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An application of the weak coupling model to the negative-parity states of the A = 15 nuclei
N&ear
Physics
A169 (197 1) 6 17-628 ; @ North-Ho~iand
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AN APPLICATION OF THE WEAK COUPLING MODEL TO THE NEGATIVE-PARITY STATES OF THE A = 15 NUCLEI S. LIE and T. ENCELAND Institatte
of Physics,
University
of Oslo, Blindern,
Norway
t
Received 15 March 1971 The negative-parity states for A = 1.5 have been investigated in a weak coupling model. Based on particles in the sd shell and holes in the p-shell the final eigenstates are found to be admixtures ofOp-lh, lp-2h r, = 0 and lp-2h T,, = 1 configurations. Good agreement is obtained with the experimentaf energy levels below 11 MeV. The correspondence between levels in ‘5N and I50 is discussed.
Abstract:
1. Introduction In a previous paper “) (hereafter referred to as II) we investigated the positiveparity states with T = 3 of the A = 15 nuclei. A weak coupling model developed by Ellis and Engeland (ref. ‘), hereafter referred to as I) was used. Within the framework of the spherical shell model the lp and the 2sld single-particle orbitals were included in the construction of the basic states. The final eigenfunctions were given in terms of particle-hole (p-h) excitations of the closed and spherical lp shell. With this procedure a systematic study of the possible p-h excitations were performed. Good agreement was obtained with the experimental positive-parity states below 10 MeV. In the present paper the study of the A = 15 nuclei is completed with a similar investigation of the negative-parity i” = 4 levels. The two lowest negative-parity states have usually been regarded as the pi ’ and p+ ’ single-hole states, see for example refs * 3*41%This conclusion is supported by pick-up reactions 5-9) on ’ 6O which show that these levels nearly exhaust all the p-l strength. Furthermore, in the region of 6.5 to 13 MeV excitation energy Warburton et al. “) did not find any cross section larger than f of the cross section to the 3; level. On the other hand the Ml/E2 mixing ratios for the 3; -4; y-transition in both 1“0 and ’ ‘N indicate collective admixtures in the single-hole states. Also additional negative-parity states have been established which indicate a more complex structure. Several *O- 12) suggestions have been made in an effort to give a consistent description of the negative-parity states. Particle-hole configurations which include up to five holes in the p-shell have been proposed and also excitations to the pf shell. By use of the weak coupling model discussed in I we investigate the possible p-h excitations to find those configurations which are important for the low-lying spect Work supported in part by the Norwegian Research Council for Science and Humanities. 617
618
S. LIE AND
T. ENGELAND
trum. The unperturbed energy position of the different p-h excitations are used as a method to reduce the many-particle configuration space in which the final eigenfunctions are expanded. With the present formulation of the model only the lp and the 2sld single-particle orbitals are considered. In sect. 2 the basic equations are given together with the results for the energy levels and the y-transitions. The results are discussed and compared with experiments in sect. 3. Our conclusion is given in sect. 4 where both positive- and negative-parity states are included. A discussion of the oneto-one correspondence between levels in r5N and IsO below 11 MeV is presented.
2. Calculations
and results
To establish our notation and the form in which the final results are presented we give the basic equations which have to be solved. Further details are found in I and II. We write the Hamiltonian in the form
H = H,+H,+I;,,, where HP and Hh are the Hamiltonian and VP,, the particle-hole interaction. I@C%,.)
(1)
for the sd and the p-shell particles respectively Our wave functions are defined by
= [I(sdP’~,J,
~,)l~“%Jt,
Th)l.m~-
(2)
The square bracket implies coupling of angular momenta JP and J,, to J and isospin TP and T,, to T.The additional quantum numbers y are chosen such that H,l(sdPyp
J, Tp M,J
= EIJ;~J,~,~Jsd)“‘~,Jp
Tp GP)’
H,IP”“Y,J~ ThMT,) = E;:J,,T,,M~,)P”%J, ThM,;).
(3)
These equations imply a separate diagonalization of the nP particle problem in the sd shell and the nh hole problem in the p-shell. For the sd shell part the corresponding eigenfunctions are expanded in a basis given by
IW”“~p J, T,,bP>
=
@4’W-,1& &K, L, S,; J, TpMT,).
(4)
We have used the L-S coupling scheme and further classified the wave functions by the partition [f,] which specifies the orbital symmetry and the SU, quantum numbers (Q,,) introduced by Elliott 13) with the orthogonal Kp label of Vergados 14). The classification of the p-shell wave functions is similar except that only one of the labels [f] and (1~) is required. Having performed the calculation indicated by eqs. (3), a set of p-h states l@Zt;nTMT) can be constructed and a matrix for the total Hamiltonian H built up. The expression for these matrix elements are given in II, eq. (5). This procedure incorporates the main correlations of particles and holes in the solutions of eqs. (3). Thus the final eigenfunctions which take the form
A = 15 NUCLEI
are in most cases well described
619
by a few terms. The eigenfunctions
of HP and Hh
necessary in the present calculation are tabulated in I, tables 1 and 2. The p-she11 matrix elements of Cohen and Kurath “) and a revised version of the sd shell matrix elements of Kuo and Brown ’ “) were used.
w
--._
._._
llC2 7094 _:/2+i:2 -
4
1
3/ (-
1 54
5;,:
1091
712 +
1045
757
712 *
7.55
l/2 +,
' 730
3/2+ 5127
' 728
712+' I 5/2f!
y 679 I
312 -
’ 632
-------._.j 61a
530
i/2+
527
512 +
L
3/z+
3/2-I
f 112-
I-
'5N Fig. 1. The experimental spectra for 15N and l5O. The data are taken from ref. I”) with some modifications which are explained in the text. In “0 our suggested spin values are given in brackets. Mirror levels are connected by solid lines if the correspondence is well established lo) and by dashed lines if the correspondence is uncertain or speculative.
The calculation of the negative T = f states are performed for the I50 case, but due to the one-to-one correspondence between the levels in the two nuclei ’ “) the wave
S. LIE AND T. ENGELAND
620
functions are assumed to be valid for ’ 5N as well. The energy level schemes with identification of mirror levels are shown in fig. 1. All the p-h configurations with unperturbed energy positions below 20 MeV are included in the calculation. The basis for the diagonalization of H in eq. (1) then con12
11/z
7'2- 5'2 g12-
32
-
32 -
-
5/2
-3/2
3/2
-512
912.L J/2 -
112 -
11
7/Z\ 11/2-
it2 -
512
I’
,-
112
712 -
10
312
312 ,_.__ 512 -
9
0
112 -
-
3/2
7
3/2 -----
6
-3/2
(2.53MeV) 112 l/2 -------
0 Op-lh
2p-3h
Zp-3h
Tp=O
Tp=l
MIXED
112
EXPERIMENT
‘50
Fig. 2. The calculated and experimental positive parity levels for 150. The first three columns show the results of separate diagonalization of the Op-lh, 2p-3h TD = 0 and 2p-3h TD =z t configurations respectively. Correspondence between experimental and theoretical levels is indicated by dashed lines.
sists of Op-lh, 2p-3h and 4p-5h configurations. An estimate of the lowest 6p-7h more than 30 MeV and is therefore excluded. For the particle-hole interaction VP, the Gillet potential 16)
gives
621
A = 15 NUCLEl
is used where V, = -40 MeV, W = M = 0.33, B = -0.10, H = 0.40 and b/,u = l.O(b = (ii/m@)*). As discussed in I and II the problem of spurious C.M. excitations has been investigated. Highly spurious states are removed from the basis before the diagonalization of N. In some cases the p-h basis has been expanded above the limit of 20 MeV in order to obtain a meaningful projection. By the use of experimental binding energies (see I) our energy scale is chosen such that the physical ground state comes at zero energy. The unperturbed positions of the Op-lh states are not fixed by this procedure. In the present case they are treated as parameters and determined by the requirement that the lowest +- and $- eigenstates should come at the experimental energy. The calculated and the experimental energy levels are shown in fig. 2. The three types of excitations Op-lh, 2p-3h Tp = 0 and 2p-3h T, = 1 are also diagonalized separately in order to show the importance of the various modes. The 4p-5h states come above 12 MeV and are not shown in the figure. The final eigenstates are given in table 1
TABLE
The excitation energies and the wave functions J”
Excitation energy
-~ lh
(MeV)
Wave functions “1
Strength ( %) 4p-5h
Zp-3h
Fe = 0 r, = 1 -
61-
0.00 b)
4X-1 -
8.57
;:-
10.72 6.18 bf
85
10
5
0
0.92]12,15>+0.21]
1,14)-0.15]10,13?
