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The Kuo interaction and band shifts in the sd-shell
Volume 45B, number 5
PHYSICS LETTERS
THE KUO INTERACTION
20 August 1973
AND BAND SHIFTS IN THE sd-SHELL
B.J. COLE, A. WATT and R.R. WHITEHEAD Department o[ Natural Philosophy, University of Glasgow, Glasgow,Scotland Received 26 June 1973 Evidence is presented to show that the major defect of the Kuo interaction in untruncated shell-model calculations.in the sd-shell is that it produces shifts of whole rotational bands relative to one another, the calculated spacings within the bands being well reproduced even where they do not follow the J(J+ 1) rule. The results of calculations in 23Na, 24Na, 26A1and 33p are used as illustrations. Our present ability to do shell-model calculations in the full basis throughout the sd-shell [ 1 - 3 ] allows us to look for systematic trends in the calculated spectra which may be related directly to the effective interaction being used. The use of the full basis is important because the effects o f truncation may well differ from nucleus to nucleus. Following previous work by the present authors and others [ 2 - 5 ] on 24Mg, where a very striking shift in the position o f the K = 2 band is seen in the calculated spectrum, we have done calculations for a variety of nuclei around the middle o f the sd-shell using the Kuo interaction [6]. This interaction, with 170 single particle energies, gives excellent results for 24Mg, apart from the band shift already mentioned, and quite tolerable results for other even-even nuclei [3]. In odd-odd nuclei, on the other hand, the Kuo interaction already gives trouble in 22Na, where it predicts the wrong spin for the ground state [7], and the difficulties continue for the heavier odd-odd nuclei in truncated calculations [8]. These shortcomings have led to the feeling that there is something radically wrong with the Kuo matrix elements which, for some reason, does not upset the even-even calculations too much. Examination of the calculated spectrum of 26A1 (fig. la), in which the first J = 5 state, corresponding to the observed ground state, appears as the eleventh excited state, tends to reinforce doubts about the adequacy of the Kuo interaction. Closer study of the spectrum reveals, however, that the distoction may be largely a consequence of band shifts of the same kind as is seen in 2~Mg. The rotational structure o f 26A1 is fairly well established [9, 10], the ground and first excited states belonging to K = 5 and 3, T = 0, bands respectively. In fig. lc these bands have been moved
MeV 4 4. ,',-
4 (2,3) i- -
3,4,5
1-'] ~l
2
65
6
K=5
5
K=3
3 2
(2,3)
3-0-
_
2 1
1
1
3
(2,3"t 3
3
~
3
1
1 1
2.O -
-
K= 3
4
(3,4)
|
Z
2
z
v/
1
1
1
1
1
3
3
K=3
5
5
K=5
1.0-
0.0K* WO Q
EXPT. b
C
Fig. 1. Spectrum of 26A1: (a) calculated, Kuo + 170 single partiele energies; (b) experiment, positive parity states only; (c) as for (a) but with K = 5 and K = 3 bands, indicated by arrows in (a), shifted downwards. The spectra are aligned vertically using the position of the first 1+ state. 429
Volume 45B, number 5
PHYSICS LETTERS
downwards so as to give the correct spacings for the first three states. Apart from the absence of a 3 + state at about 2.5 MeV, the resulting spectrum is in remarkably good agreement with experiment. It should be noted that the 6 + member of the K = 5 band has been brought to within 300 keV of its observed position, while the 4 + member of the K = 3 band now lies very close to a possible 4 + state at 2.07 MeV. The calculated spectrum of 24Na, another odd-odd nucleus, is shown in fig. 2a. Here the observed 4 ÷ ground state appears as the eighth excited state. There seems to be not much known about the rotational structure of this nucleus, and so we cannot produce as complete a picture as in 26A1. It is known, however, that in 24Mg an increase in the strength of the spinorbit interaction has the approximate effect of shift~\
MeV
(1234 5) ',
(0)
,5 3.
