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Numerical study of DWBA in a three-body model of stripping
Volume 42B, number 2
PHYSICS LETTERS
27 November 1972
NUMERICAL STUDY OF DWBA IN A THREE-BODY MODEL OF STRIPPING D.P. BOULDIN* and F.S. LEVIN**
Physics Department, Brown University, Providence, Rhode Island 02912, USA Received 26 September 1972 Spectator, adiabatic, and DWBAwave functions have been compared for a separable potential, three-body model of deuteron stripping. The spectator and adiabatic wave functions are shown to be identical in this model; their use in the stripping amplitude is exact. Detailed calculations are presented and analyzed for an incident energy of 6.7 MeV; these are similar to results obtained at energies of 1.78, 11.2, and 15.12 MeV. Although the resulting distributions are in good agreement, the adiabatic and DWBA wave functions and partial wave amplitudes do not agree too well, as in the many-body calculations of Harvey and Johnson. A major assumption made in deriving the distorted wave Born approximation (DWBA) amplitude for deuteron stripping is that the three-body wave function if0 describing the deuteron in the presence o f the unexcited target nucleus may be replaced by a product o f the deuteron ground state wave function ~d and the deuteron elastic scattering wave function u d def'med by u d = <~dl~b0):
Ud = (%1¢'0): ~0 ~ ~dUd"
(1)
Neglect of continuum deuteron states in (1) makes DWBA difficult to justifyt. Recently Johnson and Soper [2] have included some effects of continuum states in an extension of DWBA denoted the adiabatic approximation (AA). Use of AA generally provides better agreement with experiment (particularly (p, d) reactions) than does DWBA[3]. In our notation, the adiabatic wave function ~A is given by ~0 ~ ~ g = ~bdX"
(2)
where
× = [5d3rVnp(r)q~tl(r)]-lfd3rVnp(r)qJo(r, R).
(3)
Here Imp is the neutron-proton interaction binding the deuteron, r is the neutron-proton relative coordinate (r = r n - rp), and R is the deuteron center-ofmass (CM) coordinate (2R = r n + rn). We have shown recently [ 1 ] that an exact form o f * NDEA fellow, 1969-1971. Present address: Department of Theoretical Physics, The University, Manchester 13. ** Work supported in part by the U.S. Atomic Energy Commission.
DWBA can be derived if Vnp is taken to be a separable, S-wave potential. Our result is that if the replacement ~0 -~ I~ND = ~bdF
(4)
is made, then the resulting stripping amplitude is unchanged, i.e., (4) is exact. The spectator function F like u d (and X) is a CM wave function: F = F ( R ) ; it" includes effects of breakup. As noted by Johnson and Soper [4], F - X when Vnp is separable. This is easily shown by writing Vnp as Vnp= ~.[f)(Jq and substituting into (3). Hence in this case X is also exact, a result that hods for any form of the interaction between the deuteron and the target. In general neither ~0, ~ND,nor ~A can:be calculated exactly, and ~A has been computed only in the adiabatic approximation [3]. However these functions can be calculated for an S-wave, separable potential, three-body model (the Mitra model [5]). For this model, Reiner and Jaffe [6] showed that the angular distribution obtained by use o f ( l ) (DWBA) in the exact matrix element was in good agreement with the exact angular distribution. As noted in [ 1], this implies that u d ~ F and also, of course, that u d ~ ×. We have undertaken calculations both to test this hypothesis and to examine the behavior o f F ( a n d X), and we report some of our results in this note. We have calculated a special case o f the Mitra model for the stripping of deuterons incident with kinetic energies o f 1.78, 6.7, 11.2 and 15.12 MeV. The nucleons o f the deuteron interact with one another via a Yamaguchi separable potential [7] with range parameter/3 = 1.45 fm -1 ; this yields a bound state of t See [ 1 ] for a discussion of, and references to the literature on, problems in DWBA.
