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Further comment on spurious center-of-mass motion
Volume 60B, number 1
PHYSICS LETTERS
22 December 1975
F U R T H E R COMMENT ON SPURIOUS C E N T E R - O F - M A S S M O T I O N J.B. McGRORY Oak Ridge National Laboratory*, Oak Ridge, Tennessee 37830, USA and
B.H. WlLDENTHAL Michigan State University, East Lansing, Michigan 48823, USA Received 1 October 1975 We present arguments which indicate the Gloeckner-Lawson prescription for eliminating states with spurious center-of-mass motion from highly truncated shell-model calculations is not generally useful. The problem associated with spurious center-ofmass motion is one of the oldest and most irritating problems associated with nuclear shell-model calculations. Recently, Gloeckner and Lawson (G-L) [ 1] have suggested that an approximate treatment of this problem [2, 3] in truncated shell-model spaces can be quite effective, and have claimed that such a treatment significantly affects core-excited shell-model calculations [ 4 - 7 ] in the 160 region. We present here resuits and arguments which we feel cast doubts on their conclusions. A simple statement of the spurious-state problem is that the particle coordinates of a shell-model wave function are defined with respect to a fixed coordinate center. For an A-particle system, the coordinate system can be transformed to a system where the coordinates are the center-of-mass position of the system, andA - 1 position vectors relative to this center-ofmass. Thus, an arbitrary shell-model wave function is a linear combination of states
~SM
--
//.
J
where the ¢i are a complete set of functions in the space of (A - 1) relative position vectors and the •CM are a complete set o f functions in the space of the center-of-mass coordinate R. Most physically interesting quantities involve only the intrinsic coordinates. In * Research sponsored by the U.S. Energy Research and Development Administration under contract with Union Carbide Corporation.
the sheU-model functions above, the same intrinsic wave function can occur with states o f different center-of-mass motion. Thus, there is the possibility of overcounting. In most cases, there is no problem if there is some way to guarantee that in every wave function, the center of mass is in the same state. Shellmodel calculations are most often carried out in the harmonic oscillator representation. Talmi [8] has shown that if, in a given shell-model space, the configurations are restricted to those in which all particles are in the lowest oscillator orbits allowed by the Pauli principle, then the center-of-mass is in the 0S state for all states, and there is no spurious-state problem. The problem only exists when particles are excited from their lowest allowed oscillator shell to higher shells. If such excitations are allowed, there is one case known where one can effect an exact separation of spurious and non-spurious states. If one works in a harmonic oscillator representation, and if one allows all possible states of all possible configurations for which the oscillator energy is within N h ~ of the lowest energy allowed configuration, then it is possible to construct a basis in which every basis state is an eigenstate of the center-of-mass Hamiltonian. (To construct such a basis, one could diagonalize a translationally invariant interaction). One then selects all states with the same center-of-mass eigenstate to form a good model space. G-L have applied a prescription [1 ] for handling this problem approximately for those cases where one does not include all states within a given N h w of the lowest configuration. The prescription is simply to add to the residual shell-model Hamiltonian the Hamilton-
Volume 60B, n u m b e r 1
PHYSICS LETTERS
ian of the center-of-mass multiplied by a large coefficient, i.e. use A
H=HsM +~H(R),
~>~ 1
'
~ = 1 ~ 5" Ai=l
When/~ is a large positive number, those states with significant admixtures of excited center-of-mass states should be pushed to a very high energy and thus essentially eliminated from the calculation. This method has been used previously [3] by Philpott and Szydlik as a practical method of separating out the spurious states in a calculation of the states in 160 in a model space in whichall configurations within 2/~6o of the (s 4, p12) configuration were included. Since they worked in a complete space, they were assured of being able to construct eigenstates of the center-of-mass Hamiltonian. However, Gloeckner and Lawson suggest that the prescription is useful in highly truncated spaces. In such spaces, it is not clear that one can construct a state which has a significant overlap with any excited center-of-mass eigenstates. They applied this method to the Zuker-BuckMcGrory (ZBM) calculation [4] of the states in 