3
5
91
1
0.881 7,13>+0.31]
8,14)--0.16]12,15)
0
95
3
2
0.931 1,13)-0.16]
4,14)
79
II
10
0
0.89/12,16>+0.23/
7,14>-0.211
-0.21j a2-
3 -
23
%I-
9.45
10.65
9.77
12
1
0
29
57
26
57
42
73
2
0
1
0.621 8,13)+0.43] -0.341
7,14)+0.23]
+0.19]
5,13)
0.601 1,13>-0.481
10.61
0
77
23
0
11.34
0
31
69
0
4,14>+0.17/
%*-
8,14)+0.231
0.811 2,13>+0.33]
8,14) 2,14>
6,13> 2,13>+0.28/
2,14)
4,13) 9,13)+0.29]
3,14)
3,13)+0*31/
3,14) 3,14>
8,141
0.79j 9,13)1-0.45[ -0.25/
+I-
8,13)-0.341
+0.19/
-0.291 &-
1,13)+0.34/12,16) 2,14>-0.20]10,14>
4,13>-0.22]11,14>+0.20]
0.781 8,13>+0.34]
1,13>
8,13)
+0.28/
-0.28/ %I--
1,14>-0.17/
9,14>
12.58
0
66
34
0
0.691 3,13)-0.581
9,13>+0.39]
11.79
0
94
6
0
0.971 3,13>-0.241
9,14)
“) The wave functions are labehed according to the numbers in table 2. Amplitudes less than 0.15 are omitted. b, The energy is fitted to the experimental value.
622
S. LIE AND T. ENGELAND
I. As a reference to the individual components in the wave functions we use the form Ik,, kh ) where k, and k, are the numbers labelling the particle and the hole states respectively according to table 2. TABLE
2
The basic particle and hole states “) Reference number
1
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
n,
J”
T
Eigenvalue
(MeV) 2 2 2 2 2 2
1’ 3+ 5+ 2+ 12$
2 2 2 2 2 0
0+ 2+ 4’ 02 + 22+ 0+ 33-
13 +
3 3 1 1
“) Further details about the wave functions
0 0 0 0 0
0.00 1.15 1.46 3.33 3.76
0 1 1 1 1 1 0 t t 4 *
6.51 0.00 2.01 3.46 3.65 3.98 0.00 0.00 3.68 0.00 6.18
are given in ref. *).
From the negative-parity states below 11 MeV we have calculated the y-transition strengths to all lower-lying levels. The necessary wave functions for the positive-parity states are taken from II where further details of the calculation are givers. As before an additional polarization charge of 0.5e for both protons and neutrons is used in the calculation of the E2 and E3 transition rates. In evaluating the radial matrix elements the harmonic oscillator functions are used with a length parameter b = (~~~~~~ = 1.7 fm. The results are shown in table 3 which contains all transitions where either the experimental or the calculated branch is larger than 1 %. 3. Discussion Except for the lowest negative-parity levels the experimental situation in 150 is rather unclear. In a recently published compilation 1‘) five levels, at 8.98, 9.48, 9.61, 9.66 and 11.02 MeV in the region between 4.2 and 1I .l MeV have been assigned negative parity. The spin of the 9.48,9.61 and 11.02 MeV levels seem to be well established as 3-, $- and $- respectively. For the remaining ones the experimental information does not allow a unique determination. In addition to the levels mentioned above Honsaker et al. I*) concluded from their r3Ct3He, n)“O experiment that the level
623
A = 15 NUCLEI TABLE 3
Electromagnetic Ji”
JF
transitions
Multipolarity
from
Experiment
(eV)
Ml El Ml E2 El Ml E2 El El Ml E2 El E2 El El El Ml E2 El El Ml E2 El El
>
1.4.10-Z
Theory
Branching ratio
Branching ratio
100 %
100%
10-Z 10-Z
21 % 23 %
42 % 31 %
9.4.
10-2
30 %
25 %
8.2.
1O-2
26 %
3%
94 %
97 %
0.74 10-z
2.1 0.15 0.02
6%
85 % 7% 0.7 %
0.08 0.11
3% 5%
4.0
79 %
1.0
19%
0.1
1
3%
82 % 7% 4% ( 2 % 2% 5% 78 % 15 %
1
2%
5% 2%
El E2 M2 El El
84 % 2% 4% 2% 7%
Ml El Ml El
“) “) ‘) d, ‘) ‘) =)
in A = 15 nuclei for I50
‘5N
6.6. 7.2.
4.6.
states
“)
‘50 r-width
Ml E2
the negative-parity
The experimental data are taken from The width in W.U. Identified with the 8.92 MeV level in Identified with the 8.98 MeV level in Identified with the 9.48 MeV level in Identified with the 9.61 MeV level in Identified with the 9.66 MeV level in
r-width
1.1 2.4. 6.1 2.6. 8.0. 1.3
2.8 . 10-z
the the the the the
9.23 9.152 9.76 9.93 9.83
100%
IO-’
. lo-*
21 % 54 %
1O-z IO-“
24 %
10-S
1%
0.81 7.4. 10-d 0.15 0.55 2.8. 10-z 4.1 4.7. 2.0. 3.1 . 1.3. 5.2. 3.1 3.3 9.1
52 % 10 % 36 % 2%
10-S
7% 81 %
10-z 10-4 10-d 10-S 1O-3
3% 9%
. 10-Z 6%
10-S . 10-Z 0.37 2.0. 10-Z 9.3 . 10-b 6.1 . lo-* 9.1 . 10-3
16 % 63 % 4% 10% 1%
0.42 1.2. 10-X not talc. 4.3 . 10-a 2.7. 1O-3
99 %
0.24 1.7. 10-z 3.4. 10-Z 2.8 . lo-’
75 % 5% 11% 9%
ref. I’). In some cases average I50 and I50 and I50 and I50 and I50 and
Branching ratio
(eV)
MeV MeV MeV MeV MeV
level level level level level
1%
values in in in in in
lSN. lSN. lSN. lSN. 15N.
are used.
r/r,
“)
0.56 0.66 1.6. 1o-3 2.8. 1O-3 5.7. 1o-2 2.6 3.2. IO-“ 5.4 7.0 6.6. 2.6. 6.1
10-Z 1o-3 1o-3 lo-’ lo-’
2.9. lo-’ 1.5 1o-3 0.27 3.8 . 1O-5 1.8. 1O-4 1.2. 1o-3 1.6. 1O-3 2.2. 1o-2 2.5. 1O-3 1.1 . 10-Z 2.4. 1o-2 1.1 6.7 . 1O-3 1.1 . 10-S 1.2. lo-* 1.3 not talc. 4.7 10-S 4.8. 1O-4 8.4. 2.1 . 1.5. 9.0.
