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ΛN-interaction and spin-orbit coupling in hypernuclei
Volume 69B, number 2
AN-INTERACTION
PHYSICS LETI'I-RS
1 August 1977
AND SPIN-ORBIT COUPLING
IN H Y P E R N U C L E I
R. BROCKMANN and W. WEISE
Institute of Theoretical Physics, University of Regensburg. D-84 Regensburg, W. Germany Received 19 April 1977 The spin-orbit interaction for a A in a hypernucleus is discussed within the framework o f a relativistic Hartree model using effective scalar and vector boson exchange interactions as input. Meson exchange models of such AN-interactions indicate that the relevant parts of them should be rnuch weaker than the corresponding pieces of the nucleon-nucleon force. As a consequence, the central Hartree potential for a A is considerably less attractive than that for a nucleon, and the spin-orbit splitting in the p-shell of 160 is rcduced to about 1 MeV.
Recent theoretical interpretations of the structure of hypernuclear excited states as seen in (K, n) reactions [ 1 - 3 ] have focused on the relevance of strangeness analogue states [ 4 - 6 ] , quasifree mechanisms [7] as well as on the discussion of single particle properties of a A-hyperon inside a nucleus [ 8 - 9 ] . We would like to concentrate here on this latter point. According to the phenomenological analysis of ref. [8], the single particle central potential well for a A is expected to be only about half as deep as the corresponding one for a neutron. Moreover, tile spin-orbit potential U~o" for a A appears to be much weaker than the spin-orbit interaction UNo. for a nucleon:
IU :o.t < 0.31UsN.o.I.
(i)
The purpose of this work is to try to relate such properties back to some basic differences between the Anucleon and the nucleon-nucleon interaction. We recall first that a microscopic understanding of the spin-orbit interaction in nuclei is possible on the basis of relativistic boson exchange mechanisms. Our approach will be based on a Hartree-Dirac description of finite nuclei [ 10, 11 ], using as an input the exchange of isoscalar effective scalar and vector bosons. This approach will be used to describe the single particle motion of either a nucleon or a A-hyperon inside the nucleus. In either case the relevant scalar and vector exchange interactions have to be specified. We shall point out that, according to currently existing meson exchange pictures of the baryon-baryon force, both the scalar and vector exchange A-nucleon interaction should be considerably weaker than the corresponding pieces of the nucleon-nucleon interaction. This will have consequences for the depth of the ten-
tral A-Hartree potential as well as for the spin-orbit coupling. The Hartree-Dirac equation for a baryon (nucleon or A) of mass M moving in a fintie nucleus is l - i v" V - 70E= + ~,~(r) + M I ¢,~(r) = 0,
(2)
where ~,,~ denotes the Dirac spinor, a specifies the single particle orbit and E~ is the corresponding single particle energy. Furthermore, c.y~, is the Hartree potential which is chosen to be of the form [I 0]
QY~(r) = ~ / d 3 r ' ~ ( r ' ) I V S ( I r - r ' [ ) t3-~a
(3)
+ VV(Ir - r'l)] ~bo~r'). Here V s,v are two-body interactions generated by isoscalar effective scalar and vector boson exchange, VS(rl 2) .
gs~s e - m s r l 2
. 4n.
