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Wave functions of 0+ levels in 38Ca
Volume 104B, number 6
PHYSICS LETTERS
17 September 1981
WAVE FUNCTIONS OF 0 + LEVELS IN 38Ca H.T. FORTUNE and L. BLAND Physics Department, University of Pennsylvania Philadelphia, PA 19104, USA Received 27 March 1981
A simple model of the first three 0 ÷ levels in 38Ca gives a good account of the excitation energies and (3He, n) cross sections. The hamiltonian derived is reasonable and the mixing is similar to that needed for other nearby nuclei.
The presence of simple modes of core-excitation in 40Ca and in nuclei just above 4°Ca has been recognized for a long time [ 1] and is now quite well understood [2]. F o r nuclei just below 40Ca, the situation should be similar, but up to now no simple results have emerged. In particular, i n A = 38 we expect configurations with two or more nucleons in the fp shell, in addition to the zeroth order ( s d ) - 2 component. The most sensitive test of such admixtures is twonucleon transfer onto 36Ar, because the cross section for putting two nucleons into the fp shell is very much larger than that for adding them to the sd shell. The reactions 36Ar(t, p)[3] and 36Ar(3He, n) [4] have been performed, but with no definitive results concerning the structure of the final states. In ref. [3], it was reported that excited 0 + strength was missing, i.e. no state was observed with even a reasonable fraction of the 36Ar ® (fp)2(t,p) strength. There are three low-lying 0 ÷, T = 1 states in mass 38 - in 38Ca their energies are [5] 0.00, 3.10, and 4.75 MeV. In 36Ar(3He, n) all three have measurable
cross sections, as do their mirrors in 38Ar, via 36Ar(t, p). These cross sections are listed in table 1. In 4°Ca (p,t) [6], the 3.1 MeV state is very weak - its cross section is only about one percent of that for the ground state. And the 4.75 MeV state is not observed. Neither of these excited 0 + levels is present in an (sd) - 2 shell-model calculation, which puts the second 0 + level at about 6 MeV [7]. From its (p, t) strength [6], this presumably corresponds to the experimental state at 6.27 MeV in 38Ca. For purposes of understanding the 36Ar(3He, n) reaction this second (sd) - 2 shell-model state can be ignored since its predicted (3He, n) cross section (using amplitudes from Chung and Wildenthal [7] is only about two percent o f that predicted for the ground state. It is tempting to consider the 0 + states of 38Ca to be of the form 36Ar ® (sd) 2 [i.e. ( s d ) - 2 ] and 36Ar ® 42Ti. In 38Ar we would then have (sd) - 2 and 36Ar @ 42Ca. This idea is reinforced by the observation that the splitting between the second and third 0 + levels of mass 38 is very similar to that between
Table 1 Low-lying 0 + states in 3SAr, 38Ca. Ex(38Ar) a) (MeV)
o(t, p) b) (a.u.)
Ex(3SCa) a) (MeV)
o(3He, n) c) (mb/sr)
o(p, t) d) (mb/sr)
Ex(sd)2 e) 0VleV)
0.0 3.38 4.71
0.20 0.08 0.04
0.0 3.10 4.75 6.27
2.8 0.42 0.18 -
0.55 0.008 0.040
0.00 5.83
a) Ref. [5]. 426
b) Ref. [3].
c) Ref. [4].
d) Ref. [6].
e) Ref. [9].
