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Structure of weak nucleon currents and charge asymmetry in nuclear beta decay
NUCLEAR PHYSICS A ELSEVIER
Nuclear Physics A588 (1995) 165c--170c
S t r u c t u r e of w e a k n u c l e o n c u r r e n t s a n d c h a r g e a s y m m e t r y
in n u c l e a r
beta decay K. Koshigiri a , R. Morita b and M. Morita ¢ aDepartment of Physics, Osaka Kyoiku University, Kashiwara, Osaka 582, Japan bWomen's Junior College, Josai University, Sakado, Saitama 350-02, Japan eJosai International University, Gumyo, Togane, Chiba 283, Japan Structure of weak nucleon currents is studied from the beta decays in the A--12 system. We have investigated the experimental data on the beta-ray angular distribution in the aligned 12B and 12N recently given by the Osaka group with our formulas to find a new limit for the induced tensor current. A possible difference of the ratio y of the axial charge to Gamow-Teller matrix elements between the beta decays of 12B and 12N is calculated theoretically in connection with the ft-value asymmetry, since it affects the evaluation of the strength of the induced tensor current. 1. I N T R O D U C T I O N Low energy nuclear weak processes are described by the current-current type interaction based on the standard model. We shall adopt the following forms of the vector and axial vector currents of the nucleons [1]: HI = (V~ + Ax)(¢~7~(1 +
~s)e~)/v~+ h.c.
(I)
with
(2) Aa = epTS(fAT~ + fTa:,pkp + ifpk:~)~b,~.
(3)
Six terms in (2) and (3) are called the main vector, weak magnetism, induced scalar, main axial vector, induced tensor, and induced pseudoscalar currents, respectively. The induced scalar and the induced tensor terms have G-parities different from those of the main vector and axial vector currents, respectively, and are called the second-class currents [2]. Since the CVC theory implies the isotriplet nature of the electromagnetic current and the weak vector current, the induced scaler current should be equal to zero. The G-parity conservation of the weak currents gives fT = 0. There are, however, several possibilities of having a nonzero but very small value of fT by charge asymmetry in strong interactions due to quark-mass difference [3], exchange current effects due to w -+ rr + e + v [4], electromagnetic corrections, etc. 0375-94741951509.50 © 1995 Elsevier Science B.V. All rights reserved. SSDI 0375-9474(95)00118-2
166c
K. Koshigiri et aL / Nuclear Physics A588 (1995) 165c-170c
We would like to introduce here the results of our recent theoretical analysis of the new experimental data on the beta-ray angular distributions in aligned 12B and 12N which have been provided us by the Osaka group [5]. It gives a new limit for fT and a renewed value of the axial charge matrix element [6,7] in Sec. 2. It is interesting to note that there is a possibility to produce a small difference in the axial charge matrix elements between electron and positron decays by assuming the separation-energy difference between the respective beta-transforming nucleons. It is also an origin of the ft-value asymmetry and the degree of charge asymmetry could be different between the Gamow-Teller matrix elements and the axial charge matrix elements. This is discussed in Sec. 3. 2. B E T A - R A Y A N G U L A R D I S T R I B U T I O N
I N T H E A----12 S Y S T E M
The beta-ray angular distribution in the 1+ -4 0+ transition of the aligned 12B and 12N is given by
N( E, O) = Bo( E) + B2( E) A P2(cos 0),
(4)
where .4 is the degree of nuclear alignment and E is the total energy of the electron. For an illustrating pro'pose, we give an approximate formula for anisotropy coefficients a T in the following, where the full formula is given elsewhere [1,8]:
B2(E)/Bo(E) ,~ c~÷ E .~ [ i x T (2M fT/fA) -- y ] /3M x E
(5)
with X
- [ - f v (1 + ( f r
× .//o')
) + 2M/w] /IA ~ -4.706 fV/IA
(6)
and
y = - 2 M i f %v/ ]o'.
(7)
Here the upper and lower signs refer to the electron and positron decays, respectively. The parameter x is the ratio of the space component of the vector and axial vector current matrix elements, while the parameter y is the ratio of the time component (axial charge) to space component of the main axial vector current. The beta-ray angular distribution function, B2(E)/Bo(E), is measured experimentally as a function of the electron energy E and its slope gives the anisotropy coefficient a~:. In our recent work [6,7], we adopt the formalism which includes the effect of higher order partial waves of leptons, Coulomb corrections of the finite-size nucleus and radiative corrections. A chi-square analysis of the experimental data [5] has been made with the two parameters fT/fh and 8~. The latter is defined by y = yIA (1 + ~ ) + y E c .