1
2 2.0--
2 (2,3) 4
(1,2)
2 3
3
i
~s I
s¸ t f 1.0--
- - f
3 ~,
r r
t
] 2
0.0-
K÷170 a
EXP~ b
c
Fig. 2. Spectrum of 24Na: (a) calculated, Kuo + 170; (b) calculated, Kuo + modified spin-orbit splitting (see text); (c) Experiment, positive parity. 430
20 August 1973
ing the rotational bands relative to one another [ 11 ], and it might be possible to use this fact to make tentative assignments. Accordingly we changed the d3/2-ds/2 splitting from 5 MeV to 8 MeV and repeated the calculation, obtaining the spectrum shown in fig. 2b. The most striking effect is a lowering of the 4 ÷ state together with a 5 + state which was previously outside the range of the figure. We suppose, therefore, that these are the 4 + and 5+ members of a K = 4 ground-state band. The other major effect is to raise the calculated 0 ÷ state near 0.9 MeV by about 1 MeV, and to shift one of the 2 + states in the neighbourhood of 2 MeV out of the figure. These two states could be the beginning of a K = 0 band. The situation here is not nearly so clear as in 26A1, but if the above assignments are correct the only serious discrepancy, apart from the band shifts, is the prediction of an additional 2 + state near 1 MeV which has no counterpart in the experimental spectrum. We present results for the odd-A nuclei 23Na and 33p in figs. 3 and 4. In each case the shell-model calculation repreoduces the correct spin for the ground state, but the excited states are more or less jumbled. For 23Na we show in fig. 3c the result o f shifting the position of the K = 3/2 ground-state band downwards by about 2 MeV. Once more we get a very reasonable fit to the data. Shifting the K = 1/2 band upwards leaves a 1/2 + and a 5/2 + state too low and results in a worse fit. In the case of 33p the calculated level sequence (fig. 4a) looks good, but the entire spectrum is much too compressed. A rather better fit may be obtained by shifting a hypothetical rotational band, consisting of the lowest calculated 3/2 +, 5/2 + and 7/2 + states, upwards by 2 MeV. The calculated spectrum contains additional 7/2 + states in the vicinity of 4.5 MeV, one of which may be the continuation of the ground-state band. The situation in 33p, in common with a number of other nuclei in the second half of the shell, is obscured by the uncertainties inherent in trying to assign states to rotational bands with nothing but spins to go by, particularly in a region of poorly developed rotational structure. From the foregoing selection of results it appears that the major defect of the Kuo interaction in the lower half of the shell, where rotational behaviour is strong, is a marked tendency to produce band shifts. From the rotational point of view the upper half of the shell exhibits strong band mixing, which largely
Volume 45 B, number 5
PHYSICS LETTERS
20 August 1973
5/2
MeV
MeV
7/2
5.0--
3/2 ?/2
5.0--
1/2
(7/2)
3/2
3/2 512 312.512
!!2
_
4-0--
5/2 4.0 -
5/2
1/2
1/2 5/2
7/2 5/2
5/2
7/2
5/2
3/2,5/2 3.0--
3.0--
9/2 112
~'
3/2
9/2
~/2
3/2
1/2
1/2
7/t
3/2
512 2.0-
7/~ 7,2
_[
5/2 5/2
3/2 512
1
312
3/2
_~
1.0--
1,0--
5/2 1/2
0.0 --
_T
3/2 5/2 q
3/2 K,,.170 Q
5/2
0.0-
3/2 "1,
112 K.170 a
3/2
112
112
EXPT. b
EXPT. b
c
Fig. 3. Spectrum b f 23Na: (a) calculated, Kuo + tTO; (b) experiment, positive parity states; (c) as (a) but with K = 3/2 band, indicated by arrows in (a), shifted downwards, and the ground states re-aligned.