Volume 104B, number 6
PHYSICS LETTERS
the first and second 0 + states in mass 42. The excited 0+ states in mass 38 are not simply (fp)Z(sd)-4, however, because the mass-42 states contain appreciable core excitation. It is also apparent that considerable mixing is present because it is the g.s. that is strongest in two-nucleon transfer [3,4] while in the absence of mixing, it would be the first excited 0 + state that is strongest [(fp)2 transfer is considerably stronger than (sd) 2 transfer]. A recent systematic study [8] of (t, p) reactions throughout the (sd) shell noted that the 36At -+ 38Ar(gs) cross section was enhanced more than a factor of two over that calculated in a pure (sd) n basis. This enhancement is consistent with the mixing proposed above, i.e. with admixtures of (fp)2 into the 38Ar ground state. In what follows we attempt to understand the first three 0 + levels in mass 38 in terms of simple core excitations. If we take as basic states (sd)~-?, (fp)021 (sd)0-4 , and (fp)40(sd)~-6 where the subscripts denote JT, then we have a model that is similar to that successfully applied to 42Ca [1,2, 9]. One set of wave functions that gives reasonable agreement with available data is listed in table 2. If we require these eigenfunctions to correspond to excitation energies of 0, 3.10 and 4.75 MeV in 38Ca, then the hamiltonian is
042-115 H= UTEU = i--1.15 L0
where E is the diagonal energy matrix and the rows of U are the eigenfunctions. These mixing matrix elements are very similar to those obtained [9] for a similar problem in 42Ca. To check the values of the diagonal matrix elements, we note that in weak coupling [10] for 38Ca, Ex(2 p - 4h) - E x (0p - 2h) = M(42Ti) + M(36Ar) - M ( 3 8 C a ) + 8a - 4 c , where the M ' s are ground-state binding energies relative to 40Ca. Thus, AEx(2 p -- 4h) = 1.52 MeV + 8a - 4 c . In 42Ca, the unperturbed position of the 4 p - 2 h 0 + level was calculated [9] to be 2.14 MeV. But, Ex(4 p - 2h) =M(44Ti) +M(38Ar)
M(42Ca) + 8 a - 4 c
= 1.12MeV+8a-4e, .'. 8a - 4c = 1.02 MeV, so that A E x ( 2 p - 4 h ) = 1.52 + 1.02 = 2.54 MeV in 38Ca, reasonably close to the value 2.9 MeV in the hamiltonian above. An energy difference of 0.8 MeV between unperturbed 2 p - 4 h and 4 p - 6 h states also appears quite reasonable. We now proceed to calculate the 36Ar(3He, n) cross sections using the wave functions derived above. The DWBA calculations were performed with the code DWUCK [11], using the optical-model parameters listed in table 3.
0
3.31 -0.86
17 September 1981
--0.86~, 4.12j
Table 2 Wave fun~ions for 38Ca(0+) st~es and two-nuc~on transfer amplitudes for 36Ar ~ 38Ca(0+). Ex (MeV)
0p-2h
2p-2h
4p-6h
(lds~) 2
0 3.10 4.75
0.936 0.314 0.157
0.344 -0.729 -0.592
0.072 -0.608 0.791
-0.2866 0.0974 0.0465
(2Sl/2) 2
(ld3/2) 2
-0.1812 0.0616 0.0294
-0.6077 0.2064 0.0987
(lf7/2) 2
(2p3/2) 2
0.3263 0.6925 0.5597
0.1088 0.2308 0.1866
Table 3 Optical-model parameters used in DWBA calculations for 36Ar(3He, n)aSCa. Strengths in MeV, lengths in fro. .
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Channel
V
ro
a
W
W' = 4WD
ro'
a'
Vso
3He n b.s.
150 54.5 -
1.20 1.25 1.26
0.72 0.65 0.60
38.2 0.53 .
0 45.0
1.40 1.26
0.88 0.58
2.5 6.2 X= 25
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427
Volume 104B, number 6
PHYSICS LETTERS
o
!
3eAr(3He,n)38Ca, E(3He):ll.5 MeV
,oo )\
17 September 1981
Table 4 Percentage of 2p-4h in lowest three 0÷ levels of 3SCa. Ex (MeV)
Present (sd)-4 (fp)2 (%)
Haspera) (ld3n)-4 (lfTn)2 (%)
0.00 3.10 4.75
12 53 35
14 53 40
a) Ref. [12].
0
30
60
e c.m.(deg )
90
Fig. 1. Angular distr~utions from ref. [4] for 36Ar(3He, n)38Ca to the lowest three 0 + levels of 38Ca. Curves are results of DWBA calculations using the wave functions derived herein, and the optical-model parameters listed in table 2. All three curves have the same normalization.
