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Parity-mixing intershell coupling in hypernuclei and properties of ΛN interactions
NUCLEAR PHYSICS A Nuclear Physics A639 (1998) 135c-146~
Parity-mixing interactions
intershell coupling in hypernuclei
and properties
of AN
Toshio Motobaa aLaboratory
of Physics, Osaka Electra-Communication
University,
Neyagawa 572, Japan
Results of the conventional analyses of the ‘!O and ‘,$ structures are briefly summarized together with properties of AN G-matrix interactions. As a new dynamical aspect of hypernuclear spectra, it is pointed out that the A induces coupling of positive- and negative-parity nuclear core states through the .s~-p~ transition. The two components appear in the same energy region, so that this aspect is generally expected in hypernuclei and such effects should be taken into account in spectroscopic calculations. 1. INTRODUCTION More than ten years ago Dalitz et al. [l] showed that the old emulsion data have a remarkable power in identifying the details of hypernuclear energy levels due to the emulsion’s high-resolution characteristics. They assigned particle-hole multiplets of the [lp~lp,‘](J = O+, 2tz) configuration in ‘,$ from such data. Recently, a similar analysis of emulsion data on the K- capture reaction, K- +I6 0 -+liN + p + V, has been made [2], providing two peaks of the [lpnlp;‘] configuration which are proton-emitting A-bound states. On the other hand, after the pioneering experiments done at Brookhaven [3], the energy resolution of the (x+,K+) reaction measurements has been improved by using the SKS spectrometer at KEK [4,5]. In fact, the SKS experiments disclosed the A single-particle energies deep inside heavy hypernuclei up to “:Pb . For typical p-shell hypernuclei such as iBe, ‘:C and ‘:O, these reactions provide interesting detailed information on the coreexcited states and their formation rates. Moreover, the (K-,r-y), (K-,7r”), and the (e,e’K+) reactions will provide unique future sources of information on unnatural-parity hypernuclear states. We emphasize that such high-resolution spectroscopy will enable us, on one hand, to interpret hypernuclear structure to reveal details of the AN interactions, on which there is very limited information from the YN elementary scattering data. On the other hand, high-resolution spectroscopy will disclose the unique aspects of strangeness many-body problems. As one such aspect, we point out the unique role of the A particle in inducing parity-mixing intershell coupling in hypernuclear excited states. Here, we focus our attention on the structure of TO and YC for which we have relatively ‘rich’ experimental data. In each case, first we summarize typical results of the conventional shell-model analysis and then we proceed to an extended model in order to take the intershell coupling into account. Such an extended model has been exploited 0375-94741981% 19.00 0 1998 Elsevier Science B.V. All rights reserved PII SO375-9474(98)00262-O
T. Motoba/Nuclear Physics A63Y (1YY8) 135~~146~
136~
and applied to :Li by Majling et al. [6], who called it the translationally invariant shell model (TISM) - see also [7,8] and references therein. Millener [9] has also pointed out possible fragmentation of the ‘%(7r+, K+) strength if one takes into account theoretically the admixture of the positive-parity states of rlC as well. In the calculations we use several AN G-matrix interactions called ‘YNG’, which are derived by Yamamoto et al. [lo] from the Nijmegen [11,12] and Jiilich [13] meson-theoretical YN potentials. The YNG’s are expressed in the three-range Gaussian form, VAN
=
ca ‘h(r)
with
va(r) = c(ui
(1)
+ bikr + c&)exp[-(r/Pi)‘].
i
Here Q distinguishes the central, tensor, LS, and antisymmetric-LS components (vc, Q-, TJLS,UALS). Corresponding to the original potential models, these effective interactions are denoted symbolically as JA, JB, ND, NF and NS, respectively, for which one may refer to Ref. [lo]. In addition, we also use modified versions of the Nijmegen soft-core model which are called NSC97d, NSC97e, etc. [14,15]. 2.