. rl 2
g2 VV(rl 2) = ~ n 7u(1 )
,
(4)
e._mvrl 2 rl2
7u(2)"
(5)
In the Hartree approximation, such an interaction is supposed to represent most of the relevant features of tile baryon-baryon force [12], since other spin-isospin dependent mechanisms average out almost completely in eq. (3). As in ref. [14], in case of ordinary nuclei, the boson exchange parameters will be adjusted such that the binding energy and equilibrium density of nuclear matter is reporduced in tile same (relativistic Hartree) approximation [12]. This determines the ratio of boson-nucleon coupling constants, gSN and gVN, to 167
Volume 69B, number 2
PItYSI('S LE'VFI'RS
1 August 1977
boson masses,
ms/
\ m v / M N =14.0,
(6)
where M N is the nucleon mass. For effective boson masses m s = 560 MeV, m v = 780 MeV, we have g2N/ 4rr = 7.5,g~rN/4rr = t0.8, respectively. With these parameters, a self-consistent solution of eqs. (2), (3) gives very reasonable values for the neutron single particle energies in 160 and, in particular, a spin-orbit splitting in the p-shell of about 7 MeV, according to ref. [ 11 ]. The magnitude of the spin-orbit coupling is quite stable with respect to variation of the effective scalar boson mass [11 ], as long as the ratio of eq. (6) remains fixed. Almost the same results have been obtained in a lowest-order Hartree calculation using four-spinors ~k~ in eq. (3) calculated with a realistic Woods-Saxon potential to construct the Hartree potential cy~. We shall use this simplified procedure if we now consider a A instead of a nucleon to move in the single particle orbit a. St, ch a procedure is well justified according to ref. [91. We would furthermore like to discuss the nature of (isoscalar) scalar and vector boson exchange in the NNand AN-interaction. For that purpose, we note that the parameters eq. (6) are quite close to coupling constants and masses usually used for the " o " and "co" meson in one-boson exchange potentials [13]. Thus we are encouraged to study the corresponding lowest order boson exchange interactions, assuming that higher order ladder sunnnations will affect our conclusions about the magnitude of the spin-orbit force quantitatively, but not qualitatively. The scalar exchange NN-interaction in lowest order proceeds through two-pion exchange (TPE). A large part of it goes through direct and crossed (2rr)-exchange involving a A-isobar in the intermediate state [16, 17]. The corresponding TPE-AN-interaction is shown schematically in fig. la. If we neglect the small mass splitting between A and ?2, most of the difference between the NN- and AN-TPE-interaction comes from isospin factors for the NNTr-vertices in one case and for the AZTrvertices in the other case. In fact, such a TPE model would give for the scalar exchange interaction, eq. (4): (2rr-exchange): 168
VAN --3 \G~.Nn J
NN'
®
@
Fig. 1. Important isoscalar t-channel contributions to the ANinteraction: (a) two-pion exchange; (b) combined exchange of pion and p-meson. Not shown here are the corresponding crossed meson exchange pieces. The shaded area on the right hand side of each diagram contains mostly a nucleon and a Aisobar. where the 1/3 is an isospin factor and the G's are coupling constants at the respective vertices. Experimental values are .) G2z~r/47r = 1 2 - 1 3 (ref. [14] ), together with G~qN~/4n = 14. Thus, the scalar exchange AN interaction should be weaker by roughly a factor of 3 compared to the corresponding piece of the NN force. Ttfis simple feature holds also in more refined treatmen ts of the TPE-AN-potential [ 15 ]. Corrections to the simple relation, eq. (7), are expected in a nucleus because of the Pauli exclusion principle acting on tile nucleon intermediate state in VSNN (ref. [ 17 1), whereas no such effect occurs for the intertnediate .E in fig. la. This should lead to a r~-S /V NN, S which we slight enhancement of the ratio %xN/ shall, however, disregard in our context. lsoscalar vector meson exchange is generated by 3zr exchange degrees of freedom. According to recent work of Durso et al. [l 6 ] , only a minor piece of this comes from bare co meson exchange :1:1, whereas the larger part is due to combined 7r,o-exchange, in a way similar to TPE, with a p meson substituted for one of the rt's. It is then very suggestive to think of the corresponding piece in the AN-interaction in the way shown in fig. lb. This would immediately give:
- ,w - 1 [G:xz,~ .Gazp~ ('up-exchange):
V~ N --~ ~G--~N~ GNN. .) V ~ .
(8)
The couplings of a p-meson to the baryon octet are rather badly determined. If tensor coupling is assumed to dominate, then SU(3) gives GA~,JGNN p = 2aM/X/~-, where a M ~ 0.6. From there one would obtain V~x~ 0.2 V ~ . For the bare co meson exchange, we have VW AN = COS 0v'V~, N , • 2 ) 1 In fact, the simplest quark model would give g~NN/47r < 4.5•
(9)
Volume 69B, number 2 30
PHYSICS LI"ITI-RS t*.0
50
ms/m n
'°^o
1 August 1977
m v = 780 MeV. We observe that the spin-orbil splitting in the p-shell is now very small, only about 1 MeV, and quite stable against variations of ms. For m s = 4mrr = 560 MeV, the A-single particle potential has a depth of about - 3 0 MeV :t:3 . Increasing m v to 900 MeV raises the single particle energies by only about 10%, with almost no change of the spin orbit splitting, as long as is kept constant. A final remark concerns other possible choices of boson exchange parameters. Assume, for example, that the effective vector boson is identified with just V a bare co-meson, so that V~N = cos 0 v • VNN -~0.6 V~N according to eq. (9), with = 10.8. Together with eq. (7), this leads to an overly strong repulsion in the AN-interaction such that the A c a n n o t even be bound. In s u m m a r y , we come to the conclusion that if the mechanisms of fig. 1 are o f i m p o r t a n c e in the scalar and vector boson exchange AN-interaction, then the central Hartree potential as well as the spin-orbit interaction for a A in a hypernucleus should be considerably reduced in comparison with standard nuclei. In essence, these results support the phenomenological analysis of ref. [8].