wipes out the bands, and we cannot expect any really clear picture to emerge, although we believe that a shifted band is discemable in 33p. It is attractive to ascribe the cause o f the generally poor fits obtained in untruncated calculations in this region to the tendency of the interaction to try to shift rather thoroughly mixed bands relative to one another. It is natural to ask what it is about the Kuo interaction that gives rise to this wholesale band shifting. Feldmeier et al. [5] have drawn attention to the socalled antisymmetric spin-orbit (ALS) part of the effective nucleon-nucleon interaction. They have shown that if the ALS contribution to the Kuo matrix ele-
Fig. 4. As fig. 3 but for 33p. The shifted band has K = 3/2 here also. ments is increased by a factor of about 5 the K = 2 band in 24Mg is shifted upwards relative to the groundstate band by just the right amount. They also report preliminary calculations which indicate similar behaviour for the K = 1/2 band in 23Na, which has been discussed above. This is rather attractive but there seems to be no obvious connection between the Als interaction and rotational bands. The same can be said for the device of increasing the one-body spin-orbit interaction used in our discussion of 24Na. The ALS interaction appears to produce the more satisfactory band shifts, without unduly disarranging the rest of the spec trum, in the case of 24Mg, and therefore seems to be the most promising possibility. We may conclude that the Kuo interaction seems capable of producing surprisingly good results for a 431
Volume 45B, number 5
PHYSICS LETTERS
variety of the less extensively studied sd-shell nuclei, provided the band shifts are taken into consideration. This raises the tantalising possibility that, if the band shift problem can be resolved satisfactorily, we may be in a position to expect calculations using parameter-free realistic interactions to yield fits comparable to those presently obtained with heavily parameterised effective interactions and least squares fitting. We wish to thank Drs. B.H. Wildenthal, H. Feldmeier and N. MacDonald for valuable discussions, and the Computing and A u t o m a t i o n Division RHEL for access to their 360/195 on which the calculations were done.
432
20 August 1973
References [1] R.R. Whitehead, Nucl. Phys. A182 (1972) 290. [2] R.R. Whitehead and A. Watt, Phys. Lett. 35B (1971) 189. [3] R.R. Whitehead and A. Watt, Phys. Lett. 41B (1972) 7. [4] D. Strottman, Phys. Lett. 39B (1972) 457. [5] H. Feldmeier, P. Manakos and T. Wolff, Preprint (1972). [6] T.T.S. Kuo, Nucl. Phys. A103 (1967) 71. [7] E. Halbert et al., in Advances in Nuclear Physics ed. M. Baranger and E. Vogt (Plenum, New York, 1971) Vol. IV. [8] B.H. Wildenthal, private communications. [9] R.R. Betts and H.T. Fortune, Phys. Rev. C7 (1973) 1257. [10] J. Sharpey-Schaffer et al., Phys. Rev. Lett. 27 (1971) 1463. [11] V. Pucknell, private communications.
PHYSICS LETTERS
THE KUO INTERACTION
20 August 1973
AND BAND SHIFTS IN THE sd-SHELL
B.J. COLE, A. WATT and R.R. WHITEHEAD Department o[ Natural Philosophy, University of Glasgow, Glasgow,Scotland Received 26 June 1973 Evidence is presented to show that the major defect of the Kuo interaction in untruncated shell-model calculations.in the sd-shell is that it produces shifts of whole rotational bands relative to one another, the calculated spacings within the bands being well reproduced even where they do not follow the J(J+ 1) rule. The results of calculations in 23Na, 24Na, 26A1and 33p are used as illustrations. Our present ability to do shell-model calculations in the full basis throughout the sd-shell [ 1 - 3 ] allows us to look for systematic trends in the calculated spectra which may be related directly to the effective interaction being used. The use of the full basis is important because the effects o f truncation may well differ from nucleus to nucleus. Following previous work by the present authors and others [ 2 - 5 ] on 24Mg, where a very striking shift in the position o f the K = 2 band is seen in the calculated spectrum, we have done calculations for a variety of nuclei around the middle o f the sd-shell using the Kuo interaction [6]. This interaction, with 170 single particle energies, gives excellent results for 24Mg, apart from the band shift already mentioned, and quite tolerable results for other