A CONVENTIONAL
DESCRIPTION
OF ‘li”O EXCITED
STATES
A reanalysis of the emulsion data on the Kitomic +160+‘,5N + p + x- reaction yielded two peaks corresponding to ‘:O excited states [2]; their binding energies and the intensity ratios are Bjzup) = 1.54 f 0.09MeV
and
By)
= 3.10 f 0.08MeV
with
+
= 0.29 & 0.14 1OW
These excited states should be described naively figurations, so that their energies reflect both the residual AN p-h interactions. It is interesting to with the various potentials adopted. Therefore, let has no p-h interaction.
(2)
as corresponding to [lp$,lpn] conA p-state spin-orbit splitting and the see the spin-orbit splitting calculated us start with the i60+A system which
2.1. A spin-orbit splittings for 160+A calculated with YNG interactions Since all the nucleons in 160 are bound quite strongly (B, = 12.1 MeV, B, = 15.7 MeV for lpi/z), the I60 core itself can be well described with the single configuration of the harmonic oscillator (HO) double-closed shell [(l~$~)(lp~,,lp‘$]. The HO size parameter of bN = 1.76 fm is adopted so as to be consistent with the charge distribution from electron scattering data. In order to describe the weakly-bound lpn states, we allow the A to occupy higher nodal HO orbits (nej,) until we get stable solutions under the restriction kfNb& = p*bi, where PA denotes the reduced mass. We diagonalize a Hamiltonian consisting of the A kinetic energy th and the AN two-body interaction from . + 16~) YNG: H = TV+ &N VNA. We find that the HO model space of (1~) + 12~) + is large enough to describe the A radial wave functions at large distances (we obtain < ri >li2= 3.29 fm for 1p3/2 and 3.39 fm for @l/z). We also remark that the radial wave functions thus obtained within the HO basis are quite similar to those of an appropriate Woods-Saxon potential, 4(rA) ws [21. The ke parameter in each YNG(AN) interaction is determined so as to reproduce Bpwer) for the state which will be assigned 2+ in YO.
7: Motoba/Nuclear
‘60 + A -2.
s
0
____
__...__...___.____..._
. . . .._.
splitting
__.._
___
‘. ‘.-
-.
.. I.34
I. 48
0. 82
w -3.5z
,-4. 0 - -I’
_.___._
-..‘_3.
+A
ND
CTL
+A
NF
+A
0. 77
;-
CTL
NS
t/2 ;
'-
0. 78 -’
CTL
I
1. 19
I. 2I
-..,G
. ..____....__
-.
1. II
I. 06
-3. 85
: CTL
___
~-
-3.0 1 ~,i_ : I.22 w is
___...____
-.
-.
3
d
spin-orbit . .
.\
-2.5 -
137c
Physics A639 (1998) 135c-146~
-3.74
+A
NS 97e
,-3. 76
-”
CTL
+A
j ““<
/
NS 97d
Figure 1. Spin-orbit splittings calculated for the A p-states in ‘;O. The ‘CTL’ denotes the result with 21,+ ?.@+ VLS, while ‘+A’ adds ‘u&S. The italic numbers denote the splittings.
The calculated results for the spin-orbit Nijmegen potentials. The following relations I’ULSI: (VALSI
:
splittings are shown in Fig. 1 for several are obtained for the LS and ALS strengths:
NS97d N NS97e 21 ND < NF < NS
(3)
NS97d N NS97e N ND > NS > NF
(4)
The splitting due to VLS is largest for NS (1.48MeV), while the ND and NS97 versions have values around 1.2 MeV. It is notable that the reduction of the splitting due to VALS amounts to about 30% (NF) through 40% (ND and NS97’s). Thus the use of the Nijmegen potentials leads to the theoretical splittings 6(YNG) = EA(P~,~) - EA(P~,~) = 0.8 MeV (ND,NS97’s); A
-1.1
MeV (NF,NS)
.
