- Email: [email protected]
10+ Resonance at Ec.m. = 19.0 MeV in 12C(12C, α)20Ne
Volume 63B, number 4
PHYSICS LETTERS
particles to align, which causes the high angular momenta and indeed these are the same single particle states which cause the backbending in the present formalism. Therefore it may be that the present approach describes the K = 0 component of the rotational aligned state. It is well known that a minimization may fail in cases where the overlap of the relevant trial functions are too large. In such cases the backbending effect is known to disappear in the calculations, if one allows dynamical effects by mixing the states rather than treating them by minimization [4]. In the present case such a mixing corresponds to the GCM +QPapproach applied successfully to the high spin states of pf-shell nuclei [9]. In order to make sure that the present backbending in 166yb is not caused by the failures of the minimization, such a GCM +QP calculation has been performed in the following approximate way: for each angular momentum the Hamiltonian is diagonalized in a space spanned by the states projected from the BCS ground state and from the a = 5/2 + 2qp state. Since both states are taken with the parameters A~°) and A~5/2), the basis is given by altogether four states, which are of course different for each J. The techniques for such a diagonalization in a nonorthogonal basis'ar~ deseribed in refs. [4, 9], The results of the mixing are presented in fig: lb-, where the solid curves represent the two lowest solutions of the mixing, and the others of the minimization. One realizes from the crossing point onwards a noticeable mixing, which smoothes the crossing of the bands. However, the mixing is not strong enough to spoil the backbending features found by minimization, as one can see in fig. 2a, dashed curve. In conclusion one can say that the present method gives like the usual plojection approach a remarkable agreement for the low spin region. The essential improvement, however, takes place at the high spin region due to the inclusion of two quasiparticle excitations. We do not intend to overemphasize the good
402
16 August 1976
agreement between the present approach and the experimental data for the following reasons: first the force dependence of the results has not been investigated, second the inclusion of more than one 2 qpband in the mixing may change the results, and third nonaxial components of the wave functions have been omitted. Nevertheless, one may learn that the 2 qp excitations near the Fermi surface play an important role for the backbending effect, whereas they can be neglected for the low spins. This can be described in terms of a variational procedure or even a dynamical mixing in a space spanned simultaneously by projected BCS ground state and projected two quasiparticle configurations. One of the authors (F.G.) thanks Professor V.G. Soloviev for warm hospitality during his stay at the J.I.N.R. in Dubna. He is also grateful to CERN for financial support.
References [1 ] A. Johnson, H. Ryde and J. Sztarkier, Phys. Lett. 34B (1971) 605; IH.W. Beuscher, W.F.Davidson, R.M. Lieder and C. MayerB~Sricke,phys.-Lett.~40B (1972) 449; P. Thieberger et al., Phys. Rev. Lett. 40B (1972) 440. [2] B.R. Mottelson and J.C. Valatin, Phys. Rev. Lett. 5 (1960) 511. [3] F.S. Stephens and R.S. Simon, Nucl. Phys. A183 (1972) 257. [4] F. Griimmer, K.W. Schmid and A. Faessler, Z. Physik A 275 (1975) 391. [5] F. Gr/immer, K.W. Sehmid and A. Faessler, Nucl. Phys. A239 (1975) 289. [6] P. Ring, H.J. Mang and B. Banerjee, Nuel. Phys. A225 (1974) 141; A. Faessler et al., Nucl. Phys. A256 (1976) 106. [7] S.Y. Chu et al., Phys. Rev. C12 (1975) 1017. [8] I. Morrison et al., Phys. Lett. 60B (1975) 29. [9] H. M~ther, K. Goeke, A. Faessler and K. Allaart, Phys. Lett. 60B (1976) 427.
Volume 63B, number 4 10
PHYSICS LETTERS Ec.rn.(~2C) = 19.0 Me~
=ZC (=ZC'c~)Z°Ne
2
- - - % (cos8 > - -
BEST
FIT
g.s.
// ',',
I'
4
II
"\ ~-. b
~!