167
Volume 42B, number 2
PHYSICS LETTERS
2.225 MeV binding energy. Each o f the nucleons interacts with the core via the same Yamaguchi potential with range parameter/3 = 1.06 fm -1, giving a bound state of 3.300 MeV binding energy. Thus, the two stripping channels lead to identical, though distinguishable, results in our model. The two intermediate values o f the energy were chosen so as to be able to compare our results with thoses o f Reiner and Jaffe, who used different nucleoncore interactions V/i.e. V1 :/: V2 in their calculations. The high and low energy values were selected to extend the range over which exact and DWBA could be compared. Comparison o f the exact and DWBA resuits leads to similar conclusions in all four cases and we present detailed results here for the 6.7 MeV case only. This particular case is the most interesting in that the exact and DWBA angular distributions are very close; for the 11.2 MeV case, the DWBA is not as good. This is in contrast to the results reported by Reiner and Jaffe for these two energies. In fig. 1, the exact, the DWBA, and the plane wave stripping cross sections are presented. These first two cross sections are c o m p u t e d from T ND and T Dw, defined in [ 1]. The agreement between the DWBA and the exact results seems to confirm the results o f Reiner and Jaffe that for the separable-potential threebody model, DWBA is a good approximation. How-
,oo
i
4,00
II
E 0 - 6.7 M e V
300
r =
(2t + 1)r,e,(kd-k,),
l shows that this closeness of fit is somewhat fortuitous: T/Dw 4: T/ND except for l > 2. Similar results hold for the other three energies as well, although in these cases, the magnitudes o f the exact and DWBA cross sections are in worse agreement. This is striking evidence that agreement between exact and approximate cross sections need not imply agreement between the corresponding partial wave amplitudes. F r o m tafile 1, we can conclude that the main reason for such good agreement between the angular distributions is that the l --- 2 and 1 = 3 partial wave amplitudes are the most important in that their multiplicative factors o f (21 + 1) are much greater than the similar factor for l = 0, even though ITS0D[ and ITDWl are reasonably' large. The CM states IF) and [Ud) are the only quantities that differ in T ND and T Dw, and we examine typical behavior in figs. 2 and 3. We plot the l = 0 and l = 3 partial wave components of RF(R) - R ( R IF) and Rud(R ) --R (R lu d) where for example we have expanded F ( R ) via
--~--OWBA
-----PW
Table 1 Partial wave stripping matrix elements, E = 6.7 MeV
IOO
0
40
80
120
160
ANGLE (DEGREES) Fig. 1. Comparison of exact, DWBA, and plane wave stripping cross sections f o r e = 6.7 MeV.
168
~ = k/lk l)
A similar expansion is used to define Ud,l Two points are evident from these figures. First, the major differences between the elastic and spectator functions occur in the region R < 1.5 fm -1.
MODEL (d,p):
EXACT
~
T/ND
ever, examination of the partial wave amplitudes d T/Dw o f table 1, where T/is in general defined b y
F(R) = ~ FI(R )PI(kd"R). l
|
,..
27 November 1972
1
ReTND
im/ff/D
ReTD/W imT?W
/ff/ORN
0 1 2 3 4 5 6 7
1.403 0.471 -1.947 -1.352 -0.709 -0.356 -0.178 -0.089 -0.044 -0.021
0.450 0.017 -0.108 0.006 0.005 0.001 -
0.792 0.173 -1.709 -1.300 -0.699 -0.354 -0.178 -0.089 -0.044 -0.021
-10.595 - 5.868 - 2.963 - 1.461 - 0.720 - 0.357 - 0.178 - 0.089 - 0.044 - 0.021
8
9
-1.923 -0.450 -0.425 -0.072 -0.009 -0.001 -
Volume 42B, number 2
PHYSICS LETTERS
27 November 1972
@I
(a) 0.2
2.0
$
1.0
:
O,’
f
0
0
-0.1
-1.0
g $
X
K
-0.2
-2.0
0
2
I
3
0
I
2
3
0
I
R(infm)
range between 5% and lo%, with IRe(F IRe(ud,o)l and IIm(Fu)~> jIm(ud u) [ in this region. Now, as we have shown, flD and pWare given by [I] differences
and
i
(-)($jlfud),
where I,,0is the form factor for the nucleon-nucleon interaction and 3 is the bound state of particle j. These matrix elements could be computed in a space representation either by a d3rd3R integration or a d3r,d3rP integration, Since stripping is a rearrangement process, both sets of coordinates enter into the matrix elements, and it is not obvious which of the factors, 3 and f,is controlling the range of integration. Let us consider d3rd31?. Then ~~~~)and f(r) have ranges of about 1.0 and 0.7 fm, respectively, implying that only those regions where F@) and u&Q differ (I? < I;!? fm ) are important in their matrix elements. However, the effect of the weighting factors ui@ as given in a r, R representation is not easy to assess. What one can say is that in such a representation even <$$ will have an infiite series, oscillatory, partial wave expansion, each term of
