a fully microscopic calculation. The starting point involves the solution of Alt, Grassberger, and Sandhas, equations [1] for the transition operators involving all (2) + (2) and (3) + 1 channels. For local NN potentials such equations are three-vector variable integral equations which after partial wave decomposition reduce to a set of coupled equations in three continuous variables. Similar equations were recently solved for 4He [2] requiring more than one hundred hours of computing time on a supercomputer. Since scattering calculations require a greater number of channels, some technical compromises must be made. One follows an approach based on the separable representation of subsystem amplitudes in order to reduce the equations to two (or one) continuous variable, Although this procedure increases the number of channels by a factor of three or four, there is a considerable net gain in computing time by the reduction of dimensionality. Our approach is based on the formalism developed by Fonseca [3] more than two decades ago and subsequently used by Haberzettl and Sandhas [4] to express the 2 + 2 subamplitudes in terms of a convolution integral involving two noninteracting pair-propagators. The integral equations we solve are obtained from the original AGS equations after one has: (a) represented the original NN t-matrix by an operator of rank one; (b) represented the resulting 3N t-matrix by a finite rank operator and taken as many terms as needed for convergence. The sole approximation in this approach involves a rank one representation of the 2N t-matrix which may be obtained from the well-known method of Ernest, Shakin, and Taylor [5]. The multi-term representation of the 3N t-matrix is done using the EDPE method developed by Sofianos, McGurk and Fiedeldey [6]. This latest approximation for the 3N t-matrix is well under control since one can, in addition to comparing the finite rank approximation with the original t-matrix results for the 3N observables (cross sections and analyzing powers), check on the convergence rate of 4N observables for increasing rank of the 3N representation. As 0375-9474/98/$19.00 © 1998 Elsevier Science B.V. All rights reserved. PI1 S0375-9474(98)00089-X
676c
A.C. Fonseca/Nuclear Physics A631 (1998) 675c-679c
mentioned betbre the 2 + 2 subsystem is taken care by convolution and theretore it is treated in an exact form. This is particularly important at energies above breakup where moving singularities have to be well taken care. This method was first used by Fonseca [7] in the binding energy calculation of 4He and later confirmed to be accurate by the exact work of Kamada and GI6eckle [8]. The calculations use Yamaguchy-like separable interactions, as well as rank one EST expansions of the Paris, Bonn-A and Bonn-B potentials in channels ISo, 3S1- 3D 1, IP I, 3P0` 3P 1 and 3P 2. At the three-nucleon level we include all 3N subalnplitudes up toj = 7/2+ in all subchannels that are consistent with the above NN partial-wave force components. Idential procedure is used for the 2+2 subamplitudes. All 4N observables are calculated with total angular momentum up to J = 6. In Fig. 1 and 2 we use the Yamaguchy potential in channels iS0, 3S l- 3D1 (corresponding to 4% d-state probability in the deuteron) to study the convergence of the calculation as one increases 3N total angular momentum j from 1/2+ to 7/2*. We denote r = 6666666 a calculation that includes j = 1/2", 1/2", 3/2+, 3/2", 5/2+, 5/2", 7/2+, with 6 terms in the EDPE expansion of each one of the corresponding 3N subamplitudes. This clearly shows that the shape of the experimental data can only be reproduced when higher j's are included.
0 . 4 0
-
•
,
-
-
,
-
,
-
-
•
-
-
•
-
•
x Exp. "~\
",',,\\
~ ---
rr=~6
~'
0.20
~:
~=6~6
//,.',~
,~ ",, )",.a:
\
E
E O
.xx.
0.00
/,<":
,' I ;/ •
-0.20
::=H
\ II /
10.0
30.0
60.0
90.0
120.0
150.0
O~ (d=g.)
Fig. 1. Differential cross section for the 2H(d,p)3H reaction at Ed = 6.1 MeV. The data is from ref. [9]. The calculation uses the Yamaguchy interaction in channels IS0 and 3S 1- 3D 1 alone. The r = 6 curve corresponds to including 6 terms in the EDPE expansion of the I/2 + 3N subamplitudes alone, while other curves correspond to adding 3N subamplitudes with higher j and rank Nj
180.0
"0"400.0
,
,
J , 30.0
i# i
~,
.