grimy
-15
t M eVI
i
i
Fig. 2. Single particle energies ~ = Ec~ - MA of a A in I~O, calculated in a relativistic Dirac-Hartree model. The Hartree potential is generated from scalar and vector boson exchange interactions according to eqs. (3) and (10). A vector boson mass m v = 780 MeV has been chosen, and the scalar mass m s (given here in units of the pion mass ran) has been varied, keeping the ratio fixed as described in the text.
gs/ms
with a mixing angle 0 v = 40 G, (i.e. cos 0 v ~ 0.6) assuming vanishing ~NN-coupling. Now, if we identify VNN v O9 = VNN + V~q~ and use an effective coupling strength ~ 7 for the vector boson coupling due to no-exchange, according to ref. [16], then for the corresponding V~N = V~N + V ~ our result would again be a reduction of roughly = 1/3, assuming a p p r o x i m a t e l y equal range for w- and no-exchange. Thus, if the mechanisms shown in fig. I are ilnportant in the scalar and vector exchange AN-interaction, as their counterparts are in the corresponding pieces of the NN-force, we can speak of a c o m m o n reduction factor of roughly 3 being involved. This will then reduce the strength o f both the central Hartree potential as well as of the spin-orbit interaction for a A moving in a hypernucleus. In fig. 2 we show results o f a relativistic Hartree calculation where the Dirac e q u a t i o n , eq. (2), is solved for a A-hyperon in 160. The Hartree potential, eq. (3), is generated by nucleons described by spinors calculated with a Woods-Saxon potential :t:2. The input two-body interactions in eq. (3) are chosen to be
G~,~N/47r
VVAN/V~N
-_ " jl V N N ,
(10)
s,v
where the parameters of VNN are those given by eq. (6). The scalar boson mass m s has been varied keeping 2 2 fixed, and the vector mass m v was chosen to be
gs/ms
,2 The parameters or the potential are: radiusR = 3 fro, surface thickness a = 0.66 fm appropriate/'or 160.
g~y/47r
,3 We refer here to the equivalent single-particle potential to be used in a Schr6dinger equation which reproduces the same single particle energies as eq. (2).
J 1 ] B. Povh, Rep. Prog. Phys. 39 (1976) 823. [2} W. BriJckner et al., Phys. Lett. 62B (1976) 481. [3] K. Kilian, Proc. Conf. on Meson-nuclear interactions, Pittsburgh, 1976, eds. P.D. Barnes et al., (AIP Conf. Proc. No. 33, 1976). [4] A.K. Kerman and H.J. Lipkin, Ann. of Phys. 66 (1971) 738. [5] A. Gal, Adv. Nucl. Phys. 8 (1975) 1. [61 R.tt. Dalitz and A. Gal, Phys. Rev. Lett. 36 (1976) 362. [7[ R.H. Dalitz and A. Gal, Phys. Lett. 64B (1976) 154. [8] A. Bouyssy and J. ltiifner, Phys. Lett. 64B (1976) 276; A. Bouyssy, to be published. [9] M. Rayet, Ann. of Phys. 102 (1976) 226. [10] L.D. Miller, Ann. of Phys. 91 (1975) 40. [I1 ] R. Brockmann and W. Weise, submitted to Phys. Rev. C. [121 J.D. Walecka, Ann. of Phys. 83 (1974) 491. [ 13] K. Erkelenz, Phys. Reports 13C (1974) 191. [14] M.M. Nagels et al., Nucl. Phys. B 109 (1976) 1. [15] D.O. Riska, Nucl. Phys. B 56 (1973) 445. [16] J.W. Durso, M. Saarela, G.E. Brown and A.D. Jackson, Nucl. Phys. A278 (1977) 445. [171 A.M. Green, Rep. Prog. 39 (1976) 1109.