160 in a highly truncated model space. ZBM assumed an s4p8/2 12C core, and they included all configurations of four particles in the 0Pl/2, 0ds/2, lSl/2 orbits. This space includes some excitations which are 4/~w above the lowest Pauli-allowed configuration (s4p12). For the positive parity states, the "complete" space necessary to allow an exact separation of center-of-mass eigenstates would include the excitation of two particles from the 0Sl/2, Pl/2, or P3/2 orbits through 2 h ~ of excitation, as well as one particle from the same orbits through 47~w. The complete space would be immense, with dimensions in the thousands. In contrast, for any (J, T) pair, the ZBM model space never contains more than 20 states. When the spurious states are "eliminated" by the G-L prescription, the ZBM resuits are strongly distorted for several values of J and T, so that which previously appeared as good agreement with experiment disappears. We have looked, in some detail, at the application of the G-L prescription to the ZBM calculation. For the reasons we now give, we feel that the use of the prescription in this case is not justified. The first thing we did was simply to diagonalize the center-of-mass Hamiltonian by itself in the truncated 12C core space
22 December 1975
described above. We used a/ST~ of 1400. The eigenvalues we obtained for all 11 J = 0 +, T = 0 states in this space are listed in table 1. As we noted above, the model space includes excitations within 4 h w of lowest state. If we worked in the complete 4h¢o space, we would obtain the eigenvalues 2100, 3500, 4900, 6300, and 7700 MeV. The results listed in table 1 indicate there is very little part of the model space we use which overlaps with a center-of-mass excitation of lhw or greater. We can put a limit on the admixture of excited center-of-mass states in the eigenstates corresponding to the eigenvalues listed in table 1. The spurious-state admixture implied by these eigenvalues would be maximized if we assume that there are only 0h¢o and 1/~6~ states involved. Then the eigenvalues listed above, E~M, can be written as
E~M = Ot2EOS + (1 -- a2)Eip = a/2(2100) + (1 - a 2) 3500. The upper limit on the admixture of spurious states is for the case with ECM = 2333. In this case, the implied admixture is 17%. The state with E = 2158 has at most a 4% admixture. The eigenvalue 2333 is the largest one for any of the J 7r, T values in t h e A = 16 system, so that this 17% admixture is an upper limit for all states in this calculation. These admixtures would be reduced significantly if there were admixtures of 2 - 4 / i w center-of-mass excitations. Thus, in the entire space of j~r, T = 0 +, 0 states in the ZBM calculations, there is no more than one-half of a spurious state. The G-L prescription identifies as "spurious" states all five of the states with any spurious-state admixture at all, when, in fact, none of them has much overlap with a true "spurious state", but, in fact, have very large overlaps with non-spurious states. Table 1 Eigenvalues of center-of-mass Hamiltonian in an oscillator well for ~/i¢o = 1400 MeV, in J = 0 +, T = 0 s~stem for 160. EoS = 3/2 ~/w = 2100 MeV Eigenvalue (MeV)
Multiplicity
2100 2158 2217 2294 2333
6 1 2 1 1
Volume 60B, number 1
PHYSICS LETTERS
We next asked what is the nature o f the six nonspurious states. There is an alternative way [9] to construct a set of spurious center-of-mass states. This involves acting on the full set of 0 h ~ states with the center-of-mass energy raising operator, R ÷. This operator is a vector operator, so it can connect states which differ by at most 1 unit of angular momentum. In the model space that we are considering here for 160, the orbits involved are the Pl/2, the Sl/2, and the d5/2 orbits. Spurious states arise from exciting Pl/2 particles. Because of the vector nature o f the center-of-mass raising operator, it cannot excite a particle from the Pl/2 orbit to the ds/2 orbit. From this it follows that c~4 ~8 _4 - n for any configuration of the form ~.~1/2, e3/2,1-'1/2 , dn5/2) the center-of-mass is in the 0S state. In the J~, T = 0 +, 0 space of 160, there are five states which do not involve excitations to the lSl/2 orbit. This accounts for five non-spurious states. The remaining six states in the J = 0 +, T = 0 space are mixtures of states with 0S center-of-mass motion and higher center-ofmass states, although we have seen there is relatively little of the latter. As a result of diagonalizing the center-of-mass Hamiltonian in this truncated space, one more state with pure 0S center-of-mass motion is formed. By introducing the center-of-mass Hamiltonian into the shell model Hamiltonian with a very large coefficient, states with relatively small admixtures of non-0S motion are pushed to a relatively high energy. What this