3
0
1
2
3
Rtinfm)
Fig. 2. Comparison of a) the real and b) thi: imaginary parts of RFo(-m-) of RF3C-.-) and Ruall(-);
Second,themaximum
2
and Ru@-)
and c) the real and d) the imaginary parts
which will be complex. Con~quen~y, differences in Ud,{ and F1 may not produce important differences in partial wave stripping amplitudes, as is evident on comparison of, for example, the real parts of the I = 0 and E= 3 amplitudes of table 1 with the behavior exhibited in fig. 2. Results similar to these hold for the other energies. Not only do the exact and DWBA partial wave amplitudes differ for the low partial waves, we also see that real and imaginary parts of FIand Ud,l may differ by nearly an order of magnitude (fig. 2 c,d). However, as indicated above, such differences need not lead to auy obvious conclusions concerning the sizes of the correspon~ng matrix elements. This behavior of Fl (or ud,$ occurs for other energies as well, as does the fact that for both the real and the imaginary PatiSOf+fldUd,~ their maxima and minima occur at roughly the same values of R, which indicates that F1 and Ud I.essenti&y differ only in SCak in the regionR< i;S fm. From the fact that the DWBA cross section is greater than the exact one in the model, and also from the fact that tits to experimental data using AA do not need a radial cutoff [3], we might infer that F (or d is damped as compared to Ud for small R. The calculations of Harvey and Johnson [3] did not show this, and our exact results do not confirm this inference either. Since this occurs both for the optical 169
Volume 42B, number 2
PHYSICS LETTERS
potential calculations of Harvey and Johnson as well as for our separable S-wave potential calculations, we conclude that it is a characteristic of the AA, and is to be understood in terms of the preceding arguments on the rearrangement nature of the stripping process. Since we have calculated do/dI2 for energies ranging from 1.78 MeV to 15.12 MeV, we are able to examine, over this range, the convergence properties of DWBA as a function of energy. The results of Reiner and Jaffe, in which DWBA more closely approximated the exact cross-section at the higher of the two energies, suggested that DWBA might converge to the exact result as the bombarding energy increased. Our results do not support such a suggestion, since the DWBA angular distributions (forward directions) are in slightly worse agreement with the exact ones as we go from 6.7 MeV to 15.12 MeV. This behavior is not seen in comparisons of integrated cross sections (table 2), because the sin0d0 factor gives the least weight to the forward angles where the cross sections are largest in both magnitude and in their differences. We may draw several conclusions from these model results. First, F(R) and Ud(R) are equal except in a region inside of about 1.5 to 2.0 fm around the origin, R = 0. Second, the differences in F and u d occur where these functions make their main contributions to the stripping matrix elements. Third, obvious differences in FI(R ) and Ud,l(R ) cannot be used to predict that the corresponding T~ D and T/Dw will differ. Fourth, one cannot predict form near equality of angular distributions that partial wave matrix elements
170
27 November 1972
Table 2 Exact and DWBAintegrated cross sections in mb E=1.78 (MeV)
E=6.7 (MeV)
E=ll.2 (MeV)
E=15.12 (MeV)
Exact 352.46 DWBA 418.23
231.70 218.23
147.17 135.30
125.84 117.43
must be nearly equal. Hence, we cannot conclude that in the model, DWBA is a good approximation in detail, although it does produce reasonable angular distributions. Furthermore, we also cannot conclude that DWBA converges towards the exact result as the energy increases. Hence, we suggest that AA calculations be used wherever possible to analyze stripping data and in particular to extract spectroscopic factors. We thank Peter Levin for his assistance in plotting some of the results of these calculations.
References [1] D. Bouldin and F.S. Levin, Phys. Lett. 37B (1971) 145; Nuclear Physics A189 (1972) 449. [2] R.C. Johnson and P.R. Soper, Phys. Rev. C1 (1970) 976. [3] J.D. Harvey and R.C. Johnson, Phys. Rev. C3 (1971)636; G.R. Satchler, Phys. Rev. C4 (1971) 1485; B.M. Preedom, Phys. Rev. C5 (1972) 587. [4] R.C. Johnson, private communication. [5] A.N. Mitra, Phys. Rev. 139 (1965) B1472. [6] A.S. Reiner and A.I. Jaffe, Phys. Rev. 161 (1967) 935. [7] Y. Yamaguchi, Phys. Rev. 95 (1954) 1628.