E x - 6.72 MeV
1.0
2
I
~,I
10"2
~
,,:II II
J
I
0
I
30
t
,I II.[t tli, 1 I II
L
/,t
1
t'-
~, I
L
~
60
J
I Ill [I I
Table 1 kegendre coefficientsa) for the 12C(12C, a) 20Ne(g.s.) angular distribution at Ec.m. = 19.0 MeV.
l
alia 0b)
2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32
1.215 ± 0.04 0.289 ± 0.03 0.598 ± 0.08 0.863 ± 0.05 1.313 ± 0.14 1.451 ± 0.08 1.027 ± 0.06 0.897 ± 0.06 2.684 ± 0.03 2.865 ± 0.08 -1.598 ± 0.04 -0.487 +- 0.12 -0.001 +- 0.10 0.129 ± 0.12 -0.015 ± 0.12 -0.206 ± 0.30
[
90
ec.m.
Fig. 2. Angular distributions for the 12C(12C,a)20Ne reactions leading to the g.s. and 6.72 MeV 0+ state of 2°Ne, together with P~o(cos 0) and best-fit curves. The g.s. exhibits a resonance at this energy of 19.0 MeV, whereas the 6.72 MeV state does not.
of unknown al's in eq. (1). Given a set of b/'s, it is always possible to compute the al's, but the converse is not true. For a unique set of a/'s, there are 2 N - I equivalent sets of bl'S where here N is the number o f terms in the sum in eq. (2). Unfortunately, it is the al's that are determined by the data, though the bl'S most easily display the resonance information. However, in fortunate circumstances, the resonant amplitude in one/-value (say I = lo) may so dominate the cross section that its presence is detectible even with eq. (2). In that case
+ ~ blbl'PlPl'. l,l'=~ l o A 12C(12C, a)20Ne yield curve has been measured (see fig. I) for several final states in 2°Ne, using the 12C5+ beam from the Argonne tandem. The target was a 20 gtg/cm 2 self-supporting carbon foil. Outgoing 404
16 August 1976
2L
a)°(O)=ao( 1+ lo=C2 U-a~/p/(c°s0))" / l even b) The uncertainties are mostly statistical.
a-particles were m o m e n t u m analyzed in a split-pole spectrograph and detected in a proportional counter. The experimental setup has been described previously [51. Angular distributions for the g.s. and the 6.72 MeV 0 + state at E(12C) = 38.0 MeV are displayed in fig. 2. The curves shown there are P120(cos 0) and a best-fit curve obtained using eq. (1). The dominance o f L = 10 is readily apparent by inspection o f the data. The bestfit curve differs only slightly from the curve for P120, even at very large angles. The coefficients a I that yield the best-fit curve are displayed in table 1. The fit from which these coefficients arose used 2L = 32 in eq. (1). The values ofa l (and the theoretical curve) are vktually identical for 2L = 32, 28, and 24 and only very slightly different for 2L = 22. A plot o f ×2 versus 2L is displayed in fig. 3. All the coefficients a l for l > 24 are zero within the uncertainties. This would imply that the maximum l value involved is l = 12, except for the negative value o f a24, which can arise only from a term o f the form
Volume 63B, number 4 ~g
5
I
PHYSICS LETTERS i
I
O ,-, bJ
nu_ LL O
;
I
I
I
t
ZL
4
~
,,-..o(o,)-v~oO~P,(~o, o, )i
y2=
"
3
Ao" (O i )
I
2
16 August 1976
oTO T = 47r~.2(2l + 1) F c F a / P 2 where F c, F a and F are the entrance-channel, exitchannel, and total widths, respectively. For the present resonance, the g.s. yield is
<..6 LtJ c-~
1 -
°~o, I
.><
0
20
~
o, I
1
24
I
oTO T = 4~" a 0 = 3.24 mb.