even-even nuclei [3]. In odd-odd nuclei, on the other hand, the Kuo interaction already gives trouble in 22Na, where it predicts the wrong spin for the ground state [7], and the difficulties continue for the heavier odd-odd nuclei in truncated calculations [8]. These shortcomings have led to the feeling that there is something radically wrong with the Kuo matrix elements which, for some reason, does not upset the even-even calculations too much. Examination of the calculated spectrum of 26A1 (fig. la), in which the first J = 5 state, corresponding to the observed ground state, appears as the eleventh excited state, tends to reinforce doubts about the adequacy of the Kuo interaction. Closer study of the spectrum reveals, however, that the distoction may be largely a consequence of band shifts of the same kind as is seen in 2~Mg. The rotational structure o f 26A1 is fairly well established [9, 10], the ground and first excited states belonging to K = 5 and 3, T = 0, bands respectively. In fig. lc these bands have been moved
MeV 4 4. ,',-
4 (2,3) i- -
3,4,5
1-'] ~l
2
65
6
K=5
5
K=3
3 2
(2,3)
3-0-
_
2 1
1
1
3
(2,3"t 3
3
~
3
1
1 1
2.O -
-
K= 3
4
(3,4)
|
Z
2
z
v/
1
1
1
1
1
3
3
K=3
5
5
K=5
1.0-
0.0K* WO Q
EXPT. b
C
Fig. 1. Spectrum of 26A1: (a) calculated, Kuo + 170 single partiele energies; (b) experiment, positive parity states only; (c) as for (a) but with K = 5 and K = 3 bands, indicated by arrows in (a), shifted downwards. The spectra are aligned vertically using the position of the first 1+ state. 429
Volume 45B, number 5
PHYSICS LETTERS
downwards so as to give the correct spacings for the first three states. Apart from the absence of a 3 + state at about 2.5 MeV, the resulting spectrum is in remarkably good agreement with experiment. It should be noted that the 6 + member of the K = 5 band has been brought to within 300 keV of its observed position, while the 4 + member of the K = 3 band now lies very close to a possible 4 + state at 2.07 MeV. The calculated spectrum of 24Na, another odd-odd nucleus, is shown in fig. 2a. Here the observed 4 ÷ ground state appears as the eighth excited state. There seems to be not much known about the rotational structure of this nucleus, and so we cannot produce as complete a picture as in 26A1. It is known, however, that in 24Mg an increase in the strength of the spinorbit interaction has the approximate effect of shift~\
MeV
(1234 5) ',
(0)
,5 3.
1
2 2.0--
2 (2,3) 4
(1,2)
2 3
3
i
~s I
s¸ t f 1.0--
- - f
3 ~,
r r
t
] 2
0.0-
K÷170 a
EXP~ b
c
Fig. 2. Spectrum of 24Na: (a) calculated, Kuo + 170; (b) calculated, Kuo + modified spin-orbit splitting (see text); (c) Experiment, positive parity. 430
20 August 1973
ing the rotational bands relative to one another [ 11 ], and it might be possible to use this fact to make tentative assignments. Accordingly we changed the d3/2-ds/2 splitting from 5 MeV to 8 MeV and repeated the calculation, obtaining the spectrum shown in fig. 2b. The most striking effect is a lowering of the 4 ÷ state together with a 5 + state which was previously outside the range of the figure. We suppose, therefore, that these are the 4 + and 5+ members of a K = 4 ground-state band. The other major effect is to raise the calculated 0 ÷ state near 0.9 MeV by about 1 MeV, and to shift one of the 2 + states in the neighbourhood of 2 MeV out of the figure. These two states could be the beginning of a K = 0 band. The situation here is not nearly so clear as in 26A1, but if the above assignments are correct the only serious discrepancy, apart from the band shifts, is the prediction of an additional 2 + state near 1 MeV which has no counterpart in the experimental spectrum. We present results for the odd-A nuclei 23Na and 33p in figs. 3 and 4. In each case the shell-model calculation repreoduces the correct spin for the ground state, but the excited states are more or less jumbled. For 23Na we show in fig. 3c the result o f shifting the position of the K = 3/2 ground-state band downwards by about 2 MeV. Once more we get a very reasonable fit to the data. Shifting the K = 1/2 band upwards leaves a 1/2 + and a 5/2 + state too low and results in a worse fit. In the case of 33p the calculated level sequence (fig. 4a) looks good, but the entire spectrum is much too compressed. A rather better fit may be obtained by shifting a hypothetical rotational band, consisting