1O-3 10-d 1o-z 1o-4
624
S. LIE AND T. ENGELAND
at 8.92 MeV has spin -t_-. The angular distribution showed a distinct peak in the forward direction which was interpreted as an L = 0 transfer. The final spin value was obtained using a model-dependent argument for the reaction mechanism. The same conclusion of spin 3- was found for the 8.98 and 9.66 MeV levels. However, in these cases their angular distributions were more uncertain. From a theoretical point of view it is very difficult to understand that the spectrum should have four excited +- states below 11.1 MeV. Our calculation which involves a large number of p-h excitations produces only two such states. In addition, a $- and a s- state is found in this energy region. By comparing all theoretical and experimental information 17) known to us we will therefore suggest the following spin assignments as the most probable values, which also are given in brackets in fig. 1: +- for the 8.92 MeV level, $- for the 8.98 MeV level and f- for the 9.66 MeV level. As discussed below a clear difference is seen in the ground state transitions from the 8.92 and 8.98 MeV levels. This is well reproduced with our assumptions for the spin values. Furthermore, the $- assignnlent for the 9.66 MeV level agrees with the conclusions obtained by Lambert and Durand ’ “) from their elastic proton scattering experiment on 14N, It is also supported by the existence of a corresponding i- state in 1'N at 9.83 MeV. Based on these assumptions our theoretical energy spectrum shown in fig. 2, reproduces the experimental spectrum to within I MeV and the calculation accounts for all the known levels below 11.1 MeV. The effect in the energy spectrum of the different excitation modes are demonstrated in fig. 2. For the levels below 12 MeV it is necessary to include both the 2p-3h T, = 0 and rP = 1 configurations in addition to the Op-lh states. The lowest 4p-5h configuration (with J” = +-) has an unperturbed energy of 13.9 MeV. Due to small offdiagonal matrix elements to the 2p-3h states none of the eigenfunctions below If MeV contain more than 2 % of the 4p-5h configurations. In a paper by Shukla and Brown I*) the +- and the $- states were calculated neglecting the 2p-3h TP = 0 configurations. According to our results such a procedure may be justified for the +- but not for the +- levels where we find near degeneracy between the lowest Tp = 0 and Tp = 1 configurations. The spurious c.m. excitations were found to have an important effect on the eigenstates for J = +-, 3-, I- and z7 - . Except for the +- case the projection of spurious states from our p-h basis turned out to be difficult. The basis was expanded above the limit of 20 MeV but still some amount of spurious components were left in our final eigenfunctions. This may be a possible source of error in the results. 3.1. THE “ONE-HOLE”
STATES
The +; ground state and the 3; excited state at 6.18 MeV are in a simple shell model assumed to be one-hole states. Our results preserve this feature as our wave functions contain about 80 % of the Op-lh configurations 112, 15) and 112, 16) respectively. This is in good qualitative agreement with pick-up experiments ‘-‘) and also with the calculation of Shukla and Brown 12>_In a paper by Zuker et al.20) the
A = 15 NUCLEI
625
A = 15 states were constructed with the assumption of a closed 12C core and with the three particles in the Ip,, Id, and 2s, single-particle orbitals. The j$; level is excluded in such a model whereas they obtained an amplitude of 0.83 for the Op-I h configuration in the 3; level, significantly less than in our case. This is most probably due to the fact that the c.m. excitations were not investigated. By taking out the spurious states in our basis we increased the amplitude from 0.85 to 0.92. In spite of the good agreement between different theoretical calculations for the magnitude of the Op-lh component there are clear differences with respect to the nature of the p-h excitations. This mainly reflects the choice of basis in which the wave functions are expanded. Shukla and Brown 12) took only 2p-3h with Tp = 1 into account where we find about 10 y0 admixture from the 2p-3h T = 0 configurations. The admixture of multi-hole excitations has different effects in the +r and the 31 states. As is seen from fig. 2 we find an increase in binding energy of 2.53 MeV for the 3; level and 1.32 MeV for the $; level. This corresponds to a spin-orbit splitting of only 4.95 MeV between the pure one-hole states. An exact determination of this parameter depends strongly on the dimension of the basis. It is, however, interesting to note that in several calculations 4*i’s 21-23) t h’IS p arameter is found to be less than the experimental energy difference between the two levels. The 3; model state has a negligible transition to the 5 MeV doublet and a ground state width in good agreement with experiment and with the calculation of Shukla “N our results for the Ml components are and Brown 12). By a comparison with reasonable, see the discussion in ref. 24). The E2/Ml mixing ratio of the fy -+ 4; transition is measured ““) to -0.16f0.016 for I50 and +0.12&-0.015 for “N. With our wave functions we find -0.061 and +0.13 respectively. The discrepancy for the ’ 5O case has been discussed by Poletti et al. 24) on the basis of pure Op- 1h configuration. They found -0.048 for “0 and $0.12 for “N. In our calculation some improvement has been obtained by including excitations to the sd shell but the discrepancy still exists. It seems impossible that such excitations can produce the necessary E2 enhancement. An alternative explanation could be that the radial wave functions are different in the two cases due to the difference in particle separation energy for i 5N and i 50. This may increase the E2 transition in ’ 5O compared to “N. 3.2. LEVELS
BELOW
11 MeV
As discussed in the preceding section the main strength of the Op-lh configurations is concentrated in the lowest +- and j- states. Furthermore, our wave functions show that the remaining strength is distributed among many levels in good agreement with the 160(p, d)i50 pick-up experiments 5, “). In addition, if one includes 2p-211 configurations in the I60 target wave function transitions to these levels show destructive interference between the Op-1 h and 2p-3h components. This has earlier been demonstrated by Shukla and Brown 12). Based on our wave functions an investigation of the electromagnetic transition
626
S. LIE AND
T. ENGELAND
rates reveals several interesting features. From table 3 we see that reasonably good agreement is obtained for the decay of the 4; and 2; levels. However, the weak El transitions from the 3; levelare overestimated. In particular, the branch to the 3: level is predicted a factor of 100 too large. A similar situation is found for other El decays to this level. This may reflect the difficulties associated with the c.m. impurities as discussed above and in ref. ’ “)_ Of sp eciat interest are the Ml transitions from these two levels to the +- ground state. With our assumption of the spin values we reproduce the large width of 0.74 eV for the 8.98 MeV level whereas none of the calculated +- levels show such large strength for transition to the ground state. This favours our spin assignments for the 8.92 and 8.98 MeV levels to +- and $- respectively. For the next group of levels 3; at 9.48 MeV, 3; at 9.61 MeV and 3; at 9.64 MeV serious discrepancies are found between the experimental and theoretica decay properties. From the y-width 26) and branching ratio “) measurements the 3; -+ +; E2 ground state transition is found to be 14.8 W.U. Similarly, the 3; + 5; transition has a strength of 27 W.U. if we assume pure E2 and 0.21 W.U. for pure Ml. In a theoretical description of the wave functions as linear combinations of p-h excitations it is not possible to reproduce such large transition strengths with reasonable choice of the basic parameters. Similar difficulties are found for the transitions to the 3; level. In connection with our study of the positive-parity states in II we f’ound similar difficulties for the 9.50 MeV 3’ level. These exp erimental y-widths 2 “) are dated back to 1951 and it would be of great interest to have them remeasured in the light of our theoretical predictions. For the 3, and +; states no decay data is known in 150. A comparison with the corresponding $1 level in 15N at 9.83 MeV shows good agreement for the branching ratios (see table 3). The calculated 3; state does probably correspond to the experimental 11.02 MeV level. 3.3. HIGH-SPIN
STATES (.I,
$-)
using the (CI, d) In a paper by Lu et al. 28) high-spin states in “N were investigated reaction. Several spectroscopic studies of this reaction for A 5 40 have suggested that the most strongly populated states are those in which the captured proton and neutron enter the same single-particle orbital and couple to maximum J with T = 0. The reaction 13C(sr, d)l’N should therefore populate levels in 15N with the structure [j(sd)2JP = 5, 7” = 0>lp3J,, = 3, Th = +)]J,T = + (13, 13) in our notation) with J” = %- and -‘_z’-. Two levels were observed at 11.95 and 13.03 MeV with assumed spin 4- and -“z”_-respectively. In our calculation three high-spin states are found in this energy region, two 4levels at 11.34 MeV and 12.58 MeV and one J& level at 11.79 MeV. The wave functions which are given in table 2 show large amplitudes for the above-mentioned structure. The X2_ state contains 94 “/, of the 13, 13) component whereas the two $- states are mixtures of 13, 13) and 19, 13) ([l(sd)‘J, = 4, Tp = I)JP”J,~ = f, Th = &)]J,T=f). The strength of these two configurations are nearly equally distributed among the
A = 15 NUCLEI
627
two lowest $- levels. This is due to the space symmetric term in the p-h interaction one should see two strong $- levels in the as discussed in ref. 29). As a consequence r3C(c(, d)“N reaction. It would therefore be of interest to look for a second $- level at a somewhat higher energy. According to the wave function given in table 2 this state should have a significant cross section in the reaction 13C(c1, d)i5N.