means for this 160 case is that it essentially reduces the calculations to a pure Pl/2 - d5/2 calculation. Thus, we have demonstrated that the ZBM model space can essentially be divided into two parts. One part is the (Pl/2, d5/2) 4 space, which is a rigorously non-spurious space, and the remaining part of the space in which the states have very small admixtures of excited center-of-mass states. When we add the center-of-mass Hamiltonian with a large multiplicative factor, these states with small spurious-state admixtures are eliminated from the calculation. The G-L prescription is then a further extreme truncation of an already highly truncated space. None of the states which are eliminated can reasonably be labelled as spurious. On the other hand, the eliminated states have very large overlap with non-spurious states which should be included in the model space. It is well known [ 11 ] that one can construct an effective residual shell-model Hamiltonian to compen-
22 December 1975
sate for truncation effects. If the truncation effects are small, there i~ ample evidence [12] one can construct simple (i.e., one- and two-body, mass independent) effective interactions. If the truncation effects are large, one must use [13] complicated effective many-body interactions. The existence of spurious admixtures here is a truncation effect which is relatively small, and one can hope to find simple operators which compensate for the spurious-state effects. The success of the 12C core models for A = 1 3 - 2 0 nuclei is evidence that it is so possible. The truncation of the (Pl/2, d5/2, Sl/2) 4 space which is introduced by the G-L prescription is a large one (essentially half the space is eliminated) and one could anticipate difficulties in finding a simple effective interaction which compensates for this truncation. We have found that this anticipation is also realized, as we discuss next. We have repeated the 12C core calculations of the nuclei withA = 1 6 - 2 0 in a model space in which the "spurious" states have been eliminated by the G-L prescription. We have treated the matrix elements of the effective interaction as free parameters, and fit these parameters to 80 observed excitation energies in the A = 1 6 - 2 0 region. To be specific, we treated linearlyindependent combinations of the matrix elements as parameters, and varied only those combinations to which the calculations were sensitive. The results were very poor. The r.m.s, deviation was on the order of 1 MeV, and we had difficulty obtaining any distinct minimum in the x-squared space. The states which were the source of most of the difficulties were the so-called "intruder" or deformed states. These results are consistent with our discussion above. The "spurious-state projection" reduces the calculation from a model space of (p 1/2, d5/2, Sl/2) A - 12 configurations to one of (Pl/2, d5/2) A-12 configurations. The virtual elimination of the Sl/2 orbit significantly reduces the size of the quadrupole deformation which any state can acquire. It is generally believed that it is just this quadrupole deformation of multiparticle multi-hole states which accounts for their existence at relatively low excitation energies. Thus, it appears that the contributions of the Sl/2 orbit are so large that the omission of the orbit cannot be compensated for by a renormalization o f the interaction. G-L argue that the inclusion of states with small spurious-state admixtures have strong effects on the energies of the model states. We would argue that the strong effects arise because the non-spu-
Volume 60B, number 1
PHYSICS LETTERS
rious parts o f the " s p u r i o u s " states, which comprise greater than 90% o f the " s p u r i o u s " states, are an important part o f the m o d e l space, and there is no reason t h e y should be eliminated.
References [1] D.H. Gloeckner and R.D. Lawson, Phys. Letts. 53B (1974) 313. [2] J.P. EUiott and B.H. Flowers, Proc. Roy. Soc. (London) A242 (1957) 57; F. Palumbo, Nucl. Phys. A99 (1967) 100. [3] R.J. Philpott and P.P. Szydlik, Phys. Rev. 153 (1967) 1039.
22 December 1975
[4] A.P. Zuker, B. Buck and J.B. McGrory, Phys. Rev. Lett. 21 (1968) 39. [5] J.B. McGrory, Phys. Lett. 31B (1970) 339. [6] J.B. McGrory and B.H. Wildenthal, Phys. Rev. 7C (1973) 974. [7] B.S. Reehal and B.H. Wildenthal, Particles and Nuclei 6 (1973) 137. [8] I. Talmi, Helv. Phys. Acta 25 (1952) 185. [9] E.U. Baranger and C.W. Lee, Nucl. Phys. 22 (1961) 157. [10] K.H. Bhatt and J.B. McGrory, Phys. Rev. C3 (1971) 2293. [11] T.T.S. Kuo and G.E. Brown, Nucl. Phys. 85 (1966)40. {121 E.C. Halbert, J.B. McGrory, B.H. Wildenthal and S.P. Pandya, Advances in Nuclear Physics, Vol. 4, eds. M. Baranger and E. Vogt (Plenum Press, New York, 1969). [13] B.R. Barrett, E.C. Halbert and J.B. McGrory, Ann. Phys 90 (1975) 321.