For the two-nucleon transfer amplitudes for 36Ar -+ 0 p - 2 h , we use those from Chung and Wildenthal [7] for 36Ar-+ 38Ca(g.s.). For 36Ar-~ 2 p - 4 h , we assume the 4h are the 36Ar(g.s.) and the 2p wave function to be of the form 0vCoffo.90(lf7/2)2 + 0,v/~.10(2P3/2) 2, as needed for nuclei just above 4°Ca. Of course, the amplitude for 36Ar ~ 4 p - 6 h is zero in the present model. The resulting transfer amplitudes are listed in table 3. Calculated angular distributions are compared with the data of ref. [4] in fig. 1. All three curves have been plotted with the same normalization. Thus, the present wave functions give an excellent account of the 36Ar(3He, n) reaction. The amount of 2 p - 4 h in the 38Ca g.s. is only about 14%, but it is sufficient to make the g.s. the strongest 0 + state in 2p transfer. This g.s. wave function also gives the correct factor-of-
428
two enhancement for 36At ~ 38Ar(g.s.) noted in ref, [8]. It is interesting to compare the 2p - 4h components of these wave functions with those from a recent shell-model calculation of Hasper [ 12], who considers only (1d3/2)n(lf7/2) m . This is done in table 4. We see that both calculations give about the same amount of 2 p - 4 h configuration. However, a big difference is in the remainder of the wave functions, since Hasper considers only zero and two particles excited into the lf7/2 orbital. It would be of interest to calculate the 36Ar(3He, n) cross sections with Hasper's wave functions. Unfortunately, the necessary transfer amplitudes are currently not available for such a calculation. We wish to acknowledge financial support by the National Science Foundation. References [1] W.J. Gerace and A.M. Green, Nucl. Phys. A93 (1967) 110. [2] H.T. Fortune, in: Proc. Intern. Conf. on Nuclear structure (Tokyo, 1977) ed. T. M~umori, Vol. II (Physical Soc. of Japan, Tokyo, 1978) p. 99. [3] E.R. Flynn et al., Nucl. Phys. A246 (1975) 117. [4] W. Bohne et al., Nucl. Phys. A284 (1977) 14. [5 ] P.M. Endt and C. van der Leun, Nucl. Phys. A300 (1978) 1 [6] S. Kubono et al., Phys. Lett. 49B (1974) 37; Nucl. Phys. A276 (1977) 201. [7] W. Chung and B.H. Wildenthal, private communication. [8] H.T. Fortune et al., Phys. Lett. 87B (1979) 29. [9] H.T. Fortune and M.E. Cobern, Phys. Lett. 77B (1978) 21. [10] R. Bansal and J.B. French, Phys. Lett 11 (1964) 145; L. Zamick, Phys. Lett. 19 (1965) 580. [11] P. D. Kunz, private communication. [12] H. Hasper, Phys. Rev. C19 (1979) 1482.
PHYSICS LETTERS
17 September 1981
WAVE FUNCTIONS OF 0 + LEVELS IN 38Ca H.T. FORTUNE and L. BLAND Physics Department, University of Pennsylvania Philadelphia, PA 19104, USA Received 27 March 1981
A simple model of the first three 0 ÷ levels in 38Ca gives a good account of the excitation energies and (3He, n) cross sections. The hamiltonian derived is reasonable and the mixing is similar to that needed for other nearby nuclei.
The presence of simple modes of core-excitation in 40Ca and in nuclei just above 4°Ca has been recognized for a long time [ 1] and is now quite well understood [2]. F o r nuclei just below 40Ca, the situation should be similar, but up to now no simple results have emerged. In particular, i n A = 38 we expect configurations with two or more nucleons in the fp shell, in addition to the zeroth order ( s d ) - 2 component. The most sensitive test of such admixtures is twonucleon transfer onto 36Ar, because the cross section for putting two nucleons into the fp shell is very much larger than that for adding them to the sd shell. The reactions 36Ar(t, p)[3] and 36Ar(3He, n) [4] have been performed, but with no definitive results concerning the structure of the final states. In ref. [3], it was reported that excited 0 + strength was missing, i.e. no state was observed with even a reasonable fraction of the 36Ar ® (fp)2(t,p) strength. There are three low-lying 0 ÷, T = 1 states in mass 38 - in 38Ca their energies are [5] 0.00, 3.10, and 4.75 MeV. In 36Ar(3He, n) all three have measurable
cross sections, as do their mirrors in 38Ar, via 36Ar(t, p). These cross sections are listed in table 1. In 4°Ca (p,t) [6], the 3.1 MeV state is very weak - its cross section is only about one percent of that for the ground state. And the 4.75 MeV state is not observed. Neither of these excited 0 + levels is present in an (sd) - 2 shell-model calculation, which puts the second 0 + level at about 6 MeV [7]. From its (p, t) strength [6], this presumably corresponds to the experimental state at 6.27 MeV in 38Ca. For purposes of understanding the 36Ar(3He, n) reaction this second (sd) - 2 shell-model state can be ignored since its predicted (3He, n) cross section (using amplitudes from Chung and Wildenthal [7] is only about two percent o f that predicted for the ground state. It is tempting to consider the 0 + states of 38Ca to be of the form 36Ar ® (sd) 2 [i.e. ( s d ) - 2 ] and 36Ar ® 42Ti. In 38Ar we would then have (sd) - 2 and 36Ar @ 42Ca. This idea is reinforced by the observation that the splitting between the second and third 0 + levels of mass 38 is very similar to that between
Table 1 Low-lying 0 + states in 3SAr, 38Ca. Ex(38Ar) a) (MeV)
o(t, p) b) (a.u.)