4. Conclusions In the present work we have investigated the negative-parity states in the A = 15 nuclei. The spectrum is well explained. In addition we have reproduced the main properties of the positive-parity levels in a similar calculation (see II). Taking both investigations into consideration some comments about the experimental situation and the theoretical understanding may be given. On the basis of the general overah agreement between theory and experiment we conclude that the spectra of the A = 15 nuclei below 10 MeV is well explained. With our description of the wave functions as a weak coupling between particles in the sd shell and holes in the p-shell all experimental levels are reproduced. Thus no additional degrees of freedom seem to have any noticeable effect. In particular, no need for admixtures of excitations into the pf shell can be found. This is consistent with the work of Bohigas ‘l). Similarly, the importance of excitations from the 1s shell must be small as earlier demonstrated by Halbert and French 30) and for the 14N case by Cooper and Eisenberg 31). On the basis of our calculations of the A = 15 nuclei we suggest a one-to-one correspondence between all levels below 10 MeV. This is displayed in fig. 1 and implies spin and parity assignments of 8’ for the 8.31 MeV level, +- for the 9.23 MeV level and +- for the 9.39 MeV level in 1 5N. The J = 5 member of the 9.15 MeV doublet in 15N has at present no counterpart in the “0 spectrum but should correspond to the predicted 3’ level around 9 MeV as discussed in II. This one-to-one correspondence is partly in contradiction with the compilation of ref. 17) but is consistent with the work of Lambert and Durand 19). In addition, the correspondence between the 9.93 MeV level in ’ 5N and the 9.61 MeV level in I50 is strongly supported by the electromagnetic branching ratios from the two levels which are almost identical (see table 3). The two levels at 10.28 MeV and 10.46 MeV in “0 have not been yet assigned spin and parity. From our calculations we predict these levels to have 3’ and 3’. We further predict j-+ for the 10.45 MeV level in r ‘N. As shown in fig. 1 the 10.45 MeV and 10.54 MeV levels in 15N should therefore correspond to the 10.28 and 10.46 leveis in “0 as earlier suggested in refs. ‘a, 1“). As shown in the present and the earlier paper (see II) the Ellis-England weak coupling model is well suited for describing the A = 15 spectra. In a forthcoming paper 32) it is found that the model also reproduces the 14N spectrum.
628
S. LIE AND T. ENGELAND
References 1) S. Lie, T. Engeland and G. Dahll, Nucl. Phys. 156 (1970) 449 2) P. J. Ellis and T. Engeland, Nucl. Phys. Al44 (1970) 161 3) D. Kurath, Phys. Rev. 101 (1956) 216 4) S. Cohen and D. Kurath, Nucl. Phys. 73 (1965) 1 5) D. Bachelier, M. Bernas, I. Brissand, C. Detraz, N. K. Ganguhy and P. Radvangi, Compt. Rend. du Congr. Intern. de Physique Nucleaire (Paris 1964) vol. II, p. 429 6) E. K. Warburton, P. D. Parker and P. F. Donovan, Phys. Lett. 19 (1965) 397 7) J. C. Hiebert, E. Newman and R. H. Bassel, Phys. Rev. 154 (1967) 898 8) K. H. Purser, W. P. Alford, D. Cline, H. W. Fulbright, H. E. Gove and M. S. Krick, Nucl. Phys. Al32 (1969) 7.5 9) W. Bohne, H. Homeyer, H. Morgenstern and J. Scheer, Nucl. Phys. All3 (1968) 97 10) E. K. Warburton, J. W. Olness and D. E. Alburger, Phys. Rev. 140 (1965) B 1202 11) 0. Bohigas, Int. nuclear physics Conf. (Gatlinburg 1966) p. 940 12) A. P. Shukla and G. E. Brown, Nucl. Phys. All2 (1968) 296 13) J. P. Elliott, Proc. Roy. Sot. A245 (1958) 128, 562 14) J. D. Vergados, Nucl. Phys. All1 (1968) 681 15) T. T. S. Kuo and G. E. Brown, Nucl. Phys. 85 (1966) 40 16) V. Gillet, Nucl. Phys. 51 (1964) 410 17) F. Ajzenberg-Selove, Nucl. Phys. A152 (1970) 1 18) J. L. Honsaker, W. J. McDonald, G. C. Neilson and T. H. Hsu, Contributions ICPNS (Montreal 1969) p. 299 and private communication 19) M. Lambert and M. Durand, Phys. Lett. 24B (1967) 287; J. de Phys. 28 (1968) 39 20) A. P. Zuker, N. Buck and J. B. McGrory, Phys. Rev. Lett. 21 (1968) 39 21) H. J. Rose, 0. Haiisser and E. K. Warburton, Rev. Mod. Phys. 40 (1968) 591 22) C. W. Wong, Nucl. Phys. A108 (1968) 481 23) N. de Takacsy, Can. J. Phys. 46 (1968) 2091 24) A. R. Poletti, E. K. Warburton and D. Kurath, Phys. Rev. 155 (1967) 1096 25) P. J. Ellis and T. Engeland, to be published 26) D. B. Duncan and J. E. Perry, Phys. Rev. 82 (1951) 809 27) A. E. Evans, Phys. Rev. 155 (1967) 1047 28) C. C. Lu, M. S. Zisman and B. G. Harvey, Phys. Rev. 186 (1969) 1086 29) K. T. Hecht, P. J. Ellis and T. Engeland, Phys. Lett. 27B (1968) 479 30) E. C. Halbert and J. B. French, Phys. Rev. 105 (1957) 1563 31) B. S. Cooper and J. M. Eisenberg, Nucl. Phys. All4 (1968) 184 32) S. Lie, to be published
Physics
A169 (197 1) 6 17-628 ; @ North-Ho~iand
Not to be reproduced
by photoprint
P~bZ~~bin~ Co., Amsterdam
or microfilm without written permission from the publisher
AN APPLICATION OF THE WEAK COUPLING MODEL TO THE NEGATIVE-PARITY STATES OF THE A = 15 NUCLEI S. LIE and T. ENCELAND Institatte
of Physics,
University
of Oslo, Blindern,
Norway
t
Received 15 March 1971 The negative-parity states for A = 1.5 have been investigated in a weak coupling model. Based on particles in the sd shell and holes in the p-shell the final eigenstates are found to be admixtures ofOp-lh, lp-2h r, = 0 and lp-2h T,, = 1 configurations. Good agreement is obtained with the experimentaf energy levels below 11 MeV. The correspondence between levels in ‘5N and I50 is discussed.