--
r~86
....
r=6666666
4
i
'--"
. . . . . . . . . 60.0 90.0 120.0 Ang. (C.M.)
, . 150.0
.
180.0
Fig. 2. Same as in Fig. for the tensor analyzing power T22
Next we study rank convergence in the EDPE expansion of the 3N subamplitudes by showing in Fig.3 and Fig.4 results of our calculations for rank 6 (solid curves) and rank 8 (dashed curves) expansions of the 3N subamplitudes. Each pairs of curves correspond to the
677c
A. C. Fonseca /Nuclear Physics A631 (1998) 675c-679c
same number of 3N subamplitudes included such that r = 6 and r = 8 correspond to j = 1/2 + alone, while r = 66 and r = 88 include j = I/2 + and 1/2-. Although there are some small changes rank 6 expansions may be considered adequate. 0.40
-
,
•
,
-
,
"
,
"
"
.
"
"
0,020
I III ::12212
0.2O
"3- /i
i zl
.x : I S
0.010 hl
0.00 ::z=
- -
r=
~//7 0000
--- - - -
r=8
r=88 r=6868
_0.400 0
.
.
i
30.0
.
,
i 60.0
,
,
.
90.0
.
.
.
. . 120.0
.
.
. 150.0
Ang. (C.M.)
Fig.3. Tensor analyzing power T22 for the 2H(d,p)3H reaction at Ed = 6. l MeV. The data is from ref. [9]. The calculation uses the Yamaguchy interaction in channels IS0 and 3S l- 3D b The solid curves correspond to rank 6 expansions while the dashed curves to rank 8 expansions.
180.0
-0.0100
0
,
,
, 30.0
,
, 60.0 Ang. (C.M.)
,
,
,
I\ 90.0
,
, t20.0
Fig.4. Same as in Fig.3 for the tensor analyzing power T21 for the 2H(d,d)2H reaction as 6.1 MeV. The data is from ef. [10].
Subsequently in Figs. 5-8 we show the results of our calculations for rank one EST representations of the Paris, Bonn-a and Bonn-b potentials in channels tS 0 and 3S I- 3D1, and compare with the corresponding calculation for the Yamaguchy potential. To save computing time we use rank two for the EDPE expansion o f j = 5/2 + and 5/2- 3N subamplitudes and rank one for j = 7/2 +, since previous tests indicated r = 6666221 to be almost identical to r = 6666666. Because the Paris potential leads to considerable less binding for 3H than the other three potentials we use, most of the difference may result from on-shell N-(3N) momentum missmatch between calculations corresponding to different potentials. Nevertheless Fig.5 shows that only realistic interactions can reproduce the 90 ° maximum in the differential cross section for 2H(d,p)3H. Finally in Fig.9-10 we show the effect of adding NN p-wave force components to the existing IS 0 and 3S l- 3D 1 components. Although the calculation is preliminary due to the number of 3N subamplitudes included (j < 3/2) and resulting 3N channels subject to particlepair maximum orbital angular momentum I _<3, we find d d ~ dd tensor observables to be very sensity to the NN p-waves. When all needed (3N) subamplitudes are added the curves will have a different shape but the sensitivity to NN p-waves will remain. Given that these calculations are very complicated, we find the need for benchmark comparisons between different groups now interested in the four nucleon system. Nevertheless from the results presented we find that 4N scattering observables are a much greater challenge than 3N scattering observables.
678c
,4. C. Fonseca/Nuclear Physics A 631 (1998) 675c-679c
100
. ~', ~'~" ~,.
E 0
x --........ ---
Exp. r=6666221 Yankag. r=6666221 Bonn-b r=6666221 Bonn,a
," ,,'~
.. . 0.20
10
- r - t - TT T-I--rT
0.00 I
.% -0.20
--..... 1
.
,
0.0
,
,
,
30.0
,
,
,
60.0
i
.
.
g0.0
.
.
.
.
120.0
.
r--6666221 Bonn-a r=6666221 Paris
.
150.0
180.0
-0.40
0.0
'
,
' ' 30.0
'
' ' 60.0
e~ (deg.)