Ex(3SCa) a) (MeV)
o(3He, n) c) (mb/sr)
o(p, t) d) (mb/sr)
Ex(sd)2 e) 0VleV)
0.0 3.38 4.71
0.20 0.08 0.04
0.0 3.10 4.75 6.27
2.8 0.42 0.18 -
0.55 0.008 0.040
0.00 5.83
a) Ref. [5]. 426
b) Ref. [3].
c) Ref. [4].
d) Ref. [6].
e) Ref. [9].
Volume 104B, number 6
PHYSICS LETTERS
the first and second 0 + states in mass 42. The excited 0+ states in mass 38 are not simply (fp)Z(sd)-4, however, because the mass-42 states contain appreciable core excitation. It is also apparent that considerable mixing is present because it is the g.s. that is strongest in two-nucleon transfer [3,4] while in the absence of mixing, it would be the first excited 0 + state that is strongest [(fp)2 transfer is considerably stronger than (sd) 2 transfer]. A recent systematic study [8] of (t, p) reactions throughout the (sd) shell noted that the 36At -+ 38Ar(gs) cross section was enhanced more than a factor of two over that calculated in a pure (sd) n basis. This enhancement is consistent with the mixing proposed above, i.e. with admixtures of (fp)2 into the 38Ar ground state. In what follows we attempt to understand the first three 0 + levels in mass 38 in terms of simple core excitations. If we take as basic states (sd)~-?, (fp)021 (sd)0-4 , and (fp)40(sd)~-6 where the subscripts denote JT, then we have a model that is similar to that successfully applied to 42Ca [1,2, 9]. One set of wave functions that gives reasonable agreement with available data is listed in table 2. If we require these eigenfunctions to correspond to excitation energies of 0, 3.10 and 4.75 MeV in 38Ca, then the hamiltonian is
042-115 H= UTEU = i--1.15 L0
where E is the diagonal energy matrix and the rows of U are the eigenfunctions. These mixing matrix elements are very similar to those obtained [9] for a similar problem in 42Ca. To check the values of the diagonal matrix elements, we note that in weak coupling [10] for 38Ca, Ex(2 p - 4h) - E x (0p - 2h) = M(42Ti) + M(36Ar) - M ( 3 8 C a ) + 8a - 4 c , where the M ' s are ground-state binding energies relative to 40Ca. Thus, AEx(2 p -- 4h) = 1.52 MeV + 8a - 4 c . In 42Ca, the unperturbed position of the 4 p - 2 h 0 + level was calculated [9] to be 2.14 MeV. But, Ex(4 p - 2h) =M(44Ti) +M(38Ar)
M(42Ca) + 8 a - 4 c
= 1.12MeV+8a-4e, .'. 8a - 4c = 1.02 MeV, so that A E x ( 2 p - 4 h ) = 1.52 + 1.02 = 2.54 MeV in 38Ca, reasonably close to the value 2.9 MeV in the hamiltonian above. An energy difference of 0.8 MeV between unperturbed 2 p - 4 h and 4 p - 6 h states also appears quite reasonable. We now proceed to calculate the 36Ar(3He, n) cross sections using the wave functions derived above. The DWBA calculations were performed with the code DWUCK [11], using the optical-model parameters listed in table 3.
0
3.31 -0.86
17 September 1981
--0.86~, 4.12j
Table 2 Wave fun~ions for 38Ca(0+) st~es and two-nuc~on transfer amplitudes for 36Ar ~ 38Ca(0+). Ex (MeV)
0p-2h
2p-2h
4p-6h
(lds~) 2
0 3.10 4.75
0.936 0.314 0.157
0.344 -0.729 -0.592
0.072 -0.608 0.791
-0.2866 0.0974 0.0465
(2Sl/2) 2
(ld3/2) 2
-0.1812 0.0616 0.0294
-0.6077 0.2064 0.0987
(lf7/2) 2
(2p3/2) 2
0.3263 0.6925 0.5597
0.1088 0.2308 0.1866
Table 3 Optical-model parameters used in DWBA calculations for 36Ar(3He, n)aSCa. Strengths in MeV, lengths in fro. .