PHYSICS LETTERS
22 December 1975
F U R T H E R COMMENT ON SPURIOUS C E N T E R - O F - M A S S M O T I O N J.B. McGRORY Oak Ridge National Laboratory*, Oak Ridge, Tennessee 37830, USA and
B.H. WlLDENTHAL Michigan State University, East Lansing, Michigan 48823, USA Received 1 October 1975 We present arguments which indicate the Gloeckner-Lawson prescription for eliminating states with spurious center-of-mass motion from highly truncated shell-model calculations is not generally useful. The problem associated with spurious center-ofmass motion is one of the oldest and most irritating problems associated with nuclear shell-model calculations. Recently, Gloeckner and Lawson (G-L) [ 1] have suggested that an approximate treatment of this problem [2, 3] in truncated shell-model spaces can be quite effective, and have claimed that such a treatment significantly affects core-excited shell-model calculations [ 4 - 7 ] in the 160 region. We present here resuits and arguments which we feel cast doubts on their conclusions. A simple statement of the spurious-state problem is that the particle coordinates of a shell-model wave function are defined with respect to a fixed coordinate center. For an A-particle system, the coordinate system can be transformed to a system where the coordinates are the center-of-mass position of the system, andA - 1 position vectors relative to this center-ofmass. Thus, an arbitrary shell-model wave function is a linear combination of states
~SM
--
//.
J
where the ¢i are a complete set of functions in the space of (A - 1) relative position vectors and the •CM are a complete set o f functions in the space of the center-of-mass coordinate R. Most physically interesting quantities involve only the intrinsic coordinates. In * Research sponsored by the U.S. Energy Research and Development Administration under contract with Union Carbide Corporation.
the sheU-model functions above, the same intrinsic wave function can occur with states o f different center-of-mass motion. Thus, there is the possibility of overcounting. In most cases, there is no problem if there is some way to guarantee that in every wave function, the center of mass is in the same state. Shellmodel calculations are most often carried out in the harmonic oscillator representation. Talmi [8] has shown that if, in a given shell-model space, the configurations are restricted to those in which all particles are in the lowest oscillator orbits allowed by the Pauli principle, then the center-of-mass is in the 0S state for all states, and there is no spurious-state problem. The problem only exists when particles are excited from their lowest allowed oscillator shell to higher shells. If such excitations are allowed, there is one case known where one can effect an exact separation of spurious and non-spurious states. If one works in a harmonic oscillator representation, and if one allows all possible states of all possible configurations for which the oscillator energy is within N h ~ of the lowest energy allowed configuration, then it is possible to construct a basis in which every basis state is an eigenstate of the center-of-mass Hamiltonian. (To construct such a basis, one could diagonalize a translationally invariant interaction). One then selects all states with the same center-of-mass eigenstate to form a good model space. G-L have applied a prescription [1 ] for handling this problem approximately for those cases where one does not include all states within a given N h w of the lowest configuration. The prescription is simply to add to the residual shell-model Hamiltonian the Hamilton-
Volume 60B, n u m b e r 1
PHYSICS LETTERS
ian of the center-of-mass multiplied by a large coefficient, i.e. use A
H=HsM +~H(R),
~>~ 1
'
~ = 1 ~ 5" Ai=l
When/~ is a large positive number, those states with significant admixtures of excited center-of-mass states should be pushed to a very high energy and thus essentially eliminated from the calculation. This method has been used previously [3] by Philpott and Szydlik as a practical method of separating out the spurious states in a calculation of the states in 160 in a model space in whichall configurations within 2/~6o of the (s 4, p12) configuration were included. Since they worked in a complete space, they were assured of being able to construct eigenstates of the center-of-mass Hamiltonian. However, Gloeckner and Lawson suggest that the prescription is useful in highly truncated spaces. In such spaces, it is not clear that one can construct a state which has a significant overlap with any excited center-of-mass eigenstates. They applied this method to the Zuker-BuckMcGrory (ZBM) calculation [4] of the states in 160 in a highly truncated model space. ZBM assumed an s4p8/2 12C core, and they