.
........
.....
,
.
,
.
./-
•
-,
/
",
'~
iI
~',
0.010
0.000
-0.010
0.o
30.0
60.0 Ang. (C.M.)
180.0
0.020
..-
IiiIIl
-0.010
. . 150.0
, / ~ " ,
iIII \
0.010
-0.030
.
r=6666221 Yamag. r=6666221 Bonn-b 6666221 B . . . . . .
.
Fig.6. Same as in Fig.5 for the tensor analyzing power T22.
Fig.5. Differencial cross section for the ZH(d,p)3H reaction at Ed = 6.1 MeV. The data is from ref. [9]. The calculations correspond to rank one EST representations of realistic interactions in channels IS0 and 3S l - 3D 1.
0.030
' ~ • . ' . 90.0 120.0 Ang. (C.M.)
80.0
Fig.7. Tensor analyzing power T20.for the 2H(d,d)ZH reaction at Ed = 6.1 MeV. The data is from ref. [ 10].The calculations correspond to rank one EST representations of realistic interactions in channels IS0 and 3S 1 - 3D 1.
120.0
0.0
30.0
60.0 Ang. (C.M.)
Fig.8.Same as in Fig.7 for the tensor analyzing power T21.
90.0
120.0
679c
A. C, Fonseca/Nuclear Physics A631 (1998) 675c-679c
The authors would like to thank J. Heidenbauer for providing the rank one EST representation of the 2N realistic interactions, NCSC for providing the computing support needed for much of the development of these codes and TUNL for the hospitality and support during the summer visits. Finally we recognize the support of JNICT grant CERN/1101/96 and PRAXIS XXI grant that made possible the purchase of a dedicated machine.
0.030
0.020
-
•
,
•
.
,
"
p-waves
- -
no NN
---
with NN p-waves
0.010 0.010 0.000 0.000
\
~
-0.010 i /
-0.010
0.020 ""--IIIIII -0"0300.0
30.0
60,0
- -
no NN p-waves
---
with NN
I
p-waves
I
-0.020
90.0
Ang. (C.M.)
Fig.9.Tensor analyzing power T20.for the 2H(d,d)2H reaction at Ed = 6.1 MeV. The data is from ref. [10]. Only 3N subamplitudes up to j = 3/2 where included
120.0
-0.030
,
0,0
,
i
30,0
,
,
i
,
,
60.0
Ang.
i
90.0
,
,
120.0
(C.M.)
Fig. 10.Same as in Fig.9 for the tensor analyzing power T22.
REFERENCES 1. E.O. Alt, P. Grassberger, and W. Sandhas, Phys. Rev. C1, 85 (1970). 2. H. Kamada and W. Gl6eckle, Phys. Lett., B292, 1 (1993). 3. A.C. Fonseca and P.E. Shanley, Phys. Rev. D13, 2255 (1976); ibidem, Phys. Rev. C14, 1343 (1976). 4. H. Haberzettl and W. Sandhas, Phys. Rev. C24, 359 (1981). 5. J. Ernest, C.M. Shakin and R.M. Thaler, Phys. Rev. C8, 46 (1973). 6. S. Sofianos, N.J. McGurk and H. Fiedeldey, Nucl. Phys, A318, 295 (1979). 7. A.C. Fonseca, Phys. Rev. C40, 1390 (1989). 8. H. Kamada and W. G10eckle, Nucl. Phys,, A548, 205 (1992). 9. W. Gruebler et al., Nucl. Phys. A193, 129 (1972); ibidem A369, 381 (1981). 10. W. Gruebler et al., Nucl. Phys. A193, 149 (1972).
Nuclear Physics A631 (1998) 675c~79c
Microscopic Calculation of Four-Nucleon Scattering Observables in dd--+dd and dd----)p3H A.C. Fonseca Centro Fisica Nuclear da U.L., Av. Prof. Gama Pinto, 2, 1699 Lisboa, Portugal
The four-body equations of Alt, Grassberger and Sandhas are solved for a system of four nucleons, using realistic NN interactions. The results of the calculations are compared with data for the reactions and d d ~ dd and d d ~ p3H. Preliminary calculations indicate that the nucleonnucleon p-waves have a strong effect on 4N observables.