(8)
Here YIA is 3.17 given in the impulse approximation with the Hauge-Maripuu wave function, and yP.c is 1.30 due to the effect of the soft pion exchange currents [9]. The other matrix elements are also calculated in a similar way.
K. Koshigiri et aL / Nuclear PhysicsA588 (1995) 165c-170c
167c
The analysis of the experimental data of the Osaka group leads to the following numerical results:
(9)
2MfT/fA = 0.21 4- 0.14, and
(10)
y = 5.10 + 0.16.
For more detail of the axial charge parameter y, the core polarization effects should be taken into consideration. In fact, the first order core polarization effect gives us YcP = -0.44 [9], while all-order effect of the core polarization reduces the magnitude of YcP [1O]. In the chi-square fit, 8y represents core polarization and some other effects, such as the multipion exchange currents [11], which are not explicitly taken into account here. The effect is 8~ = 20%, compared with the effects of exchange currents YEc/YI, = 41%. Similar but large enhancement of the axial charge operator was reported in the first forbidden beta decays of the heavy mass region [12]. 3. C H A R G E A S Y M M E T R Y OF N U C L E A R M A T R I X E L E M E N T S I N T H E MIRROR BETA DECAY In the early days, the ft-value asymmetry of mirror beta decay was studied whether it comes fi'om the nuclear origins or fundamental interactions. Thus the mirror asymmetry of Gamow-Teller matrix elements are theoretically investigated [13-15]. However, in the investigations of beta-ray angular correlation experiments, it is usually assumed that the charge symmetry holds for the mirror beta decays, since the physical observables in correlation experiments are related to the ratio of the nuclear matrix elements. We would like to study the consequence of deleting this assumption. Blomqvist pointed out that the separation energy of the proton in I~N is 0.601 MeV while that of the neutron in :2B is 3.370 MeV, and that the beta-decaying proton wave function is spreading compared to the neutron. This binding energy effect leads to the mismatch of the overlap integral in nuclear matrix elements. The axial charge operator is nuclear momentum dependent and its derivative term for nucleon coordinate gives the different contribution for this mismatch. Employing the simple harmonic oscillator model and the Blomqvist's model which explain the ft-value asymmetry, we have studied a charge asymmetry effect on the parameter y. Since the ratio of the orbital angular momentum to Gamow-Teller matrix element is small in the A=12 system, the parameter x is almost given nuclear-model independently and its charge asymmetry is negligible. We have used the simple harmonic oscillator model for the nucleon wave function to find a tendency of the charge asymmetry effects. The relatively weak binding of the proton in 12N is simulated by the enhancement of the oscillator parameter for the Opl/2 proton wave fnnction as b --+ b (1 + :). Nucleon wave functions of the other states are set to have an equal oscillator parameter b. If we take the single particle model for the initial and final states, ft-value asymmetry is given as follows:
ft+/ft-= /rr_//
Or+ 2
=[Iop[b,b]/ I o p [ b , b ( l + e ) ] l = ~ l + 5 / 2 : ~,
(11)
168c
K. Koshigiri et al. / Nuclear Physics A588 (1995) 165c-170e
where I0p [a,/3] = fo°° R0p(r; a) Rop(r;/3) r 2 dr = (2a/J/(a 2 +/32)) 5/2
(12)
and Rop(r; a) is the nucleon radial wave function with an oscillator parameter a, while the parameter y for the electron and positron decays are given by the following form in the above approximation scheme,
y+/y-=
i
~sr / i +
%r
x(ft+/ft -)~1+5/3e-5/6e
2.