169
AN-INTERACTION
PHYSICS LETI'I-RS
1 August 1977
AND SPIN-ORBIT COUPLING
IN H Y P E R N U C L E I
R. BROCKMANN and W. WEISE
Institute of Theoretical Physics, University of Regensburg. D-84 Regensburg, W. Germany Received 19 April 1977 The spin-orbit interaction for a A in a hypernucleus is discussed within the framework o f a relativistic Hartree model using effective scalar and vector boson exchange interactions as input. Meson exchange models of such AN-interactions indicate that the relevant parts of them should be rnuch weaker than the corresponding pieces of the nucleon-nucleon force. As a consequence, the central Hartree potential for a A is considerably less attractive than that for a nucleon, and the spin-orbit splitting in the p-shell of 160 is rcduced to about 1 MeV.
Recent theoretical interpretations of the structure of hypernuclear excited states as seen in (K, n) reactions [ 1 - 3 ] have focused on the relevance of strangeness analogue states [ 4 - 6 ] , quasifree mechanisms [7] as well as on the discussion of single particle properties of a A-hyperon inside a nucleus [ 8 - 9 ] . We would like to concentrate here on this latter point. According to the phenomenological analysis of ref. [8], the single particle central potential well for a A is expected to be only about half as deep as the corresponding one for a neutron. Moreover, tile spin-orbit potential U~o" for a A appears to be much weaker than the spin-orbit interaction UNo. for a nucleon:
IU :o.t < 0.31UsN.o.I.
(i)
The purpose of this work is to try to relate such properties back to some basic differences between the Anucleon and the nucleon-nucleon interaction. We recall first that a microscopic understanding of the spin-orbit interaction in nuclei is possible on the basis of relativistic boson exchange mechanisms. Our approach will be based on a Hartree-Dirac description of finite nuclei [ 10, 11 ], using as an input the exchange of isoscalar effective scalar and vector bosons. This approach will be used to describe the single particle motion of either a nucleon or a A-hyperon inside the nucleus. In either case the relevant scalar and vector exchange interactions have to be specified. We shall point out that, according to currently existing meson exchange pictures of the baryon-baryon force, both the scalar and vector exchange A-nucleon interaction should be considerably weaker than the corresponding pieces of the nucleon-nucleon interaction. This will have consequences for the depth of the ten-
tral A-Hartree potential as well as for the spin-orbit coupling. The Hartree-Dirac equation for a baryon (nucleon or A) of mass M moving in a fintie nucleus is l - i v" V - 70E= + ~,~(r) + M I ¢,~(r) = 0,
(2)
where ~,,~ denotes the Dirac spinor, a specifies the single particle orbit and E~ is the corresponding single particle energy. Furthermore, c.y~, is the Hartree potential which is chosen to be of the form [I 0]
QY~(r) = ~ / d 3 r ' ~ ( r ' ) I V S ( I r - r ' [ ) t3-~a
(3)
+ VV(Ir - r'l)] ~bo~r'). Here V s,v are two-body interactions generated by isoscalar effective scalar and vector boson exchange, VS(rl 2) .
gs~s e - m s r l 2
. 4n.