Abstract:
1. Introduction In a previous paper “) (hereafter referred to as II) we investigated the positiveparity states with T = 3 of the A = 15 nuclei. A weak coupling model developed by Ellis and Engeland (ref. ‘), hereafter referred to as I) was used. Within the framework of the spherical shell model the lp and the 2sld single-particle orbitals were included in the construction of the basic states. The final eigenfunctions were given in terms of particle-hole (p-h) excitations of the closed and spherical lp shell. With this procedure a systematic study of the possible p-h excitations were performed. Good agreement was obtained with the experimental positive-parity states below 10 MeV. In the present paper the study of the A = 15 nuclei is completed with a similar investigation of the negative-parity i” = 4 levels. The two lowest negative-parity states have usually been regarded as the pi ’ and p+ ’ single-hole states, see for example refs * 3*41%This conclusion is supported by pick-up reactions 5-9) on ’ 6O which show that these levels nearly exhaust all the p-l strength. Furthermore, in the region of 6.5 to 13 MeV excitation energy Warburton et al. “) did not find any cross section larger than f of the cross section to the 3; level. On the other hand the Ml/E2 mixing ratios for the 3; -4; y-transition in both 1“0 and ’ ‘N indicate collective admixtures in the single-hole states. Also additional negative-parity states have been established which indicate a more complex structure. Several *O- 12) suggestions have been made in an effort to give a consistent description of the negative-parity states. Particle-hole configurations which include up to five holes in the p-shell have been proposed and also excitations to the pf shell. By use of the weak coupling model discussed in I we investigate the possible p-h excitations to find those configurations which are important for the low-lying spect Work supported in part by the Norwegian Research Council for Science and Humanities. 617
618
S. LIE AND
T. ENGELAND
trum. The unperturbed energy position of the different p-h excitations are used as a method to reduce the many-particle configuration space in which the final eigenfunctions are expanded. With the present formulation of the model only the lp and the 2sld single-particle orbitals are considered. In sect. 2 the basic equations are given together with the results for the energy levels and the y-transitions. The results are discussed and compared with experiments in sect. 3. Our conclusion is given in sect. 4 where both positive- and negative-parity states are included. A discussion of the oneto-one correspondence between levels in r5N and IsO below 11 MeV is presented.
2. Calculations
and results
To establish our notation and the form in which the final results are presented we give the basic equations which have to be solved. Further details are found in I and II. We write the Hamiltonian in the form
H = H,+H,+I;,,, where HP and Hh are the Hamiltonian and VP,, the particle-hole interaction. I@C%,.)
(1)
for the sd and the p-shell particles respectively Our wave functions are defined by
= [I(sdP’~,J,
~,)l~“%Jt,
Th)l.m~-
(2)
The square bracket implies coupling of angular momenta JP and J,, to J and isospin TP and T,, to T.The additional quantum numbers y are chosen such that H,l(sdPyp
J, Tp M,J
= EIJ;~J,~,~Jsd)“‘~,Jp
Tp GP)’
H,IP”“Y,J~ ThMT,) = E;:J,,T,,M~,)P”%J, ThM,;).
(3)
These equations imply a separate diagonalization of the nP particle problem in the sd shell and the nh hole problem in the p-shell. For the sd shell part the corresponding eigenfunctions are expanded in a basis given by
IW”“~p J, T,,bP>
=
@4’W-,1& &K, L, S,; J, TpMT,).
(4)
We have used the L-S coupling scheme and further classified the wave functions by the partition [f,] which specifies the orbital symmetry and the SU, quantum numbers (Q,,) introduced by Elliott 13) with the orthogonal Kp label of Vergados 14). The classification of the p-shell wave functions is similar except that only one of the labels [f] and (1~) is required. Having performed the calculation indicated by eqs. (3), a set of p-h states l@Zt;nTMT) can be constructed and a matrix for the total Hamiltonian H built up. The expression for these matrix elements are given in II, eq. (5). This procedure incorporates the main correlations of particles and holes in the solutions of eqs. (3). Thus the final eigenfunctions which take the form
A = 15 NUCLEI
are in most cases well described
619
by a few terms. The eigenfunctions
of HP and Hh
necessary in the present calculation are tabulated in I, tables 1 and 2. The p-she11 matrix elements of Cohen and Kurath “) and a revised version of the sd shell matrix elements of Kuo and Brown ’ “) were used.
w
--._
._._
llC2 7094 _:/2+i:2 -
4
1
3/ (-
1 54
5;,:
1091
712 +
1045
757
712 *
7.55
l/2 +,
' 730
3/2+ 5127
' 728
712+' I 5/2f!
y 679 I
312 -
’ 632
-------._.j 61a
530
i/2+
527
512 +
L
3/z+
3/2-I
f 112-
I-
'5N Fig. 1. The experimental spectra for 15N and l5O. The data are taken from ref. I”) with some modifications which are explained in the text. In “0 our suggested spin values are given in brackets. Mirror levels are connected by solid lines if the correspondence is well established lo) and by dashed lines if the correspondence is uncertain or speculative.
The calculation of the negative T = f states are performed for the I50 case, but due to the one-to-one correspondence between the levels in the two nuclei ’ “) the wave
S. LIE AND T. ENGELAND
620
functions are assumed to be valid for ’ 5N as well. The energy level schemes with identification of mirror levels are shown in fig. 1. All the p-h configurations with unperturbed energy positions below 20 MeV are included in the calculation. The basis for the diagonalization of H in eq. (1) then con12
11/z
7'2- 5'2 g12-
32
-
32 -
-
5/2
-3/2
3/2
-512
912.L J/2 -
112 -
11
7/Z\ 11/2-
it2 -
512
I’
,-
112
712 -
10
312
312 ,_.__ 512 -
9
0
112 -
-
3/2
7
3/2 -----
6
-3/2
(2.53MeV) 112 l/2 -------
0 Op-lh
2p-3h
Zp-3h
Tp=O
Tp=l
MIXED
112
EXPERIMENT
‘50
Fig. 2. The calculated and experimental positive parity levels for 150. The first three columns show the results of separate diagonalization of the Op-lh, 2p-3h TD = 0 and 2p-3h TD =z t configurations respectively. Correspondence between experimental and theoretical levels is indicated by dashed lines.