(13)
If we set the parameter s to reproduce the experimental value of the f t - a s y m m e t r y 5 = f t + / f t - - 1 = 13%, we get ¢=0.22, and substituting this value in (13) we have A y = (y+ - y - ) / 2 = 5/4e - 5/8e 2 = 0.25. In Table 1, we show the results of a harmonic
oscillator model with the general 0p-shell configuration mixing of Hauge and Maripuu, where oscillator parameters are set as b(proton) = (1 + e) b (neutron) and b(OPll2) = (1 + 2e)b (Opa/~) with e = 0.052 which is fixed from the experimental ft-value asymmetry as above. Now we follow the method of Blomqvist except for the nuclear model, which we adopt the Hauge-Maripuu model and take seven parent states for the residual A = l l system. The Woods-Saxon central potential depth parameter is varied so as to reproduce the correct separation energy for the decaying nucleon for each parent states. Potential parameters other than that of the central potential depth are the same for those of the Blomqvist model. In Table 1, the numerical results which indicate the charge asymmetry of nuclear matrix elements are summarized for three models. The W S W F model is similar to the Blomqvist model where the fixed Woods-Saxon central potential of 40.2 MeV is used by neglecting the parent dependence of the potential. Table 1 ft-value and y asymmetry in the mirror beta Model 5(%) yHOWF 12.1 3.49 WSWF 7.2 3.36 Blomqvist 10.8 3.42 We use the following notation: 5 = f t + / f t - and A y = (y+ - y - ) / 2
decay of the A=12 system y+ y~. 3.94 3.71 3.63 3.50 3.62 3.52 1, y,~ = ( y - + y+)/2
Ay 0.22 0.13 0.10
Summarizing these three cases in Table 1, we have A 9 = 0.10 ,,~ 0.22, and the induced tensor coupling constant is derived from the following formula when the charge asymmetry of the parameter 9 is included in (5), 2 M f T / fA = X + A y -- (3211/2) (a_ - oL+).
(14)
With this equation and the numerical value of A y in (14), we have to interpret (9) as = [ (0.3 ,-~ 0.4) :1:0.14 ] / 2 M .
fT/fa
K. Koshigiri et al. / Nuclear Physics A588 (1995) 165c-170c
169c
4. S u m m a r y We have deduced the induced tensor coupling constant from the experimental data of the beta-decay alignment correlation coefficients. We adopt the models which incorporate the binding energy effects of beta-decaying nucleon wave function. These models almost reproduce the ft-value asymmetry of the mirror beta decay and give the asymmetry of the parameter y which is the ratio of the axial charge to Gamow-Teller matrix elements. It is shown that the parameter y is larger for positron decay than for the electron decay by several percent and the upper limit of the induced tensor current derived from the present analysis is about 10% of the weak magnetism term, if we adopt the experimental results of Ref. [5]. Since this seems to be rather large from the theoretical point of view, those for a higher precision would be greatly appreciated. Further investigation on Ay is useful. We would like to express our hearty thanks to Professor T. Minamisono for providing us their experimental data before publication. This work is partly supported by Grant-in-Aid for Scientific Research, The Ministry of Education, Science and Culture. REFERENCES 1. M. Morita, M. Nishimura, A. Shimizu, H. Ohtsubo and K. Kubodera, Prog. Theor. Phys. Suppl. No. 60 (1976) 1. 2. S. Weinberg, Phys. Rev. 112 (1958) 1375. 3. J . F . Donoghue and B. R. Holstein, Phys. Rev. D25 (1982) 206. 4. I(. Kubodera, J. Delorme and M. Rho, Nucl. Phys. B66 (1973) 253. 5. T. Minamisono, A. Kitagawa, K. Matsuta and Y. Nojiri, Hyperfine Interactions 78 (1993) 77. 6. A part of the present work was presented in a paper by M. Morita, R. Morita and K. I(oshigiri, Proceedings of International Synposium on Spin and Isospin Responses and Weak Processes in Hadron and Nuclei, Osaka, 1994, Nucl. Phys., in press. 7. M. Morita, R. Morita and K. Koshigiri to be published. 8. K. Koshigiri, M. Nishimura, H. Ohtsubo and M. Morita, Nucl. Phys. A319 (1979) 301. 9. I(. I(oshigiri, H. Ohtsubo and M. Morita, Prog. Theor. Phys. 66 (1981) 358. 10. I(. Koshigiri, I,:. I(ubodera, H. Ohtsubo and M. Morita, (ed.) M.Morita et al., Nuclear Weak Process and Nuclear Structure, World Scientific, Singapore, 1989, p.52 11. M. Kirchbach, D. O. Riska and K. Tsushima, Nucl. Phys. A542 (1992) 616. 12. E. I(. Warburton, Phys. Rev. C44 (1991) 233. E. I(. Warburton and I. S. Towner, Phys. Lett. B294 (1992) 1. 13. J. Blomqvist, Phys. Lett. B35 (1971) 375. 14. D. H. Wilkinson, Phys. Rev. Lett. 27 (1971) 1018. 15. I. S. Towner, Nucl. Phys. A216 (1973) 589.