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Channel
V
ro
a
W
W' = 4WD
ro'
a'
Vso
3He n b.s.
150 54.5 -
1.20 1.25 1.26
0.72 0.65 0.60
38.2 0.53 .
0 45.0
1.40 1.26
0.88 0.58
2.5 6.2 X= 25
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427
Volume 104B, number 6
PHYSICS LETTERS
o
!
3eAr(3He,n)38Ca, E(3He):ll.5 MeV
,oo )\
17 September 1981
Table 4 Percentage of 2p-4h in lowest three 0÷ levels of 3SCa. Ex (MeV)
Present (sd)-4 (fp)2 (%)
Haspera) (ld3n)-4 (lfTn)2 (%)
0.00 3.10 4.75
12 53 35
14 53 40
a) Ref. [12].
0
30
60
e c.m.(deg )
90
Fig. 1. Angular distr~utions from ref. [4] for 36Ar(3He, n)38Ca to the lowest three 0 + levels of 38Ca. Curves are results of DWBA calculations using the wave functions derived herein, and the optical-model parameters listed in table 2. All three curves have the same normalization.
For the two-nucleon transfer amplitudes for 36Ar -+ 0 p - 2 h , we use those from Chung and Wildenthal [7] for 36Ar-+ 38Ca(g.s.). For 36Ar-~ 2 p - 4 h , we assume the 4h are the 36Ar(g.s.) and the 2p wave function to be of the form 0vCoffo.90(lf7/2)2 + 0,v/~.10(2P3/2) 2, as needed for nuclei just above 4°Ca. Of course, the amplitude for 36Ar ~ 4 p - 6 h is zero in the present model. The resulting transfer amplitudes are listed in table 3. Calculated angular distributions are compared with the data of ref. [4] in fig. 1. All three curves have been plotted with the same normalization. Thus, the present wave functions give an excellent account of the 36Ar(3He, n) reaction. The amount of 2 p - 4 h in the 38Ca g.s. is only about 14%, but it is sufficient to make the g.s. the strongest 0 + state in 2p transfer. This g.s. wave function also gives the correct factor-of-
428
two enhancement for 36At ~ 38Ar(g.s.) noted in ref, [8]. It is interesting to compare the 2p - 4h components of these wave functions with those from a recent shell-model calculation of Hasper [ 12], who considers only (1d3/2)n(lf7/2) m . This is done in table 4. We see that both calculations give about the same amount of 2 p - 4 h configuration. However, a big difference is in the remainder of the wave functions, since Hasper considers only zero and two particles excited into the lf7/2 orbital. It would be of interest to calculate the 36Ar(3He, n) cross sections with Hasper's wave functions. Unfortunately, the necessary transfer amplitudes are currently not available for such a calculation. We wish to acknowledge financial support by the National Science Foundation. References [1] W.J. Gerace and A.M. Green, Nucl. Phys. A93 (1967) 110. [2] H.T. Fortune, in: Proc. Intern. Conf. on Nuclear structure (Tokyo, 1977) ed. T. M~umori, Vol. II (Physical Soc. of Japan, Tokyo, 1978) p. 99. [3] E.R. Flynn et al., Nucl. Phys. A246 (1975) 117. [4] W. Bohne et al., Nucl. Phys. A284 (1977) 14. [5 ] P.M. Endt and C. van der Leun, Nucl. Phys. A300 (1978) 1 [6] S. Kubono et al., Phys. Lett. 49B (1974) 37; Nucl. Phys. A276 (1977) 201. [7] W. Chung and B.H. Wildenthal, private communication. [8] H.T. Fortune et al., Phys. Lett. 87B (1979) 29. [9] H.T. Fortune and M.E. Cobern, Phys. Lett. 77B (1978) 21. [10] R. Bansal and J.B. French, Phys. Lett 11 (1964) 145; L. Zamick, Phys. Lett. 19 (1965) 580. [11] P. D. Kunz, private communication. [12] H. Hasper, Phys. Rev. C19 (1979) 1482.