PHYSICS LETTERS
27 November 1972
NUMERICAL STUDY OF DWBA IN A THREE-BODY MODEL OF STRIPPING D.P. BOULDIN* and F.S. LEVIN**
Physics Department, Brown University, Providence, Rhode Island 02912, USA Received 26 September 1972 Spectator, adiabatic, and DWBAwave functions have been compared for a separable potential, three-body model of deuteron stripping. The spectator and adiabatic wave functions are shown to be identical in this model; their use in the stripping amplitude is exact. Detailed calculations are presented and analyzed for an incident energy of 6.7 MeV; these are similar to results obtained at energies of 1.78, 11.2, and 15.12 MeV. Although the resulting distributions are in good agreement, the adiabatic and DWBA wave functions and partial wave amplitudes do not agree too well, as in the many-body calculations of Harvey and Johnson. A major assumption made in deriving the distorted wave Born approximation (DWBA) amplitude for deuteron stripping is that the three-body wave function if0 describing the deuteron in the presence o f the unexcited target nucleus may be replaced by a product o f the deuteron ground state wave function ~d and the deuteron elastic scattering wave function u d def'med by u d = <~dl~b0):
Ud = (%1¢'0): ~0 ~ ~dUd"
(1)
Neglect of continuum deuteron states in (1) makes DWBA difficult to justifyt. Recently Johnson and Soper [2] have included some effects of continuum states in an extension of DWBA denoted the adiabatic approximation (AA). Use of AA generally provides better agreement with experiment (particularly (p, d) reactions) than does DWBA[3]. In our notation, the adiabatic wave function ~A is given by ~0 ~ ~ g = ~bdX"
(2)
where
× = [5d3rVnp(r)q~tl(r)]-lfd3rVnp(r)qJo(r, R).
(3)
Here Imp is the neutron-proton interaction binding the deuteron, r is the neutron-proton relative coordinate (r = r n - rp), and R is the deuteron center-ofmass (CM) coordinate (2R = r n + rn). We have shown recently [ 1 ] that an exact form o f * NDEA fellow, 1969-1971. Present address: Department of Theoretical Physics, The University, Manchester 13. ** Work supported in part by the U.S. Atomic Energy Commission.
DWBA can be derived if Vnp is taken to be a separable, S-wave potential. Our result is that if the replacement ~0 -~ I~ND = ~bdF
(4)
is made, then the resulting stripping amplitude is unchanged, i.e., (4) is exact. The spectator function F like u d (and X) is a CM wave function: F = F ( R ) ; it" includes effects of breakup. As noted by Johnson and Soper [4], F - X when Vnp is separable. This is easily shown by writing Vnp as Vnp= ~.[f)(Jq and substituting into (3). Hence in this case X is also exact, a result that hods for any form of the interaction between the deuteron and the target. In general neither ~0, ~ND,nor ~A can:be calculated exactly, and ~A has been computed only in the adiabatic approximation [3]. However these functions can be calculated for an S-wave, separable potential, three-body model (the Mitra model [5]). For this model, Reiner and Jaffe [6] showed that the angular distribution obtained by use o f ( l ) (DWBA) in the exact matrix element was in good agreement with the exact angular distribution. As noted in [ 1], this implies that u d ~ F and also, of course, that u d ~ ×. We have undertaken calculations both to test this hypothesis and to examine the behavior o f F ( a n d X), and we report some of our results in this note. We have calculated a special case o f the Mitra model for the stripping of deuterons incident with kinetic energies o f 1.78, 6.7, 11.2 and 15.12 MeV. The nucleons o f the deuteron interact with one another via a Yamaguchi separable potential [7] with range parameter/3 = 1.45 fm -1 ; this yields a bound state of t See [ 1 ] for a discussion of, and references to the literature on, problems in DWBA.