• J
28
I
i
32
2L
Thus F c F a / F 2 = 0.00682,
Fig. 3. A plot of X2 versus 2L. -°10-014 in eq. (2). Thus, if our errors are realistic, it would appear that l = 14 is present but to a limited extent. Optical-model calculations indicate that the grazing partial wave is l = 10 or 12. The large negative a22 presumably arises from the interference of-010 and P12" In any event the g.s. angular distribution at resonance is dominated by p120. The dominance o f L = 10 is not present for the 6.72 MeV 0 + state, which does not exhibit a resonance at 38 MeV. Furthermore, the dominance o f L = 10 vanishes [6] at energies only a few hundred keV below and above the resonance. Thus the dominance of l = 10 in the g.s. angular distribution is not merely a manifestation o f the fact that the grazing partial wave is about 10. We thus assign j~r = 10 + to this resonance. This may be the same resonance as that observed [2] at a slightly lower energy (18.8 MeV) in 12C(12C, 8Be)160 (g.s.) and tentatively assigned J ~r = 10 +. It is not the resonance observed [3] at 19.3 MeV in 12C(12C, p)23Na and thought [7] to be 12 +. That resonance is not apparent in 12C(12C, a)2°Ne [1, 8]. The angle-integrated cross section at resonance is equal to
i.e. a significant portion of the total strength is in channels other than 12C + 12C and a + 2°Ne (g.s.). In fact, much of this strength lies in the 7.83 and 9.04 MeV states, whose angle-integrated cross sections at resonance are 10.2 and 41.7 mb, respectively. Summing these three, we get
r c ~rJr
2=
0.116
nearly 1/2 the unitarity limit o f 0.25.
References [1] H.T. Fortune, LR. Greenwood, R.E. Segel and J.R. Erskine, submitted to Phys. Rev. [21 N.R. Fletcher et al., Phys. Rev. C13 (1976) 1173. [3] K. van Bibber et al., Phys. Rev. Lett. 28 (1974) 1590. [4] J.M. Blatt and L,C. Biedenharn, Revs. Mod. Phys. 24 (1952) 258. [51 L.R. Greenwood et al., Phys. Rev. C12 (1975) 156. [6] H.T. Fortune, T.H. Braid, R.E. Segel and K. Raghunathan, to be published. [7] E.R. Cosman et al., Phys. Rev. Lett. 35 (1975) 265. [8] L.R. Greenwood, H.T. Fortune, R.E. Segel and J.R. Erskine, Phys. Rev. C10 (1974) 1211.
405
PHYSICS LETTERS
particles to align, which causes the high angular momenta and indeed these are the same single particle states which cause the backbending in the present formalism. Therefore it may be that the present approach describes the K = 0 component of the rotational aligned state. It is well known that a minimization may fail in cases where the overlap of the relevant trial functions are too large. In such cases the backbending effect is known to disappear in the calculations, if one allows dynamical effects by mixing the states rather than treating them by minimization [4]. In the present case such a mixing corresponds to the GCM +QPapproach applied successfully to the high spin states of pf-shell nuclei [9]. In order to make sure that the present backbending in 166yb is not caused by the failures of the minimization, such a GCM +QP calculation has been performed in the following approximate way: for each angular momentum the Hamiltonian is diagonalized in a space spanned by the states projected from the BCS ground state and from the a = 5/2 + 2qp state. Since both states are taken with the parameters A~°) and A~5/2), the basis is given by altogether four states, which are of course different for each J. The techniques for such a diagonalization in a nonorthogonal basis'ar~ deseribed in refs. [4, 9], The results of the mixing are presented in fig: lb-, where the solid curves represent the two lowest solutions of the mixing, and the others of the minimization. One realizes from the crossing point onwards a noticeable mixing, which smoothes the crossing of the bands. However, the mixing is not strong enough to spoil the backbending features found by minimization, as one can see in fig. 2a, dashed curve. In conclusion one can say that the present method gives like the usual plojection approach a remarkable agreement for the low spin region. The essential improvement, however, takes place at the high spin region due to the inclusion of two quasiparticle excitations. We do not intend to overemphasize the good
402
16 August 1976
agreement between the present approach and the experimental data for the following reasons: first the force dependence of the results has not been investigated, second the inclusion of more than one 2 qpband in the mixing may change the results, and third nonaxial components of the wave functions have been omitted. Nevertheless, one may learn that the 2 qp excitations near the Fermi surface play an important role for the backbending effect, whereas they can be neglected for the low spins. This can be described in terms of a variational procedure or even a dynamical mixing in a space spanned simultaneously by projected BCS ground state and projected two quasiparticle configurations. One of the authors (F.G.) thanks Professor V.G. Soloviev for warm hospitality during his stay at the J.I.N.R. in Dubna. He is also grateful to CERN for financial support.