Nuclear Physics A588 (1995) 165c--170c
S t r u c t u r e of w e a k n u c l e o n c u r r e n t s a n d c h a r g e a s y m m e t r y
in n u c l e a r
beta decay K. Koshigiri a , R. Morita b and M. Morita ¢ aDepartment of Physics, Osaka Kyoiku University, Kashiwara, Osaka 582, Japan bWomen's Junior College, Josai University, Sakado, Saitama 350-02, Japan eJosai International University, Gumyo, Togane, Chiba 283, Japan Structure of weak nucleon currents is studied from the beta decays in the A--12 system. We have investigated the experimental data on the beta-ray angular distribution in the aligned 12B and 12N recently given by the Osaka group with our formulas to find a new limit for the induced tensor current. A possible difference of the ratio y of the axial charge to Gamow-Teller matrix elements between the beta decays of 12B and 12N is calculated theoretically in connection with the ft-value asymmetry, since it affects the evaluation of the strength of the induced tensor current. 1. I N T R O D U C T I O N Low energy nuclear weak processes are described by the current-current type interaction based on the standard model. We shall adopt the following forms of the vector and axial vector currents of the nucleons [1]: HI = (V~ + Ax)(¢~7~(1 +
~s)e~)/v~+ h.c.
(I)
with
(2) Aa = epTS(fAT~ + fTa:,pkp + ifpk:~)~b,~.
(3)
Six terms in (2) and (3) are called the main vector, weak magnetism, induced scalar, main axial vector, induced tensor, and induced pseudoscalar currents, respectively. The induced scalar and the induced tensor terms have G-parities different from those of the main vector and axial vector currents, respectively, and are called the second-class currents [2]. Since the CVC theory implies the isotriplet nature of the electromagnetic current and the weak vector current, the induced scaler current should be equal to zero. The G-parity conservation of the weak currents gives fT = 0. There are, however, several possibilities of having a nonzero but very small value of fT by charge asymmetry in strong interactions due to quark-mass difference [3], exchange current effects due to w -+ rr + e + v [4], electromagnetic corrections, etc. 0375-94741951509.50 © 1995 Elsevier Science B.V. All rights reserved. SSDI 0375-9474(95)00118-2
166c
K. Koshigiri et aL / Nuclear Physics A588 (1995) 165c-170c
We would like to introduce here the results of our recent theoretical analysis of the new experimental data on the beta-ray angular distributions in aligned 12B and 12N which have been provided us by the Osaka group [5]. It gives a new limit for fT and a renewed value of the axial charge matrix element [6,7] in Sec. 2. It is interesting to note that there is a possibility to produce a small difference in the axial charge matrix elements between electron and positron decays by assuming the separation-energy difference between the respective beta-transforming nucleons. It is also an origin of the ft-value asymmetry and the degree of charge asymmetry could be different between the Gamow-Teller matrix elements and the axial charge matrix elements. This is discussed in Sec. 3. 2. B E T A - R A Y A N G U L A R D I S T R I B U T I O N
I N T H E A----12 S Y S T E M
The beta-ray angular distribution in the 1+ -4 0+ transition of the aligned 12B and 12N is given by
N( E, O) = Bo( E) + B2( E) A P2(cos 0),
(4)
where .4 is the degree of nuclear alignment and E is the total energy of the electron. For an illustrating pro'pose, we give an approximate formula for anisotropy coefficients a T in the following, where the full formula is given elsewhere [1,8]:
B2(E)/Bo(E) ,~ c~÷ E .~ [ i x T (2M fT/fA) -- y ] /3M x E
(5)
with X
- [ - f v (1 + ( f r
× .//o')
) + 2M/w] /IA ~ -4.706 fV/IA
(6)
and
y = - 2 M i f %v/ ]o'.
(7)
Here the upper and lower signs refer to the electron and positron decays, respectively. The parameter x is the ratio of the space component of the vector and axial vector current matrix elements, while the parameter y is the ratio of the time component (axial charge) to space component of the main axial vector current. The beta-ray angular distribution function, B2(E)/Bo(E), is measured experimentally as a function of the electron energy E and its slope gives the anisotropy coefficient a~:. In our recent work [6,7], we adopt the formalism which includes the effect of higher order partial waves of leptons, Coulomb corrections of the finite-size nucleus and radiative corrections. A chi-square analysis of the experimental data [5] has been made with the two parameters fT/fh and 8~. The latter is defined by y = yIA (1 + ~ ) + y E c .