included all configurations of four particles in the 0Pl/2, 0ds/2, lSl/2 orbits. This space includes some excitations which are 4/~w above the lowest Pauli-allowed configuration (s4p12). For the positive parity states, the "complete" space necessary to allow an exact separation of center-of-mass eigenstates would include the excitation of two particles from the 0Sl/2, Pl/2, or P3/2 orbits through 2 h ~ of excitation, as well as one particle from the same orbits through 47~w. The complete space would be immense, with dimensions in the thousands. In contrast, for any (J, T) pair, the ZBM model space never contains more than 20 states. When the spurious states are "eliminated" by the G-L prescription, the ZBM resuits are strongly distorted for several values of J and T, so that which previously appeared as good agreement with experiment disappears. We have looked, in some detail, at the application of the G-L prescription to the ZBM calculation. For the reasons we now give, we feel that the use of the prescription in this case is not justified. The first thing we did was simply to diagonalize the center-of-mass Hamiltonian by itself in the truncated 12C core space
22 December 1975
described above. We used a/ST~ of 1400. The eigenvalues we obtained for all 11 J = 0 +, T = 0 states in this space are listed in table 1. As we noted above, the model space includes excitations within 4 h w of lowest state. If we worked in the complete 4h¢o space, we would obtain the eigenvalues 2100, 3500, 4900, 6300, and 7700 MeV. The results listed in table 1 indicate there is very little part of the model space we use which overlaps with a center-of-mass excitation of lhw or greater. We can put a limit on the admixture of excited center-of-mass states in the eigenstates corresponding to the eigenvalues listed in table 1. The spurious-state admixture implied by these eigenvalues would be maximized if we assume that there are only 0h¢o and 1/~6~ states involved. Then the eigenvalues listed above, E~M, can be written as
E~M = Ot2EOS + (1 -- a2)Eip = a/2(2100) + (1 - a 2) 3500. The upper limit on the admixture of spurious states is for the case with ECM = 2333. In this case, the implied admixture is 17%. The state with E = 2158 has at most a 4% admixture. The eigenvalue 2333 is the largest one for any of the J 7r, T values in t h e A = 16 system, so that this 17% admixture is an upper limit for all states in this calculation. These admixtures would be reduced significantly if there were admixtures of 2 - 4 / i w center-of-mass excitations. Thus, in the entire space of j~r, T = 0 +, 0 states in the ZBM calculations, there is no more than one-half of a spurious state. The G-L prescription identifies as "spurious" states all five of the states with any spurious-state admixture at all, when, in fact, none of them has much overlap with a true "spurious state", but, in fact, have very large overlaps with non-spurious states. Table 1 Eigenvalues of center-of-mass Hamiltonian in an oscillator well for ~/i¢o = 1400 MeV, in J = 0 +, T = 0 s~stem for 160. EoS = 3/2 ~/w = 2100 MeV Eigenvalue (MeV)
Multiplicity
2100 2158 2217 2294 2333
6 1 2 1 1
Volume 60B, number 1
PHYSICS LETTERS
We next asked what is the nature o f the six nonspurious states. There is an alternative way [9] to construct a set of spurious center-of-mass states. This involves acting on the full set of 0 h ~ states with the center-of-mass energy raising operator, R ÷. This operator is a vector operator, so it can connect states which differ by at most 1 unit of angular momentum. In the model space that we are considering here for 160, the orbits involved are the Pl/2, the Sl/2, and the d5/2 orbits. Spurious states arise from exciting Pl/2 particles. Because of the vector nature o f the center-of-mass raising operator, it cannot excite a particle from the Pl/2 orbit to the ds/2 orbit. From this it follows that c~4 ~8 _4 - n for any configuration of the form ~.~1/2, e3/2,1-'1/2 , dn5/2) the center-of-mass is in the 0S state. In the J~, T = 0 +, 0 space of 160, there are five states which do not involve excitations to the lSl/2 orbit. This accounts for five non-spurious states. The remaining six states in the J = 0 +, T = 0 space are mixtures of states with 0S center-of-mass motion and higher center-ofmass states, although we have seen there is relatively little of the latter. As a result of diagonalizing the center-of-mass Hamiltonian in this truncated space, one more state with pure 0S center-of-mass motion is formed. By introducing the center-of-mass Hamiltonian into the shell model Hamiltonian with a very large coefficient, states with relatively small admixtures of non-0S motion are pushed to a relatively high energy. What this means for this 160 case is that it essentially