sists of Op-lh, 2p-3h and 4p-5h configurations. An estimate of the lowest 6p-7h more than 30 MeV and is therefore excluded. For the particle-hole interaction VP, the Gillet potential 16)
gives
621
A = 15 NUCLEl
is used where V, = -40 MeV, W = M = 0.33, B = -0.10, H = 0.40 and b/,u = l.O(b = (ii/m@)*). As discussed in I and II the problem of spurious C.M. excitations has been investigated. Highly spurious states are removed from the basis before the diagonalization of N. In some cases the p-h basis has been expanded above the limit of 20 MeV in order to obtain a meaningful projection. By the use of experimental binding energies (see I) our energy scale is chosen such that the physical ground state comes at zero energy. The unperturbed positions of the Op-lh states are not fixed by this procedure. In the present case they are treated as parameters and determined by the requirement that the lowest +- and $- eigenstates should come at the experimental energy. The calculated and the experimental energy levels are shown in fig. 2. The three types of excitations Op-lh, 2p-3h Tp = 0 and 2p-3h T, = 1 are also diagonalized separately in order to show the importance of the various modes. The 4p-5h states come above 12 MeV and are not shown in the figure. The final eigenstates are given in table 1
TABLE
The excitation energies and the wave functions J”
Excitation energy
-~ lh
(MeV)
Wave functions “1
Strength ( %) 4p-5h
Zp-3h
Fe = 0 r, = 1 -
61-
0.00 b)
4X-1 -
8.57
;:-
10.72 6.18 bf
85
10
5
0
0.92]12,15>+0.21]
1,14)-0.15]10,13?
3
5
91
1
0.881 7,13>+0.31]
8,14)--0.16]12,15)
0
95
3
2
0.931 1,13)-0.16]
4,14)
79
II
10
0
0.89/12,16>+0.23/
7,14>-0.211
-0.21j a2-
3 -
23
%I-
9.45
10.65
9.77
12
1
0
29
57
26
57
42
73
2
0
1
0.621 8,13)+0.43] -0.341
7,14)+0.23]
+0.19]
5,13)
0.601 1,13>-0.481
10.61
0
77
23
0
11.34
0
31
69
0
4,14>+0.17/
%*-
8,14)+0.231
0.811 2,13>+0.33]
8,14) 2,14>
6,13> 2,13>+0.28/
2,14)
4,13) 9,13)+0.29]
3,14)
3,13)+0*31/
3,14) 3,14>
8,141
0.79j 9,13)1-0.45[ -0.25/
+I-
8,13)-0.341
+0.19/
-0.291 &-
1,13)+0.34/12,16) 2,14>-0.20]10,14>
4,13>-0.22]11,14>+0.20]
0.781 8,13>+0.34]
1,13>
8,13)
+0.28/
-0.28/ %I--
1,14>-0.17/
9,14>
12.58
0
66
34
0
0.691 3,13)-0.581
9,13>+0.39]
11.79
0
94
6
0
0.971 3,13>-0.241
9,14)
“) The wave functions are labehed according to the numbers in table 2. Amplitudes less than 0.15 are omitted. b, The energy is fitted to the experimental value.
622
S. LIE AND T. ENGELAND
I. As a reference to the individual components in the wave functions we use the form Ik,, kh ) where k, and k, are the numbers labelling the particle and the hole states respectively according to table 2. TABLE
2
The basic particle and hole states “) Reference number
1
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
n,
J”
T
Eigenvalue
(MeV) 2 2 2 2 2 2
1’ 3+ 5+ 2+ 12$
2 2 2 2 2 0
0+ 2+ 4’ 02 + 22+ 0+ 33-
13 +
3 3 1 1
“) Further details about the wave functions
0 0 0 0 0
0.00 1.15 1.46 3.33 3.76
0 1 1 1 1 1 0 t t 4 *
6.51 0.00 2.01 3.46 3.65 3.98 0.00 0.00 3.68 0.00 6.18
are given in ref. *).
From the negative-parity states below 11 MeV we have calculated the y-transition strengths to all lower-lying levels. The necessary wave functions for the positive-parity states are taken from II where further details of the calculation are givers. As before an additional polarization charge of 0.5e for both protons and neutrons is used in the calculation of the E2 and E3 transition rates. In evaluating the radial matrix elements the harmonic oscillator functions are used with a length parameter b = (~~~~~~ = 1.7 fm. The results are shown in table 3 which contains all transitions where either the experimental or the calculated branch is larger than 1 %. 3. Discussion Except for the lowest negative-parity levels the experimental situation in 150 is rather unclear. In a recently published compilation 1‘) five levels, at 8.98, 9.48, 9.61, 9.66 and 11.02 MeV in the region between 4.2 and 1I .l MeV have been assigned negative parity. The spin of the 9.48,9.61 and 11.02 MeV levels seem to be well established as 3-, $- and $- respectively. For the remaining ones the experimental information does not allow a unique determination. In addition to the levels mentioned above Honsaker et al. I*) concluded from their r3Ct3He, n)“O experiment that the level
623
A = 15 NUCLEI TABLE 3
Electromagnetic Ji”
JF
transitions
Multipolarity
from
Experiment
(eV)
Ml El Ml E2 El Ml E2 El El Ml E2 El E2 El El El Ml E2 El El Ml E2 El El
>
1.4.10-Z
Theory
Branching ratio
Branching ratio
100 %
100%
10-Z 10-Z
21 % 23 %
42 % 31 %
9.4.
10-2
30 %
25 %
8.2.
1O-2
26 %
3%
94 %
97 %
0.74 10-z
2.1 0.15 0.02
6%
85 % 7% 0.7 %
0.08 0.11
3% 5%
4.0
79 %
1.0
19%
0.1
1
3%
82 % 7% 4% ( 2 % 2% 5% 78 % 15 %
1
2%
5% 2%
El E2 M2 El El
84 % 2% 4% 2% 7%
Ml El Ml El
“) “) ‘) d, ‘) ‘) =)
in A = 15 nuclei for I50
‘5N
6.6. 7.2.
4.6.
states
“)
‘50 r-width
Ml E2
the negative-parity
The experimental data are taken from The width in W.U. Identified with the 8.92 MeV level in Identified with the 8.98 MeV level in Identified with the 9.48 MeV level in Identified with the 9.61 MeV level in Identified with the 9.66 MeV level in
r-width
1.1 2.4. 6.1 2.6. 8.0. 1.3
2.8 . 10-z
the the the the the
9.23 9.152 9.76 9.93 9.83
100%
IO-’
. lo-*
21 % 54 %
1O-z IO-“
24 %
10-S
1%
0.81 7.4. 10-d 0.15 0.55 2.8. 10-z 4.1 4.7. 2.0. 3.1 . 1.3. 5.2. 3.1 3.3 9.1
52 % 10 % 36 % 2%
10-S
7% 81 %
10-z 10-4 10-d 10-S 1O-3
3% 9%
. 10-Z 6%
10-S . 10-Z 0.37 2.0. 10-Z 9.3 . 10-b 6.1 . lo-* 9.1 . 10-3
16 % 63 % 4% 10% 1%
0.42 1.2. 10-X not talc. 4.3 . 10-a 2.7. 1O-3
99 %
0.24 1.7. 10-z 3.4. 10-Z 2.8 . lo-’
75 % 5% 11% 9%
ref. I’). In some cases average I50 and I50 and I50 and I50 and I50 and
Branching ratio
(eV)
MeV MeV MeV MeV MeV
level level level level level
1%
values in in in in in
lSN. lSN. lSN. lSN. 15N.
are used.
r/r,
“)
0.56 0.66 1.6. 1o-3 2.8. 1O-3 5.7. 1o-2 2.6 3.2. IO-“ 5.4 7.0 6.6. 2.6. 6.1
10-Z 1o-3 1o-3 lo-’ lo-’
2.9. lo-’ 1.5 1o-3 0.27 3.8 . 1O-5 1.8. 1O-4 1.2. 1o-3 1.6. 1O-3 2.2. 1o-2 2.5. 1O-3 1.1 . 10-Z 2.4. 1o-2 1.1 6.7 . 1O-3 1.1 . 10-S 1.2. lo-* 1.3 not talc. 4.7 10-S 4.8. 1O-4 8.4. 2.1 . 1.5. 9.0.