. rl 2
g2 VV(rl 2) = ~ n 7u(1 )
,
(4)
e._mvrl 2 rl2
7u(2)"
(5)
In the Hartree approximation, such an interaction is supposed to represent most of the relevant features of tile baryon-baryon force [12], since other spin-isospin dependent mechanisms average out almost completely in eq. (3). As in ref. [14], in case of ordinary nuclei, the boson exchange parameters will be adjusted such that the binding energy and equilibrium density of nuclear matter is reporduced in tile same (relativistic Hartree) approximation [12]. This determines the ratio of boson-nucleon coupling constants, gSN and gVN, to 167
Volume 69B, number 2
PItYSI('S LE'VFI'RS
1 August 1977
boson masses,
ms/
\ m v / M N =14.0,
(6)
where M N is the nucleon mass. For effective boson masses m s = 560 MeV, m v = 780 MeV, we have g2N/ 4rr = 7.5,g~rN/4rr = t0.8, respectively. With these parameters, a self-consistent solution of eqs. (2), (3) gives very reasonable values for the neutron single particle energies in 160 and, in particular, a spin-orbit splitting in the p-shell of about 7 MeV, according to ref. [ 11 ]. The magnitude of the spin-orbit coupling is quite stable with respect to variation of the effective scalar boson mass [11 ], as long as the ratio of eq. (6) remains fixed. Almost the same results have been obtained in a lowest-order Hartree calculation using four-spinors ~k~ in eq. (3) calculated with a realistic Woods-Saxon potential to construct the Hartree potential cy~. We shall use this simplified procedure if we now consider a A instead of a nucleon to move in the single particle orbit a. St, ch a procedure is well justified according to ref. [91. We would furthermore like to discuss the nature of (isoscalar) scalar and vector boson exchange in the NNand AN-interaction. For that purpose, we note that the parameters eq. (6) are quite close to coupling constants and masses usually used for the " o " and "co" meson in one-boson exchange potentials [13]. Thus we are encouraged to study the corresponding lowest order boson exchange interactions, assuming that higher order ladder sunnnations will affect our conclusions about the magnitude of the spin-orbit force quantitatively, but not qualitatively. The scalar exchange NN-interaction in lowest order proceeds through two-pion exchange (TPE). A large part of it goes through direct and crossed (2rr)-exchange involving a A-isobar in the intermediate state [16, 17]. The corresponding TPE-AN-interaction is shown schematically in fig. la. If we neglect the small mass splitting between A and ?2, most of the difference between the NN- and AN-TPE-interaction comes from isospin factors for the NNTr-vertices in one case and for the AZTrvertices in the other case. In fact, such a TPE model would give for the scalar exchange interaction, eq. (4): (2rr-exchange): 168
VAN --3 \G~.Nn J
NN'
®
@
Fig. 1. Important isoscalar t-channel contributions to the ANinteraction: (a) two-pion exchange; (b) combined exchange of pion and p-meson. Not shown here are the corresponding crossed meson exchange pieces. The shaded area on the right hand side of each diagram contains mostly a nucleon and a Aisobar. where the 1/3 is an isospin factor and the G's are coupling constants at the respective vertices. Experimental values are .) G2z~r/47r = 1 2 - 1 3 (ref. [14] ), together with G~qN~/4n = 14. Thus, the scalar exchange AN interaction should be weaker by roughly a factor of 3 compared to the corresponding piece of the NN force. Ttfis simple feature holds also in more refined treatmen ts of the TPE-AN-potential [ 15 ]. Corrections to the simple relation, eq. (7), are expected in a nucleus because of the Pauli exclusion principle acting on tile nucleon intermediate state in VSNN (ref. [ 17 1), whereas no such effect occurs for the intertnediate .E in fig. la. This should lead to a r~-S /V NN, S which we slight enhancement of the ratio %xN/ shall, however, disregard in our context. lsoscalar vector meson exchange is generated by 3zr exchange degrees of freedom. According to recent work of Durso et al. [l 6 ] , only a minor piece of this comes from bare co meson exchange :1:1, whereas the larger part is due to combined 7r,o-exchange, in a way similar to TPE, with a p meson substituted for one of the rt's. It is then very suggestive to think of the corresponding piece in the AN-interaction in the way shown in fig. lb. This would immediately give:
- ,w - 1 [G:xz,~ .Gazp~ ('up-exchange):
V~ N --~ ~G--~N~ GNN. .) V ~ .
(8)
The couplings of a p-meson to the baryon octet are rather badly determined. If tensor coupling is assumed to dominate, then SU(3) gives GA~,JGNN p = 2aM/X/~-, where a M ~ 0.6. From there one would obtain V~x~ 0.2 V ~ . For the bare co meson exchange, we have VW AN = COS 0v'V~, N , • 2 ) 1 In fact, the simplest quark model would give g~NN/47r < 4.5•
(9)
Volume 69B, number 2 30
PHYSICS LI"ITI-RS t*.0
50
ms/m n
'°^o
1 August 1977
m v = 780 MeV. We observe that the spin-orbil splitting in the p-shell is now very small, only about 1 MeV, and quite stable against variations of ms. For m s = 4mrr = 560 MeV, the A-single particle potential has a depth of about - 3 0 MeV :t:3 . Increasing m v to 900 MeV raises the single particle energies by only about 10%, with almost no change of the spin orbit splitting, as long as is kept constant. A final remark concerns other possible choices of boson exchange parameters. Assume, for example, that the effective vector boson is identified with just V a bare co-meson, so that V~N = cos 0 v • VNN -~0.6 V~N according to eq. (9), with = 10.8. Together with eq. (7), this leads to an overly strong repulsion in the AN-interaction such that the A c a n n o t even be bound. In s u m m a r y , we come to the conclusion that if the mechanisms of fig. 1 are o f i m p o r t a n c e in the scalar and vector boson exchange AN-interaction, then the central Hartree potential as well as the spin-orbit interaction for a A in a hypernucleus should be considerably reduced in comparison with standard nuclei. In essence, these results support the phenomenological analysis of ref. [8].