reduces the calculations to a pure Pl/2 - d5/2 calculation. Thus, we have demonstrated that the ZBM model space can essentially be divided into two parts. One part is the (Pl/2, d5/2) 4 space, which is a rigorously non-spurious space, and the remaining part of the space in which the states have very small admixtures of excited center-of-mass states. When we add the center-of-mass Hamiltonian with a large multiplicative factor, these states with small spurious-state admixtures are eliminated from the calculation. The G-L prescription is then a further extreme truncation of an already highly truncated space. None of the states which are eliminated can reasonably be labelled as spurious. On the other hand, the eliminated states have very large overlap with non-spurious states which should be included in the model space. It is well known [ 11 ] that one can construct an effective residual shell-model Hamiltonian to compen-
22 December 1975
sate for truncation effects. If the truncation effects are small, there i~ ample evidence [12] one can construct simple (i.e., one- and two-body, mass independent) effective interactions. If the truncation effects are large, one must use [13] complicated effective many-body interactions. The existence of spurious admixtures here is a truncation effect which is relatively small, and one can hope to find simple operators which compensate for the spurious-state effects. The success of the 12C core models for A = 1 3 - 2 0 nuclei is evidence that it is so possible. The truncation of the (Pl/2, d5/2, Sl/2) 4 space which is introduced by the G-L prescription is a large one (essentially half the space is eliminated) and one could anticipate difficulties in finding a simple effective interaction which compensates for this truncation. We have found that this anticipation is also realized, as we discuss next. We have repeated the 12C core calculations of the nuclei withA = 1 6 - 2 0 in a model space in which the "spurious" states have been eliminated by the G-L prescription. We have treated the matrix elements of the effective interaction as free parameters, and fit these parameters to 80 observed excitation energies in the A = 1 6 - 2 0 region. To be specific, we treated linearlyindependent combinations of the matrix elements as parameters, and varied only those combinations to which the calculations were sensitive. The results were very poor. The r.m.s, deviation was on the order of 1 MeV, and we had difficulty obtaining any distinct minimum in the x-squared space. The states which were the source of most of the difficulties were the so-called "intruder" or deformed states. These results are consistent with our discussion above. The "spurious-state projection" reduces the calculation from a model space of (p 1/2, d5/2, Sl/2) A - 12 configurations to one of (Pl/2, d5/2) A-12 configurations. The virtual elimination of the Sl/2 orbit significantly reduces the size of the quadrupole deformation which any state can acquire. It is generally believed that it is just this quadrupole deformation of multiparticle multi-hole states which accounts for their existence at relatively low excitation energies. Thus, it appears that the contributions of the Sl/2 orbit are so large that the omission of the orbit cannot be compensated for by a renormalization o f the interaction. G-L argue that the inclusion of states with small spurious-state admixtures have strong effects on the energies of the model states. We would argue that the strong effects arise because the non-spu-
Volume 60B, number 1
PHYSICS LETTERS
rious parts o f the " s p u r i o u s " states, which comprise greater than 90% o f the " s p u r i o u s " states, are an important part o f the m o d e l space, and there is no reason t h e y should be eliminated.
References [1] D.H. Gloeckner and R.D. Lawson, Phys. Letts. 53B (1974) 313. [2] J.P. EUiott and B.H. Flowers, Proc. Roy. Soc. (London) A242 (1957) 57; F. Palumbo, Nucl. Phys. A99 (1967) 100. [3] R.J. Philpott and P.P. Szydlik, Phys. Rev. 153 (1967) 1039.
22 December 1975
[4] A.P. Zuker, B. Buck and J.B. McGrory, Phys. Rev. Lett. 21 (1968) 39. [5] J.B. McGrory, Phys. Lett. 31B (1970) 339. [6] J.B. McGrory and B.H. Wildenthal, Phys. Rev. 7C (1973) 974. [7] B.S. Reehal and B.H. Wildenthal, Particles and Nuclei 6 (1973) 137. [8] I. Talmi, Helv. Phys. Acta 25 (1952) 185. [9] E.U. Baranger and C.W. Lee, Nucl. Phys. 22 (1961) 157. [10] K.H. Bhatt and J.B. McGrory, Phys. Rev. C3 (1971) 2293. [11] T.T.S. Kuo and G.E. Brown, Nucl. Phys. 85 (1966)40. {121 E.C. Halbert, J.B. McGrory, B.H. Wildenthal and S.P. Pandya, Advances in Nuclear Physics, Vol. 4, eds. M. Baranger and E. Vogt (Plenum Press, New York, 1969). [13] B.R. Barrett, E.C. Halbert and J.B. McGrory, Ann. Phys 90 (1975) 321.