1O-3 10-d 1o-z 1o-4
624
S. LIE AND T. ENGELAND
at 8.92 MeV has spin -t_-. The angular distribution showed a distinct peak in the forward direction which was interpreted as an L = 0 transfer. The final spin value was obtained using a model-dependent argument for the reaction mechanism. The same conclusion of spin 3- was found for the 8.98 and 9.66 MeV levels. However, in these cases their angular distributions were more uncertain. From a theoretical point of view it is very difficult to understand that the spectrum should have four excited +- states below 11.1 MeV. Our calculation which involves a large number of p-h excitations produces only two such states. In addition, a $- and a s- state is found in this energy region. By comparing all theoretical and experimental information 17) known to us we will therefore suggest the following spin assignments as the most probable values, which also are given in brackets in fig. 1: +- for the 8.92 MeV level, $- for the 8.98 MeV level and f- for the 9.66 MeV level. As discussed below a clear difference is seen in the ground state transitions from the 8.92 and 8.98 MeV levels. This is well reproduced with our assumptions for the spin values. Furthermore, the $- assignnlent for the 9.66 MeV level agrees with the conclusions obtained by Lambert and Durand ’ “) from their elastic proton scattering experiment on 14N, It is also supported by the existence of a corresponding i- state in 1'N at 9.83 MeV. Based on these assumptions our theoretical energy spectrum shown in fig. 2, reproduces the experimental spectrum to within I MeV and the calculation accounts for all the known levels below 11.1 MeV. The effect in the energy spectrum of the different excitation modes are demonstrated in fig. 2. For the levels below 12 MeV it is necessary to include both the 2p-3h T, = 0 and rP = 1 configurations in addition to the Op-lh states. The lowest 4p-5h configuration (with J” = +-) has an unperturbed energy of 13.9 MeV. Due to small offdiagonal matrix elements to the 2p-3h states none of the eigenfunctions below If MeV contain more than 2 % of the 4p-5h configurations. In a paper by Shukla and Brown I*) the +- and the $- states were calculated neglecting the 2p-3h TP = 0 configurations. According to our results such a procedure may be justified for the +- but not for the +- levels where we find near degeneracy between the lowest Tp = 0 and Tp = 1 configurations. The spurious c.m. excitations were found to have an important effect on the eigenstates for J = +-, 3-, I- and z7 - . Except for the +- case the projection of spurious states from our p-h basis turned out to be difficult. The basis was expanded above the limit of 20 MeV but still some amount of spurious components were left in our final eigenfunctions. This may be a possible source of error in the results. 3.1. THE “ONE-HOLE”
STATES
The +; ground state and the 3; excited state at 6.18 MeV are in a simple shell model assumed to be one-hole states. Our results preserve this feature as our wave functions contain about 80 % of the Op-lh configurations 112, 15) and 112, 16) respectively. This is in good qualitative agreement with pick-up experiments ‘-‘) and also with the calculation of Shukla and Brown 12>_In a paper by Zuker et al.20) the
A = 15 NUCLEI
625
A = 15 states were constructed with the assumption of a closed 12C core and with the three particles in the Ip,, Id, and 2s, single-particle orbitals. The j$; level is excluded in such a model whereas they obtained an amplitude of 0.83 for the Op-I h configuration in the 3; level, significantly less than in our case. This is most probably due to the fact that the c.m. excitations were not investigated. By taking out the spurious states in our basis we increased the amplitude from 0.85 to 0.92. In spite of the good agreement between different theoretical calculations for the magnitude of the Op-lh component there are clear differences with respect to the nature of the p-h excitations. This mainly reflects the choice of basis in which the wave functions are expanded. Shukla and Brown 12) took only 2p-3h with Tp = 1 into account where we find about 10 y0 admixture from the 2p-3h T = 0 configurations. The admixture of multi-hole excitations has different effects in the +r and the 31 states. As is seen from fig. 2 we find an increase in binding energy of 2.53 MeV for the 3; level and 1.32 MeV for the $; level. This corresponds to a spin-orbit splitting of only 4.95 MeV between the pure one-hole states. An exact determination of this parameter depends strongly on the dimension of the basis. It is, however, interesting to note that in several calculations 4*i’s 21-23) t h’IS p arameter is found to be less than the experimental energy difference between the two levels. The 3; model state has a negligible transition to the 5 MeV doublet and a ground state width in good agreement with experiment and with the calculation of Shukla “N our results for the Ml components are and Brown 12). By a comparison with reasonable, see the discussion in ref. 24). The E2/Ml mixing ratio of the fy -+ 4; transition is measured ““) to -0.16f0.016 for I50 and +0.12&-0.015 for “N. With our wave functions we find -0.061 and +0.13 respectively. The discrepancy for the ’ 5O case has been discussed by Poletti et al. 24) on the basis of pure Op- 1h configuration. They found -0.048 for “0 and $0.12 for “N. In our calculation some improvement has been obtained by including excitations to the sd shell but the discrepancy still exists. It seems impossible that such excitations can produce the necessary E2 enhancement. An alternative explanation could be that the radial wave functions are different in the two cases due to the difference in particle separation energy for i 5N and i 50. This may increase the E2 transition in ’ 5O compared to “N. 3.2. LEVELS
BELOW
11 MeV
As discussed in the preceding section the main strength of the Op-lh configurations is concentrated in the lowest +- and j- states. Furthermore, our wave functions show that the remaining strength is distributed among many levels in good agreement with the 160(p, d)i50 pick-up experiments 5, “). In addition, if one includes 2p-211 configurations in the I60 target wave function transitions to these levels show destructive interference between the Op-1 h and 2p-3h components. This has earlier been demonstrated by Shukla and Brown 12). Based on our wave functions an investigation of the electromagnetic transition
626
S. LIE AND
T. ENGELAND
rates reveals several interesting features. From table 3 we see that reasonably good agreement is obtained for the decay of the 4; and 2; levels. However, the weak El transitions from the 3; levelare overestimated. In particular, the branch to the 3: level is predicted a factor of 100 too large. A similar situation is found for other El decays to this level. This may reflect the difficulties associated with the c.m. impurities as discussed above and in ref. ’ “)_ Of sp eciat interest are the Ml transitions from these two levels to the +- ground state. With our assumption of the spin values we reproduce the large width of 0.74 eV for the 8.98 MeV level whereas none of the calculated +- levels show such large strength for transition to the ground state. This favours our spin assignments for the 8.92 and 8.98 MeV levels to +- and $- respectively. For the next group of levels 3; at 9.48 MeV, 3; at 9.61 MeV and 3; at 9.64 MeV serious discrepancies are found between the experimental and theoretica decay properties. From the y-width 26) and branching ratio “) measurements the 3; -+ +; E2 ground state transition is found to be 14.8 W.U. Similarly, the 3; + 5; transition has a strength of 27 W.U. if we assume pure E2 and 0.21 W.U. for pure Ml. In a theoretical description of the wave functions as linear combinations of p-h excitations it is not possible to reproduce such large transition strengths with reasonable choice of the basic parameters. Similar difficulties are found for the transitions to the 3; level. In connection with our study of the positive-parity states in II we f’ound similar difficulties for the 9.50 MeV 3’ level. These exp erimental y-widths 2 “) are dated back to 1951 and it would be of great interest to have them remeasured in the light of our theoretical predictions. For the 3, and +; states no decay data is known in 150. A comparison with the corresponding $1 level in 15N at 9.83 MeV shows good agreement for the branching ratios (see table 3). The calculated 3; state does probably correspond to the experimental 11.02 MeV level. 3.3. HIGH-SPIN
STATES (.I,
$-)
using the (CI, d) In a paper by Lu et al. 28) high-spin states in “N were investigated reaction. Several spectroscopic studies of this reaction for A 5 40 have suggested that the most strongly populated states are those in which the captured proton and neutron enter the same single-particle orbital and couple to maximum J with T = 0. The reaction 13C(sr, d)l’N should therefore populate levels in 15N with the structure [j(sd)2JP = 5, 7” = 0>lp3J,, = 3, Th = +)]J,T = + (13, 13) in our notation) with J” = %- and -‘_z’-. Two levels were observed at 11.95 and 13.03 MeV with assumed spin 4- and -“z”_-respectively. In our calculation three high-spin states are found in this energy region, two 4levels at 11.34 MeV and 12.58 MeV and one J& level at 11.79 MeV. The wave functions which are given in table 2 show large amplitudes for the above-mentioned structure. The X2_ state contains 94 “/, of the 13, 13) component whereas the two $- states are mixtures of 13, 13) and 19, 13) ([l(sd)‘J, = 4, Tp = I)JP”J,~ = f, Th = &)]J,T=f). The strength of these two configurations are nearly equally distributed among the
A = 15 NUCLEI
627
two lowest $- levels. This is due to the space symmetric term in the p-h interaction one should see two strong $- levels in the as discussed in ref. 29). As a consequence r3C(c(, d)“N reaction. It would therefore be of interest to look for a second $- level at a somewhat higher energy. According to the wave function given in table 2 this state should have a significant cross section in the reaction 13C(c1, d)i5N.