167
Volume 42B, number 2
PHYSICS LETTERS
2.225 MeV binding energy. Each o f the nucleons interacts with the core via the same Yamaguchi potential with range parameter/3 = 1.06 fm -1, giving a bound state of 3.300 MeV binding energy. Thus, the two stripping channels lead to identical, though distinguishable, results in our model. The two intermediate values o f the energy were chosen so as to be able to compare our results with thoses o f Reiner and Jaffe, who used different nucleoncore interactions V/i.e. V1 :/: V2 in their calculations. The high and low energy values were selected to extend the range over which exact and DWBA could be compared. Comparison o f the exact and DWBA resuits leads to similar conclusions in all four cases and we present detailed results here for the 6.7 MeV case only. This particular case is the most interesting in that the exact and DWBA angular distributions are very close; for the 11.2 MeV case, the DWBA is not as good. This is in contrast to the results reported by Reiner and Jaffe for these two energies. In fig. 1, the exact, the DWBA, and the plane wave stripping cross sections are presented. These first two cross sections are c o m p u t e d from T ND and T Dw, defined in [ 1]. The agreement between the DWBA and the exact results seems to confirm the results o f Reiner and Jaffe that for the separable-potential threebody model, DWBA is a good approximation. How-
,oo
i
4,00
II
E 0 - 6.7 M e V
300
r =
(2t + 1)r,e,(kd-k,),
l shows that this closeness of fit is somewhat fortuitous: T/Dw 4: T/ND except for l > 2. Similar results hold for the other three energies as well, although in these cases, the magnitudes o f the exact and DWBA cross sections are in worse agreement. This is striking evidence that agreement between exact and approximate cross sections need not imply agreement between the corresponding partial wave amplitudes. F r o m tafile 1, we can conclude that the main reason for such good agreement between the angular distributions is that the l --- 2 and 1 = 3 partial wave amplitudes are the most important in that their multiplicative factors o f (21 + 1) are much greater than the similar factor for l = 0, even though ITS0D[ and ITDWl are reasonably' large. The CM states IF) and [Ud) are the only quantities that differ in T ND and T Dw, and we examine typical behavior in figs. 2 and 3. We plot the l = 0 and l = 3 partial wave components of RF(R) - R ( R IF) and Rud(R ) --R (R lu d) where for example we have expanded F ( R ) via
--~--OWBA
-----PW
Table 1 Partial wave stripping matrix elements, E = 6.7 MeV
IOO
0
40
80
120
160
ANGLE (DEGREES) Fig. 1. Comparison of exact, DWBA, and plane wave stripping cross sections f o r e = 6.7 MeV.
168
~ = k/lk l)
A similar expansion is used to define Ud,l Two points are evident from these figures. First, the major differences between the elastic and spectator functions occur in the region R < 1.5 fm -1.
MODEL (d,p):
EXACT
~
T/ND
ever, examination of the partial wave amplitudes d T/Dw o f table 1, where T/is in general defined b y
F(R) = ~ FI(R )PI(kd"R). l
|
,..
27 November 1972
1
ReTND
im/ff/D
ReTD/W imT?W
/ff/ORN
0 1 2 3 4 5 6 7
1.403 0.471 -1.947 -1.352 -0.709 -0.356 -0.178 -0.089 -0.044 -0.021
0.450 0.017 -0.108 0.006 0.005 0.001 -
0.792 0.173 -1.709 -1.300 -0.699 -0.354 -0.178 -0.089 -0.044 -0.021
-10.595 - 5.868 - 2.963 - 1.461 - 0.720 - 0.357 - 0.178 - 0.089 - 0.044 - 0.021
8
9
-1.923 -0.450 -0.425 -0.072 -0.009 -0.001 -
Volume 42B, number 2
PHYSICS LETTERS
27 November 1972
@I
(a) 0.2
2.0
$
1.0
:
O,’
f
0
0
-0.1
-1.0
g $
X
K
-0.2
-2.0
0
2
I
3
0
I
2
3
0
I
R(infm)
range between 5% and lo%, with IRe(F IRe(ud,o)l and IIm(Fu)~> jIm(ud u) [ in this region. Now, as we have shown, flD and pWare given by [I] differences
and
i
(-)($jlfud),
where I,,0is the form factor for the nucleon-nucleon interaction and 3 is the bound state of particle j. These matrix elements could be computed in a space representation either by a d3rd3R integration or a d3r,d3rP integration, Since stripping is a rearrangement process, both sets of coordinates enter into the matrix elements, and it is not obvious which of the factors, 3 and f,is controlling the range of integration. Let us consider d3rd31?. Then ~~~~)and f(r) have ranges of about 1.0 and 0.7 fm, respectively, implying that only those regions where F@) and u&Q differ (I? < I;!? fm ) are important in their matrix elements. However, the effect of the weighting factors ui@ as given in a r, R representation is not easy to assess. What one can say is that in such a representation even <$$ will have an infiite series, oscillatory, partial wave expansion, each term of