(8)
Here YIA is 3.17 given in the impulse approximation with the Hauge-Maripuu wave function, and yP.c is 1.30 due to the effect of the soft pion exchange currents [9]. The other matrix elements are also calculated in a similar way.
K. Koshigiri et aL / Nuclear PhysicsA588 (1995) 165c-170c
167c
The analysis of the experimental data of the Osaka group leads to the following numerical results:
(9)
2MfT/fA = 0.21 4- 0.14, and
(10)
y = 5.10 + 0.16.
For more detail of the axial charge parameter y, the core polarization effects should be taken into consideration. In fact, the first order core polarization effect gives us YcP = -0.44 [9], while all-order effect of the core polarization reduces the magnitude of YcP [1O]. In the chi-square fit, 8y represents core polarization and some other effects, such as the multipion exchange currents [11], which are not explicitly taken into account here. The effect is 8~ = 20%, compared with the effects of exchange currents YEc/YI, = 41%. Similar but large enhancement of the axial charge operator was reported in the first forbidden beta decays of the heavy mass region [12]. 3. C H A R G E A S Y M M E T R Y OF N U C L E A R M A T R I X E L E M E N T S I N T H E MIRROR BETA DECAY In the early days, the ft-value asymmetry of mirror beta decay was studied whether it comes fi'om the nuclear origins or fundamental interactions. Thus the mirror asymmetry of Gamow-Teller matrix elements are theoretically investigated [13-15]. However, in the investigations of beta-ray angular correlation experiments, it is usually assumed that the charge symmetry holds for the mirror beta decays, since the physical observables in correlation experiments are related to the ratio of the nuclear matrix elements. We would like to study the consequence of deleting this assumption. Blomqvist pointed out that the separation energy of the proton in I~N is 0.601 MeV while that of the neutron in :2B is 3.370 MeV, and that the beta-decaying proton wave function is spreading compared to the neutron. This binding energy effect leads to the mismatch of the overlap integral in nuclear matrix elements. The axial charge operator is nuclear momentum dependent and its derivative term for nucleon coordinate gives the different contribution for this mismatch. Employing the simple harmonic oscillator model and the Blomqvist's model which explain the ft-value asymmetry, we have studied a charge asymmetry effect on the parameter y. Since the ratio of the orbital angular momentum to Gamow-Teller matrix element is small in the A=12 system, the parameter x is almost given nuclear-model independently and its charge asymmetry is negligible. We have used the simple harmonic oscillator model for the nucleon wave function to find a tendency of the charge asymmetry effects. The relatively weak binding of the proton in 12N is simulated by the enhancement of the oscillator parameter for the Opl/2 proton wave fnnction as b --+ b (1 + :). Nucleon wave functions of the other states are set to have an equal oscillator parameter b. If we take the single particle model for the initial and final states, ft-value asymmetry is given as follows:
ft+/ft-= /rr_//
Or+ 2
=[Iop[b,b]/ I o p [ b , b ( l + e ) ] l = ~ l + 5 / 2 : ~,
(11)
168c
K. Koshigiri et al. / Nuclear Physics A588 (1995) 165c-170e
where I0p [a,/3] = fo°° R0p(r; a) Rop(r;/3) r 2 dr = (2a/J/(a 2 +/32)) 5/2
(12)
and Rop(r; a) is the nucleon radial wave function with an oscillator parameter a, while the parameter y for the electron and positron decays are given by the following form in the above approximation scheme,
y+/y-=
i
~sr / i +
%r
x(ft+/ft -)~1+5/3e-5/6e
2.