4. Conclusions In the present work we have investigated the negative-parity states in the A = 15 nuclei. The spectrum is well explained. In addition we have reproduced the main properties of the positive-parity levels in a similar calculation (see II). Taking both investigations into consideration some comments about the experimental situation and the theoretical understanding may be given. On the basis of the general overah agreement between theory and experiment we conclude that the spectra of the A = 15 nuclei below 10 MeV is well explained. With our description of the wave functions as a weak coupling between particles in the sd shell and holes in the p-shell all experimental levels are reproduced. Thus no additional degrees of freedom seem to have any noticeable effect. In particular, no need for admixtures of excitations into the pf shell can be found. This is consistent with the work of Bohigas ‘l). Similarly, the importance of excitations from the 1s shell must be small as earlier demonstrated by Halbert and French 30) and for the 14N case by Cooper and Eisenberg 31). On the basis of our calculations of the A = 15 nuclei we suggest a one-to-one correspondence between all levels below 10 MeV. This is displayed in fig. 1 and implies spin and parity assignments of 8’ for the 8.31 MeV level, +- for the 9.23 MeV level and +- for the 9.39 MeV level in 1 5N. The J = 5 member of the 9.15 MeV doublet in 15N has at present no counterpart in the “0 spectrum but should correspond to the predicted 3’ level around 9 MeV as discussed in II. This one-to-one correspondence is partly in contradiction with the compilation of ref. 17) but is consistent with the work of Lambert and Durand 19). In addition, the correspondence between the 9.93 MeV level in ’ 5N and the 9.61 MeV level in I50 is strongly supported by the electromagnetic branching ratios from the two levels which are almost identical (see table 3). The two levels at 10.28 MeV and 10.46 MeV in “0 have not been yet assigned spin and parity. From our calculations we predict these levels to have 3’ and 3’. We further predict j-+ for the 10.45 MeV level in r ‘N. As shown in fig. 1 the 10.45 MeV and 10.54 MeV levels in 15N should therefore correspond to the 10.28 and 10.46 leveis in “0 as earlier suggested in refs. ‘a, 1“). As shown in the present and the earlier paper (see II) the Ellis-England weak coupling model is well suited for describing the A = 15 spectra. In a forthcoming paper 32) it is found that the model also reproduces the 14N spectrum.
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S. LIE AND T. ENGELAND
References 1) S. Lie, T. Engeland and G. Dahll, Nucl. Phys. 156 (1970) 449 2) P. J. Ellis and T. Engeland, Nucl. Phys. Al44 (1970) 161 3) D. Kurath, Phys. Rev. 101 (1956) 216 4) S. Cohen and D. Kurath, Nucl. Phys. 73 (1965) 1 5) D. Bachelier, M. Bernas, I. Brissand, C. Detraz, N. K. Ganguhy and P. Radvangi, Compt. Rend. du Congr. Intern. de Physique Nucleaire (Paris 1964) vol. II, p. 429 6) E. K. Warburton, P. D. Parker and P. F. Donovan, Phys. Lett. 19 (1965) 397 7) J. C. Hiebert, E. Newman and R. H. Bassel, Phys. Rev. 154 (1967) 898 8) K. H. Purser, W. P. Alford, D. Cline, H. W. Fulbright, H. E. Gove and M. S. Krick, Nucl. Phys. Al32 (1969) 7.5 9) W. Bohne, H. Homeyer, H. Morgenstern and J. Scheer, Nucl. Phys. All3 (1968) 97 10) E. K. Warburton, J. W. Olness and D. E. Alburger, Phys. Rev. 140 (1965) B 1202 11) 0. Bohigas, Int. nuclear physics Conf. (Gatlinburg 1966) p. 940 12) A. P. Shukla and G. E. Brown, Nucl. Phys. All2 (1968) 296 13) J. P. Elliott, Proc. Roy. Sot. A245 (1958) 128, 562 14) J. D. Vergados, Nucl. Phys. All1 (1968) 681 15) T. T. S. Kuo and G. E. Brown, Nucl. Phys. 85 (1966) 40 16) V. Gillet, Nucl. Phys. 51 (1964) 410 17) F. Ajzenberg-Selove, Nucl. Phys. A152 (1970) 1 18) J. L. Honsaker, W. J. McDonald, G. C. Neilson and T. H. Hsu, Contributions ICPNS (Montreal 1969) p. 299 and private communication 19) M. Lambert and M. Durand, Phys. Lett. 24B (1967) 287; J. de Phys. 28 (1968) 39 20) A. P. Zuker, N. Buck and J. B. McGrory, Phys. Rev. Lett. 21 (1968) 39 21) H. J. Rose, 0. Haiisser and E. K. Warburton, Rev. Mod. Phys. 40 (1968) 591 22) C. W. Wong, Nucl. Phys. A108 (1968) 481 23) N. de Takacsy, Can. J. Phys. 46 (1968) 2091 24) A. R. Poletti, E. K. Warburton and D. Kurath, Phys. Rev. 155 (1967) 1096 25) P. J. Ellis and T. Engeland, to be published 26) D. B. Duncan and J. E. Perry, Phys. Rev. 82 (1951) 809 27) A. E. Evans, Phys. Rev. 155 (1967) 1047 28) C. C. Lu, M. S. Zisman and B. G. Harvey, Phys. Rev. 186 (1969) 1086 29) K. T. Hecht, P. J. Ellis and T. Engeland, Phys. Lett. 27B (1968) 479 30) E. C. Halbert and J. B. French, Phys. Rev. 105 (1957) 1563 31) B. S. Cooper and J. M. Eisenberg, Nucl. Phys. All4 (1968) 184 32) S. Lie, to be published