3
0
1
2
3
Rtinfm)
Fig. 2. Comparison of a) the real and b) thi: imaginary parts of RFo(-m-) of RF3C-.-) and Ruall(-);
Second,themaximum
2
and Ru@-)
and c) the real and d) the imaginary parts
which will be complex. Con~quen~y, differences in Ud,{ and F1 may not produce important differences in partial wave stripping amplitudes, as is evident on comparison of, for example, the real parts of the I = 0 and E= 3 amplitudes of table 1 with the behavior exhibited in fig. 2. Results similar to these hold for the other energies. Not only do the exact and DWBA partial wave amplitudes differ for the low partial waves, we also see that real and imaginary parts of FIand Ud,l may differ by nearly an order of magnitude (fig. 2 c,d). However, as indicated above, such differences need not lead to auy obvious conclusions concerning the sizes of the correspon~ng matrix elements. This behavior of Fl (or ud,$ occurs for other energies as well, as does the fact that for both the real and the imaginary PatiSOf+fldUd,~ their maxima and minima occur at roughly the same values of R, which indicates that F1 and Ud I.essenti&y differ only in SCak in the regionR< i;S fm. From the fact that the DWBA cross section is greater than the exact one in the model, and also from the fact that tits to experimental data using AA do not need a radial cutoff [3], we might infer that F (or d is damped as compared to Ud for small R. The calculations of Harvey and Johnson [3] did not show this, and our exact results do not confirm this inference either. Since this occurs both for the optical 169
Volume 42B, number 2
PHYSICS LETTERS
potential calculations of Harvey and Johnson as well as for our separable S-wave potential calculations, we conclude that it is a characteristic of the AA, and is to be understood in terms of the preceding arguments on the rearrangement nature of the stripping process. Since we have calculated do/dI2 for energies ranging from 1.78 MeV to 15.12 MeV, we are able to examine, over this range, the convergence properties of DWBA as a function of energy. The results of Reiner and Jaffe, in which DWBA more closely approximated the exact cross-section at the higher of the two energies, suggested that DWBA might converge to the exact result as the bombarding energy increased. Our results do not support such a suggestion, since the DWBA angular distributions (forward directions) are in slightly worse agreement with the exact ones as we go from 6.7 MeV to 15.12 MeV. This behavior is not seen in comparisons of integrated cross sections (table 2), because the sin0d0 factor gives the least weight to the forward angles where the cross sections are largest in both magnitude and in their differences. We may draw several conclusions from these model results. First, F(R) and Ud(R) are equal except in a region inside of about 1.5 to 2.0 fm around the origin, R = 0. Second, the differences in F and u d occur where these functions make their main contributions to the stripping matrix elements. Third, obvious differences in FI(R ) and Ud,l(R ) cannot be used to predict that the corresponding T~ D and T/Dw will differ. Fourth, one cannot predict form near equality of angular distributions that partial wave matrix elements
170
27 November 1972
Table 2 Exact and DWBAintegrated cross sections in mb E=1.78 (MeV)
E=6.7 (MeV)
E=ll.2 (MeV)
E=15.12 (MeV)
Exact 352.46 DWBA 418.23
231.70 218.23
147.17 135.30
125.84 117.43
must be nearly equal. Hence, we cannot conclude that in the model, DWBA is a good approximation in detail, although it does produce reasonable angular distributions. Furthermore, we also cannot conclude that DWBA converges towards the exact result as the energy increases. Hence, we suggest that AA calculations be used wherever possible to analyze stripping data and in particular to extract spectroscopic factors. We thank Peter Levin for his assistance in plotting some of the results of these calculations.
References [1] D. Bouldin and F.S. Levin, Phys. Lett. 37B (1971) 145; Nuclear Physics A189 (1972) 449. [2] R.C. Johnson and P.R. Soper, Phys. Rev. C1 (1970) 976. [3] J.D. Harvey and R.C. Johnson, Phys. Rev. C3 (1971)636; G.R. Satchler, Phys. Rev. C4 (1971) 1485; B.M. Preedom, Phys. Rev. C5 (1972) 587. [4] R.C. Johnson, private communication. [5] A.N. Mitra, Phys. Rev. 139 (1965) B1472. [6] A.S. Reiner and A.I. Jaffe, Phys. Rev. 161 (1967) 935. [7] Y. Yamaguchi, Phys. Rev. 95 (1954) 1628.