(13)
If we set the parameter s to reproduce the experimental value of the f t - a s y m m e t r y 5 = f t + / f t - - 1 = 13%, we get ¢=0.22, and substituting this value in (13) we have A y = (y+ - y - ) / 2 = 5/4e - 5/8e 2 = 0.25. In Table 1, we show the results of a harmonic
oscillator model with the general 0p-shell configuration mixing of Hauge and Maripuu, where oscillator parameters are set as b(proton) = (1 + e) b (neutron) and b(OPll2) = (1 + 2e)b (Opa/~) with e = 0.052 which is fixed from the experimental ft-value asymmetry as above. Now we follow the method of Blomqvist except for the nuclear model, which we adopt the Hauge-Maripuu model and take seven parent states for the residual A = l l system. The Woods-Saxon central potential depth parameter is varied so as to reproduce the correct separation energy for the decaying nucleon for each parent states. Potential parameters other than that of the central potential depth are the same for those of the Blomqvist model. In Table 1, the numerical results which indicate the charge asymmetry of nuclear matrix elements are summarized for three models. The W S W F model is similar to the Blomqvist model where the fixed Woods-Saxon central potential of 40.2 MeV is used by neglecting the parent dependence of the potential. Table 1 ft-value and y asymmetry in the mirror beta Model 5(%) yHOWF 12.1 3.49 WSWF 7.2 3.36 Blomqvist 10.8 3.42 We use the following notation: 5 = f t + / f t - and A y = (y+ - y - ) / 2
decay of the A=12 system y+ y~. 3.94 3.71 3.63 3.50 3.62 3.52 1, y,~ = ( y - + y+)/2
Ay 0.22 0.13 0.10
Summarizing these three cases in Table 1, we have A 9 = 0.10 ,,~ 0.22, and the induced tensor coupling constant is derived from the following formula when the charge asymmetry of the parameter 9 is included in (5), 2 M f T / fA = X + A y -- (3211/2) (a_ - oL+).
(14)
With this equation and the numerical value of A y in (14), we have to interpret (9) as = [ (0.3 ,-~ 0.4) :1:0.14 ] / 2 M .
fT/fa
K. Koshigiri et al. / Nuclear Physics A588 (1995) 165c-170c
169c
4. S u m m a r y We have deduced the induced tensor coupling constant from the experimental data of the beta-decay alignment correlation coefficients. We adopt the models which incorporate the binding energy effects of beta-decaying nucleon wave function. These models almost reproduce the ft-value asymmetry of the mirror beta decay and give the asymmetry of the parameter y which is the ratio of the axial charge to Gamow-Teller matrix elements. It is shown that the parameter y is larger for positron decay than for the electron decay by several percent and the upper limit of the induced tensor current derived from the present analysis is about 10% of the weak magnetism term, if we adopt the experimental results of Ref. [5]. Since this seems to be rather large from the theoretical point of view, those for a higher precision would be greatly appreciated. Further investigation on Ay is useful. We would like to express our hearty thanks to Professor T. Minamisono for providing us their experimental data before publication. This work is partly supported by Grant-in-Aid for Scientific Research, The Ministry of Education, Science and Culture. REFERENCES 1. M. Morita, M. Nishimura, A. Shimizu, H. Ohtsubo and K. Kubodera, Prog. Theor. Phys. Suppl. No. 60 (1976) 1. 2. S. Weinberg, Phys. Rev. 112 (1958) 1375. 3. J . F . Donoghue and B. R. Holstein, Phys. Rev. D25 (1982) 206. 4. I(. Kubodera, J. Delorme and M. Rho, Nucl. Phys. B66 (1973) 253. 5. T. Minamisono, A. Kitagawa, K. Matsuta and Y. Nojiri, Hyperfine Interactions 78 (1993) 77. 6. A part of the present work was presented in a paper by M. Morita, R. Morita and K. I(oshigiri, Proceedings of International Synposium on Spin and Isospin Responses and Weak Processes in Hadron and Nuclei, Osaka, 1994, Nucl. Phys., in press. 7. M. Morita, R. Morita and K. Koshigiri to be published. 8. K. Koshigiri, M. Nishimura, H. Ohtsubo and M. Morita, Nucl. Phys. A319 (1979) 301. 9. I(. I(oshigiri, H. Ohtsubo and M. Morita, Prog. Theor. Phys. 66 (1981) 358. 10. I(. Koshigiri, I,:. I(ubodera, H. Ohtsubo and M. Morita, (ed.) M.Morita et al., Nuclear Weak Process and Nuclear Structure, World Scientific, Singapore, 1989, p.52 11. M. Kirchbach, D. O. Riska and K. Tsushima, Nucl. Phys. A542 (1992) 616. 12. E. I(. Warburton, Phys. Rev. C44 (1991) 233. E. I(. Warburton and I. S. Towner, Phys. Lett. B294 (1992) 1. 13. J. Blomqvist, Phys. Lett. B35 (1971) 375. 14. D. H. Wilkinson, Phys. Rev. Lett. 27 (1971) 1018. 15. I. S. Towner, Nucl. Phys. A216 (1973) 589.