of the lowest calculated 3/2 +, 5/2 + and 7/2 + states, upwards by 2 MeV. The calculated spectrum contains additional 7/2 + states in the vicinity of 4.5 MeV, one of which may be the continuation of the ground-state band. The situation in 33p, in common with a number of other nuclei in the second half of the shell, is obscured by the uncertainties inherent in trying to assign states to rotational bands with nothing but spins to go by, particularly in a region of poorly developed rotational structure. From the foregoing selection of results it appears that the major defect of the Kuo interaction in the lower half of the shell, where rotational behaviour is strong, is a marked tendency to produce band shifts. From the rotational point of view the upper half of the shell exhibits strong band mixing, which largely
Volume 45 B, number 5
PHYSICS LETTERS
20 August 1973
5/2
MeV
MeV
7/2
5.0--
3/2 ?/2
5.0--
1/2
(7/2)
3/2
3/2 512 312.512
!!2
_
4-0--
5/2 4.0 -
5/2
1/2
1/2 5/2
7/2 5/2
5/2
7/2
5/2
3/2,5/2 3.0--
3.0--
9/2 112
~'
3/2
9/2
~/2
3/2
1/2
1/2
7/t
3/2
512 2.0-
7/~ 7,2
_[
5/2 5/2
3/2 512
1
312
3/2
_~
1.0--
1,0--
5/2 1/2
0.0 --
_T
3/2 5/2 q
3/2 K,,.170 Q
5/2
0.0-
3/2 "1,
112 K.170 a
3/2
112
112
EXPT. b
EXPT. b
c
Fig. 3. Spectrum b f 23Na: (a) calculated, Kuo + tTO; (b) experiment, positive parity states; (c) as (a) but with K = 3/2 band, indicated by arrows in (a), shifted downwards, and the ground states re-aligned.
wipes out the bands, and we cannot expect any really clear picture to emerge, although we believe that a shifted band is discemable in 33p. It is attractive to ascribe the cause o f the generally poor fits obtained in untruncated calculations in this region to the tendency of the interaction to try to shift rather thoroughly mixed bands relative to one another. It is natural to ask what it is about the Kuo interaction that gives rise to this wholesale band shifting. Feldmeier et al. [5] have drawn attention to the socalled antisymmetric spin-orbit (ALS) part of the effective nucleon-nucleon interaction. They have shown that if the ALS contribution to the Kuo matrix ele-
Fig. 4. As fig. 3 but for 33p. The shifted band has K = 3/2 here also. ments is increased by a factor of about 5 the K = 2 band in 24Mg is shifted upwards relative to the groundstate band by just the right amount. They also report preliminary calculations which indicate similar behaviour for the K = 1/2 band in 23Na, which has been discussed above. This is rather attractive but there seems to be no obvious connection between the Als interaction and rotational bands. The same can be said for the device of increasing the one-body spin-orbit interaction used in our discussion of 24Na. The ALS interaction appears to produce the more satisfactory band shifts, without unduly disarranging the rest of the spec trum, in the case of 24Mg, and therefore seems to be the most promising possibility. We may conclude that the Kuo interaction seems capable of producing surprisingly good results for a 431
Volume 45B, number 5
PHYSICS LETTERS
variety of the less extensively studied sd-shell nuclei, provided the band shifts are taken into consideration. This raises the tantalising possibility that, if the band shift problem can be resolved satisfactorily, we may be in a position to expect calculations using parameter-free realistic interactions to yield fits comparable to those presently obtained with heavily parameterised effective interactions and least squares fitting. We wish to thank Drs. B.H. Wildenthal, H. Feldmeier and N. MacDonald for valuable discussions, and the Computing and A u t o m a t i o n Division RHEL for access to their 360/195 on which the calculations were done.
432
20 August 1973
References [1] R.R. Whitehead, Nucl. Phys. A182 (1972) 290. [2] R.R. Whitehead and A. Watt, Phys. Lett. 35B (1971) 189. [3] R.R. Whitehead and A. Watt, Phys. Lett. 41B (1972) 7. [4] D. Strottman, Phys. Lett. 39B (1972) 457. [5] H. Feldmeier, P. Manakos and T. Wolff, Preprint (1972). [6] T.T.S. Kuo, Nucl. Phys. A103 (1967) 71. [7] E. Halbert et al., in Advances in Nuclear Physics ed. M. Baranger and E. Vogt (Plenum, New York, 1971) Vol. IV. [8] B.H. Wildenthal, private communications. [9] R.R. Betts and H.T. Fortune, Phys. Rev. C7 (1973) 1257. [10] J. Sharpey-Schaffer et al., Phys. Rev. Lett. 27 (1971) 1463. [11] V. Pucknell, private communications.