References [1 ] A. Johnson, H. Ryde and J. Sztarkier, Phys. Lett. 34B (1971) 605; IH.W. Beuscher, W.F.Davidson, R.M. Lieder and C. MayerB~Sricke,phys.-Lett.~40B (1972) 449; P. Thieberger et al., Phys. Rev. Lett. 40B (1972) 440. [2] B.R. Mottelson and J.C. Valatin, Phys. Rev. Lett. 5 (1960) 511. [3] F.S. Stephens and R.S. Simon, Nucl. Phys. A183 (1972) 257. [4] F. Griimmer, K.W. Schmid and A. Faessler, Z. Physik A 275 (1975) 391. [5] F. Gr/immer, K.W. Sehmid and A. Faessler, Nucl. Phys. A239 (1975) 289. [6] P. Ring, H.J. Mang and B. Banerjee, Nuel. Phys. A225 (1974) 141; A. Faessler et al., Nucl. Phys. A256 (1976) 106. [7] S.Y. Chu et al., Phys. Rev. C12 (1975) 1017. [8] I. Morrison et al., Phys. Lett. 60B (1975) 29. [9] H. M~ther, K. Goeke, A. Faessler and K. Allaart, Phys. Lett. 60B (1976) 427.
Volume 63B, number 4 10
PHYSICS LETTERS Ec.rn.(~2C) = 19.0 Me~
=ZC (=ZC'c~)Z°Ne
2
- - - % (cos8 > - -
BEST
FIT
g.s.
// ',',
I'
4
II
"\ ~-. b
~!
E x - 6.72 MeV
1.0
2
I
~,I
10"2
~
,,:II II
J
I
0
I
30
t
,I II.[t tli, 1 I II
L
/,t
1
t'-
~, I
L
~
60
J
I Ill [I I
Table 1 kegendre coefficientsa) for the 12C(12C, a) 20Ne(g.s.) angular distribution at Ec.m. = 19.0 MeV.
l
alia 0b)
2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32
1.215 ± 0.04 0.289 ± 0.03 0.598 ± 0.08 0.863 ± 0.05 1.313 ± 0.14 1.451 ± 0.08 1.027 ± 0.06 0.897 ± 0.06 2.684 ± 0.03 2.865 ± 0.08 -1.598 ± 0.04 -0.487 +- 0.12 -0.001 +- 0.10 0.129 ± 0.12 -0.015 ± 0.12 -0.206 ± 0.30
[
90
ec.m.
Fig. 2. Angular distributions for the 12C(12C,a)20Ne reactions leading to the g.s. and 6.72 MeV 0+ state of 2°Ne, together with P~o(cos 0) and best-fit curves. The g.s. exhibits a resonance at this energy of 19.0 MeV, whereas the 6.72 MeV state does not.
of unknown al's in eq. (1). Given a set of b/'s, it is always possible to compute the al's, but the converse is not true. For a unique set of a/'s, there are 2 N - I equivalent sets of bl'S where here N is the number o f terms in the sum in eq. (2). Unfortunately, it is the al's that are determined by the data, though the bl'S most easily display the resonance information. However, in fortunate circumstances, the resonant amplitude in one/-value (say I = lo) may so dominate the cross section that its presence is detectible even with eq. (2). In that case
+ ~ blbl'PlPl'. l,l'=~ l o A 12C(12C, a)20Ne yield curve has been measured (see fig. I) for several final states in 2°Ne, using the 12C5+ beam from the Argonne tandem. The target was a 20 gtg/cm 2 self-supporting carbon foil. Outgoing 404
16 August 1976
2L
a)°(O)=ao( 1+ lo=C2 U-a~/p/(c°s0))" / l even b) The uncertainties are mostly statistical.