Unlike the three-nucleon (3N) scattering problem where the qualitative features of the observables seem to be well-described even by relatively simple models of the nucleon-nucleon (NN) interaction, the four-nucleon (4N) system has remained a challenge for many years. Our aim is to make a serious attempt to overcome this barrier and provide a framework where one can reach an understanding of the data presently available in terms of a fully microscopic calculation. The starting point involves the solution of Alt, Grassberger, and Sandhas, equations [1] for the transition operators involving all (2) + (2) and (3) + 1 channels. For local NN potentials such equations are three-vector variable integral equations which after partial wave decomposition reduce to a set of coupled equations in three continuous variables. Similar equations were recently solved for 4He [2] requiring more than one hundred hours of computing time on a supercomputer. Since scattering calculations require a greater number of channels, some technical compromises must be made. One follows an approach based on the separable representation of subsystem amplitudes in order to reduce the equations to two (or one) continuous variable, Although this procedure increases the number of channels by a factor of three or four, there is a considerable net gain in computing time by the reduction of dimensionality. Our approach is based on the formalism developed by Fonseca [3] more than two decades ago and subsequently used by Haberzettl and Sandhas [4] to express the 2 + 2 subamplitudes in terms of a convolution integral involving two noninteracting pair-propagators. The integral equations we solve are obtained from the original AGS equations after one has: (a) represented the original NN t-matrix by an operator of rank one; (b) represented the resulting 3N t-matrix by a finite rank operator and taken as many terms as needed for convergence. The sole approximation in this approach involves a rank one representation of the 2N t-matrix which may be obtained from the well-known method of Ernest, Shakin, and Taylor [5]. The multi-term representation of the 3N t-matrix is done using the EDPE method developed by Sofianos, McGurk and Fiedeldey [6]. This latest approximation for the 3N t-matrix is well under control since one can, in addition to comparing the finite rank approximation with the original t-matrix results for the 3N observables (cross sections and analyzing powers), check on the convergence rate of 4N observables for increasing rank of the 3N representation. As 0375-9474/98/$19.00 © 1998 Elsevier Science B.V. All rights reserved. PI1 S0375-9474(98)00089-X
676c
A.C. Fonseca/Nuclear Physics A631 (1998) 675c-679c
mentioned betbre the 2 + 2 subsystem is taken care by convolution and theretore it is treated in an exact form. This is particularly important at energies above breakup where moving singularities have to be well taken care. This method was first used by Fonseca [7] in the binding energy calculation of 4He and later confirmed to be accurate by the exact work of Kamada and GI6eckle [8]. The calculations use Yamaguchy-like separable interactions, as well as rank one EST expansions of the Paris, Bonn-A and Bonn-B potentials in channels ISo, 3S1- 3D 1, IP I, 3P0` 3P 1 and 3P 2. At the three-nucleon level we include all 3N subalnplitudes up toj = 7/2+ in all subchannels that are consistent with the above NN partial-wave force components. Idential procedure is used for the 2+2 subamplitudes. All 4N observables are calculated with total angular momentum up to J = 6. In Fig. 1 and 2 we use the Yamaguchy potential in channels iS0, 3S l- 3D1 (corresponding to 4% d-state probability in the deuteron) to study the convergence of the calculation as one increases 3N total angular momentum j from 1/2+ to 7/2*. We denote r = 6666666 a calculation that includes j = 1/2", 1/2", 3/2+, 3/2", 5/2+, 5/2", 7/2+, with 6 terms in the EDPE expansion of each one of the corresponding 3N subamplitudes. This clearly shows that the shape of the experimental data can only be reproduced when higher j's are included.
0 . 4 0
-
•
,
-
-
,
-
,
-
-
•
-
-
•
-
•
x Exp. "~\
",',,\\
~ ---
rr=~6
~'
0.20
~:
~=6~6
//,.',~
,~ ",, )",.a:
\
E
E O
.xx.
0.00
/,<":
,' I ;/ •
-0.20
::=H
\ II /
10.0
30.0
60.0
90.0
120.0
150.0
O~ (d=g.)