(5)
It is hard to expect data from the direct production of YO for comparison. These values, however, are incorporated when the potentials are applied to the theoretical analysis of ‘i0*. 2.2. Spin-orbit splitting necessary to explain the excited states in ‘iO* The above-mentioned two peaks in ‘jO* lie between the p+‘iN and A+150 thresholds and they should be attributed to the lp-lh configuration [(1p;;2)N(1p1,21p3,2)f-\l (J” = Ot, 2:). Here, both the A spin-orbit splitting 6~ and the p-h interactions are relevant to these energies. In order to describe the weakly-bound A p-states and the positions of 0; and 2: states as well, we naturally adopt the model space KW,/$W(lP
+ 2P + *..+6p);]
@ [(lp;;J~(lp+2p+.-.+6p);].
(6)
7: Motoba/Nuclear Physics A639 (1998) 13.S146~
13%
The Hamiltonian ?I! = tl\ +
now contains
c2)NA(r)+ iEN
c
the nucleon-hole
energies,
fN>
N-hole
with the empirical
nucleon energy difference from 150, 6~ = eN(lp$)
- e~(lp;$
= 6.18
As we do not know the SA value which MeV (we put eN( lp$) = 0 for simplicity). is necessary to reproduce the BA’s from the emulsion data, given in Eq. (2), we treat 6, = @No) + z as an adjustable parameter by changing z in every diagonalization. Before going to the results of the diagonalization, we report the spin-parity assignment for the two peaks found in the emulsion data. On the basis of the analysis for the stopped K- capture rates, the larger peak at Bz”’ = 3.10 MeV should be assigned as 2: and the smaller peak at Bip = 1.54 MeV as O+. The assignment is confirmed by the comparison shown in Fig. 2, where the theoretical formation rates for the 2: and 0:: states are displayed and the relative ratio is compared with N,,/Nr,, given by Eq. (2). The 2: state is found to be kinematically favored [2]. Thus, the emulsion data lead to the experimental energy difference LLL?(~~P) = E(Ot) - E(2t) = 1.56 & 0.12 MeV.
K-: 3d (82%)+2p(18%)
2.00 ,
160 (stopped K-, x-1 ‘20 .,9
1.50 m._.-m c3 *. 1.00
0. 50
*./
*’
,_.-m’.*
!\
*\ IJ
R(2+)
-
I
o.oo’.,,
,““N1”lll’,‘l”‘, JA
JB
“’ ND
NF
NS
,“,,I”’
(BMZ) (GK)
EXP
Figure 2. Formation rates, R(J+)/A m units of 10e3, and the ratio R(O+)/R(2+) line) calculated for the 160(stopped K-, r-)‘i 0* reaction.
(solid
The diagonalization of ?L, Eq. (7), to obtain the O+ and 2+ energies has been repeated by changing x for each of the YNG’s. All the YNG interactions lead to the right order for these two energy levels. It is notable that the 2: state hardly changes with 2, since the dominant component is [lp$lp$,]. On the contrary, the the Of state energy is sensitive to 6~ or the position of the lp$, orbit. The obtained results are summarized in Fig. 3 as a function of SA for each YNG interaction.
l? Motoba /Nuclear Physics A639 (1998) 135c-146~
139c
_,,..._,....._........... ,,.........._.....__.............~
6
B
t
Sd _______________
r,
---__---___-
F
Figure 3. Energies of O+ and 2+ states calculated as a function of 6~ = eh(pi,s) - c~(ps,2). The JA, JB,... NS97d interactions are denoted here by A, B, . . . Sd, respectively.
Here, we have an ambiguity due to the two variables, the AN p-h interaction and 6~. Nevertheless, it is quite interesting to find that four potentials (ND, NF, JA, JB) lead to very similar curves for the 0; state energy. Only the NS curve deviates considerably from the others because of its strong spin-singlet interaction, but the deviation is remedied by the recent improvement in the spin-singlet/triplet balance [14,15] as shown by the Sd curve. It is also interesting to point out that the position of 0; is drastically improved in NS97d (E = 4 MeV, shown by Sd in Fig. 3) in comparison with the NS result (E = 2.9 MeV). Thus we conclude that, in order to reproduce the empirical energy difference LL?Z(~~P) = 1.56 MeV from the emulsion data, the acceptable A spin-orbit splittings are 1.30 5 6~ 5 1.45 in MeV
(for ND, NF, JA, JB) .