a-particles were m o m e n t u m analyzed in a split-pole spectrograph and detected in a proportional counter. The experimental setup has been described previously [51. Angular distributions for the g.s. and the 6.72 MeV 0 + state at E(12C) = 38.0 MeV are displayed in fig. 2. The curves shown there are P120(cos 0) and a best-fit curve obtained using eq. (1). The dominance o f L = 10 is readily apparent by inspection o f the data. The bestfit curve differs only slightly from the curve for P120, even at very large angles. The coefficients a I that yield the best-fit curve are displayed in table 1. The fit from which these coefficients arose used 2L = 32 in eq. (1). The values ofa l (and the theoretical curve) are vktually identical for 2L = 32, 28, and 24 and only very slightly different for 2L = 22. A plot o f ×2 versus 2L is displayed in fig. 3. All the coefficients a l for l > 24 are zero within the uncertainties. This would imply that the maximum l value involved is l = 12, except for the negative value o f a24, which can arise only from a term o f the form
Volume 63B, number 4 ~g
5
I
PHYSICS LETTERS i
I
O ,-, bJ
nu_ LL O
;
I
I
I
t
ZL
4
~
,,-..o(o,)-v~oO~P,(~o, o, )i
y2=
"
3
Ao" (O i )
I
2
16 August 1976
oTO T = 47r~.2(2l + 1) F c F a / P 2 where F c, F a and F are the entrance-channel, exitchannel, and total widths, respectively. For the present resonance, the g.s. yield is
<..6 LtJ c-~
1 -
°~o, I
.><
0
20
~
o, I
1
24
I
oTO T = 4~" a 0 = 3.24 mb.
• J
28
I
i
32
2L
Thus F c F a / F 2 = 0.00682,
Fig. 3. A plot of X2 versus 2L. -°10-014 in eq. (2). Thus, if our errors are realistic, it would appear that l = 14 is present but to a limited extent. Optical-model calculations indicate that the grazing partial wave is l = 10 or 12. The large negative a22 presumably arises from the interference of-010 and P12" In any event the g.s. angular distribution at resonance is dominated by p120. The dominance o f L = 10 is not present for the 6.72 MeV 0 + state, which does not exhibit a resonance at 38 MeV. Furthermore, the dominance o f L = 10 vanishes [6] at energies only a few hundred keV below and above the resonance. Thus the dominance of l = 10 in the g.s. angular distribution is not merely a manifestation o f the fact that the grazing partial wave is about 10. We thus assign j~r = 10 + to this resonance. This may be the same resonance as that observed [2] at a slightly lower energy (18.8 MeV) in 12C(12C, 8Be)160 (g.s.) and tentatively assigned J ~r = 10 +. It is not the resonance observed [3] at 19.3 MeV in 12C(12C, p)23Na and thought [7] to be 12 +. That resonance is not apparent in 12C(12C, a)2°Ne [1, 8]. The angle-integrated cross section at resonance is equal to
i.e. a significant portion of the total strength is in channels other than 12C + 12C and a + 2°Ne (g.s.). In fact, much of this strength lies in the 7.83 and 9.04 MeV states, whose angle-integrated cross sections at resonance are 10.2 and 41.7 mb, respectively. Summing these three, we get
r c ~rJr
2=
0.116
nearly 1/2 the unitarity limit o f 0.25.
References [1] H.T. Fortune, LR. Greenwood, R.E. Segel and J.R. Erskine, submitted to Phys. Rev. [21 N.R. Fletcher et al., Phys. Rev. C13 (1976) 1173. [3] K. van Bibber et al., Phys. Rev. Lett. 28 (1974) 1590. [4] J.M. Blatt and L,C. Biedenharn, Revs. Mod. Phys. 24 (1952) 258. [51 L.R. Greenwood et al., Phys. Rev. C12 (1975) 156. [6] H.T. Fortune, T.H. Braid, R.E. Segel and K. Raghunathan, to be published. [7] E.R. Cosman et al., Phys. Rev. Lett. 35 (1975) 265. [8] L.R. Greenwood, H.T. Fortune, R.E. Segel and J.R. Erskine, Phys. Rev. C10 (1974) 1211.
405