Fig. 1. Differential cross section for the 2H(d,p)3H reaction at Ed = 6.1 MeV. The data is from ref. [9]. The calculation uses the Yamaguchy interaction in channels IS0 and 3S 1- 3D 1 alone. The r = 6 curve corresponds to including 6 terms in the EDPE expansion of the I/2 + 3N subamplitudes alone, while other curves correspond to adding 3N subamplitudes with higher j and rank Nj
180.0
"0"400.0
,
,
J , 30.0
i# i
~,
.
--
r~86
....
r=6666666
4
i
'--"
. . . . . . . . . 60.0 90.0 120.0 Ang. (C.M.)
, . 150.0
.
180.0
Fig. 2. Same as in Fig. for the tensor analyzing power T22
Next we study rank convergence in the EDPE expansion of the 3N subamplitudes by showing in Fig.3 and Fig.4 results of our calculations for rank 6 (solid curves) and rank 8 (dashed curves) expansions of the 3N subamplitudes. Each pairs of curves correspond to the
677c
A. C. Fonseca /Nuclear Physics A631 (1998) 675c-679c
same number of 3N subamplitudes included such that r = 6 and r = 8 correspond to j = 1/2 + alone, while r = 66 and r = 88 include j = I/2 + and 1/2-. Although there are some small changes rank 6 expansions may be considered adequate. 0.40
-
,
•
,
-
,
"
,
"
"
.
"
"
0,020
I III ::12212
0.2O
"3- /i
i zl
.x : I S
0.010 hl
0.00 ::z=
- -
r=
~//7 0000
--- - - -
r=8
r=88 r=6868
_0.400 0
.
.
i
30.0
.
,
i 60.0
,
,
.
90.0
.
.
.
. . 120.0
.
.
. 150.0
Ang. (C.M.)
Fig.3. Tensor analyzing power T22 for the 2H(d,p)3H reaction at Ed = 6. l MeV. The data is from ref. [9]. The calculation uses the Yamaguchy interaction in channels IS0 and 3S l- 3D b The solid curves correspond to rank 6 expansions while the dashed curves to rank 8 expansions.
180.0
-0.0100
0
,
,
, 30.0
,
, 60.0 Ang. (C.M.)
,
,
,
I\ 90.0
,
, t20.0
Fig.4. Same as in Fig.3 for the tensor analyzing power T21 for the 2H(d,d)2H reaction as 6.1 MeV. The data is from ef. [10].
Subsequently in Figs. 5-8 we show the results of our calculations for rank one EST representations of the Paris, Bonn-a and Bonn-b potentials in channels tS 0 and 3S I- 3D1, and compare with the corresponding calculation for the Yamaguchy potential. To save computing time we use rank two for the EDPE expansion o f j = 5/2 + and 5/2- 3N subamplitudes and rank one for j = 7/2 +, since previous tests indicated r = 6666221 to be almost identical to r = 6666666. Because the Paris potential leads to considerable less binding for 3H than the other three potentials we use, most of the difference may result from on-shell N-(3N) momentum missmatch between calculations corresponding to different potentials. Nevertheless Fig.5 shows that only realistic interactions can reproduce the 90 ° maximum in the differential cross section for 2H(d,p)3H. Finally in Fig.9-10 we show the effect of adding NN p-wave force components to the existing IS 0 and 3S l- 3D 1 components. Although the calculation is preliminary due to the number of 3N subamplitudes included (j < 3/2) and resulting 3N channels subject to particlepair maximum orbital angular momentum I _<3, we find d d ~ dd tensor observables to be very sensity to the NN p-waves. When all needed (3N) subamplitudes are added the curves will have a different shape but the sensitivity to NN p-waves will remain. Given that these calculations are very complicated, we find the need for benchmark comparisons between different groups now interested in the four nucleon system. Nevertheless from the results presented we find that 4N scattering observables are a much greater challenge than 3N scattering observables.
678c
,4. C. Fonseca/Nuclear Physics A 631 (1998) 675c-679c
100
. ~', ~'~" ~,.