grimy
-15
t M eVI
i
i
Fig. 2. Single particle energies ~ = Ec~ - MA of a A in I~O, calculated in a relativistic Dirac-Hartree model. The Hartree potential is generated from scalar and vector boson exchange interactions according to eqs. (3) and (10). A vector boson mass m v = 780 MeV has been chosen, and the scalar mass m s (given here in units of the pion mass ran) has been varied, keeping the ratio fixed as described in the text.
gs/ms
with a mixing angle 0 v = 40 G, (i.e. cos 0 v ~ 0.6) assuming vanishing ~NN-coupling. Now, if we identify VNN v O9 = VNN + V~q~ and use an effective coupling strength ~ 7 for the vector boson coupling due to no-exchange, according to ref. [16], then for the corresponding V~N = V~N + V ~ our result would again be a reduction of roughly = 1/3, assuming a p p r o x i m a t e l y equal range for w- and no-exchange. Thus, if the mechanisms shown in fig. I are ilnportant in the scalar and vector exchange AN-interaction, as their counterparts are in the corresponding pieces of the NN-force, we can speak of a c o m m o n reduction factor of roughly 3 being involved. This will then reduce the strength o f both the central Hartree potential as well as of the spin-orbit interaction for a A moving in a hypernucleus. In fig. 2 we show results o f a relativistic Hartree calculation where the Dirac e q u a t i o n , eq. (2), is solved for a A-hyperon in 160. The Hartree potential, eq. (3), is generated by nucleons described by spinors calculated with a Woods-Saxon potential :t:2. The input two-body interactions in eq. (3) are chosen to be
G~,~N/47r
VVAN/V~N
-_ " jl V N N ,
(10)
s,v
where the parameters of VNN are those given by eq. (6). The scalar boson mass m s has been varied keeping 2 2 fixed, and the vector mass m v was chosen to be
gs/ms
,2 The parameters or the potential are: radiusR = 3 fro, surface thickness a = 0.66 fm appropriate/'or 160.
g~y/47r
,3 We refer here to the equivalent single-particle potential to be used in a Schr6dinger equation which reproduces the same single particle energies as eq. (2).
J 1 ] B. Povh, Rep. Prog. Phys. 39 (1976) 823. [2} W. BriJckner et al., Phys. Lett. 62B (1976) 481. [3] K. Kilian, Proc. Conf. on Meson-nuclear interactions, Pittsburgh, 1976, eds. P.D. Barnes et al., (AIP Conf. Proc. No. 33, 1976). [4] A.K. Kerman and H.J. Lipkin, Ann. of Phys. 66 (1971) 738. [5] A. Gal, Adv. Nucl. Phys. 8 (1975) 1. [61 R.tt. Dalitz and A. Gal, Phys. Rev. Lett. 36 (1976) 362. [7[ R.H. Dalitz and A. Gal, Phys. Lett. 64B (1976) 154. [8] A. Bouyssy and J. ltiifner, Phys. Lett. 64B (1976) 276; A. Bouyssy, to be published. [9] M. Rayet, Ann. of Phys. 102 (1976) 226. [10] L.D. Miller, Ann. of Phys. 91 (1975) 40. [I1 ] R. Brockmann and W. Weise, submitted to Phys. Rev. C. [121 J.D. Walecka, Ann. of Phys. 83 (1974) 491. [ 13] K. Erkelenz, Phys. Reports 13C (1974) 191. [14] M.M. Nagels et al., Nucl. Phys. B 109 (1976) 1. [15] D.O. Riska, Nucl. Phys. B 56 (1973) 445. [16] J.W. Durso, M. Saarela, G.E. Brown and A.D. Jackson, Nucl. Phys. A278 (1977) 445. [171 A.M. Green, Rep. Prog. 39 (1976) 1109.
169