E 0
x --........ ---
Exp. r=6666221 Yankag. r=6666221 Bonn-b r=6666221 Bonn,a
," ,,'~
.. . 0.20
10
- r - t - TT T-I--rT
0.00 I
.% -0.20
--..... 1
.
,
0.0
,
,
,
30.0
,
,
,
60.0
i
.
.
g0.0
.
.
.
.
120.0
.
r--6666221 Bonn-a r=6666221 Paris
.
150.0
180.0
-0.40
0.0
'
,
' ' 30.0
'
' ' 60.0
e~ (deg.)
.
........
.....
,
.
,
.
./-
•
-,
/
",
'~
iI
~',
0.010
0.000
-0.010
0.o
30.0
60.0 Ang. (C.M.)
180.0
0.020
..-
IiiIIl
-0.010
. . 150.0
, / ~ " ,
iIII \
0.010
-0.030
.
r=6666221 Yamag. r=6666221 Bonn-b 6666221 B . . . . . .
.
Fig.6. Same as in Fig.5 for the tensor analyzing power T22.
Fig.5. Differencial cross section for the ZH(d,p)3H reaction at Ed = 6.1 MeV. The data is from ref. [9]. The calculations correspond to rank one EST representations of realistic interactions in channels IS0 and 3S l - 3D 1.
0.030
' ~ • . ' . 90.0 120.0 Ang. (C.M.)
80.0
Fig.7. Tensor analyzing power T20.for the 2H(d,d)ZH reaction at Ed = 6.1 MeV. The data is from ref. [ 10].The calculations correspond to rank one EST representations of realistic interactions in channels IS0 and 3S 1 - 3D 1.
120.0
0.0
30.0
60.0 Ang. (C.M.)
Fig.8.Same as in Fig.7 for the tensor analyzing power T21.
90.0
120.0
679c
A. C, Fonseca/Nuclear Physics A631 (1998) 675c-679c
The authors would like to thank J. Heidenbauer for providing the rank one EST representation of the 2N realistic interactions, NCSC for providing the computing support needed for much of the development of these codes and TUNL for the hospitality and support during the summer visits. Finally we recognize the support of JNICT grant CERN/1101/96 and PRAXIS XXI grant that made possible the purchase of a dedicated machine.
0.030
0.020
-
•
,
•
.
,
"
p-waves
- -
no NN
---
with NN p-waves
0.010 0.010 0.000 0.000
\
~
-0.010 i /
-0.010
0.020 ""--IIIIII -0"0300.0
30.0
60,0
- -
no NN p-waves
---
with NN
I
p-waves
I
-0.020
90.0
Ang. (C.M.)
Fig.9.Tensor analyzing power T20.for the 2H(d,d)2H reaction at Ed = 6.1 MeV. The data is from ref. [10]. Only 3N subamplitudes up to j = 3/2 where included
120.0
-0.030
,
0,0
,
i
30,0
,
,
i
,
,
60.0
Ang.
i
90.0
,
,
120.0
(C.M.)
Fig. 10.Same as in Fig.9 for the tensor analyzing power T22.
REFERENCES 1. E.O. Alt, P. Grassberger, and W. Sandhas, Phys. Rev. C1, 85 (1970). 2. H. Kamada and W. Gl6eckle, Phys. Lett., B292, 1 (1993). 3. A.C. Fonseca and P.E. Shanley, Phys. Rev. D13, 2255 (1976); ibidem, Phys. Rev. C14, 1343 (1976). 4. H. Haberzettl and W. Sandhas, Phys. Rev. C24, 359 (1981). 5. J. Ernest, C.M. Shakin and R.M. Thaler, Phys. Rev. C8, 46 (1973). 6. S. Sofianos, N.J. McGurk and H. Fiedeldey, Nucl. Phys, A318, 295 (1979). 7. A.C. Fonseca, Phys. Rev. C40, 1390 (1989). 8. H. Kamada and W. G10eckle, Nucl. Phys,, A548, 205 (1992). 9. W. Gruebler et al., Nucl. Phys. A193, 129 (1972); ibidem A369, 381 (1981). 10. W. Gruebler et al., Nucl. Phys. A193, 149 (1972).