For NS97’s, a slightly larger value S,, = 1.64 MeV is necessary
(8) to reproduce
AEe”P
7: Motoba/Nuclear
14oc
Ph,vsics A439 (1998) 135~~146~
An early estimate 6~ = 0.8 f 0.7 MeV was made by Bouyssy [16] from the analysis of recoilless (K-,T-) reaction data. Another estimate 0.36 f 0.3 MeV has been given by Auerbach et al. [17] from their analysis of the liC* data [18]. The bh values deduced here are substantially larger than these estimates. 3. EFFECT
OF INTERSHELL
COUPLING
IN yO*
In the preceding section, only two nuclear core states (J; = l/2- and 3/2-) are considered to describe the low-lying states of fi”O by the [‘“O(J~)@ls”] configuration and the positive-parity excited states by the [150(J;) @lp*] one. The latter configuration has lfiw excitation energy. There is another configuration with the same excitation energy in which the positive-parity nuclear core states are excited by 1Aw with the A particle being kept in the 1s1/2 orbit. Therefore the hypernuclear excited states such as 0: and 2: discussed in the preceding section should be reconsidered within the extended model space of intershell configuration mixings: [lso(J,-)
x lp”]
@
[‘50(5,+) x l.@]
(9)
One should note that it is obviously not sufficient to describe the negative- and positiveparity T=1/2 excited states of 150 by the lh and (lp-2h) configurations, because any negative-parity states other than the lps,z hole state must be due to 2fiw (or higher) configurations and 3p-4h positive-parity configurations are also known be involved at low are important for the excitation energy [19]. H owever, only the simple configurations IL hypernuclear states and it is interesting to examine the effects coming from the new couplings for just the lowest few configurations of this type. For A%, we will show the results of a full lfiw calculation for the energy levels. Here, we assume only the lpi/s nucleon holes for simplicity. Then, we have three lp2h J,’ = l/2+ states in I50 which couple with I$‘,, to form J = O+ of ?O*. Their structures may be characterized by the following wave functions: [{(pd2(1+) .2sl,z}(J, = > [{(P-2(o+). h/2)(& In the evaluation
l/2:)
x ls$,]o+
1/m
x 1s:,210+ .
[p;,\lp$&,+
= l/2$) x $,]o+
of the coupling
state, we use the experimental
excitation
> and
[{b-“(I+) . ld3/2}(Jc =
effect on the energy shift of the
energies as their unperturbed
energy
positions: EC = 5.18 (Jc = l/2?), 7.56 (l/2:) and 9.53 (1/2z)t MeV. The calculated energy shift obtained in the diagonalization within the present prescription is displayed in Fig. 4 as a function of the unperturbed position of the main term. The excitation energy of 0: is about 11.7 MeV (BA(gs) = 13.38 MeV and B*(Ot) = 1.54 MeV), the corresponding energy shift coming from the intershell coupling is about &?Z(Ot) N +0.3 MeV (ND,NF,NS) and in this case the unperturbed energy position of + should be at 11.4 MeV. Note that all of the core-excited intruder states lie the K/M,210 below this ‘main term’ having the [p~~l~p$,]~ + character. There are no other l/2+ states near the energy region, so that the coupling pushes its energy up. In order to get more quantitative energy shift, we should take into account of p;/2p$\(sd)1 @ SA configurations. tThe assignment was based on old data sheets, but this state is 3/2+ according to the recent data [20]. Thus we should use 10.94 MeV [19,20] instead of 9.53 MeV. However the essence of the present story persists when the [p~\lp~,,]o+ state lies above 11 MeV.
141c
C Motoba/Nuclear Physics A639 (1998) 135c-14612
10. 5
10
11
Figure 4. Energy shift of the O+ and 2+ states due to the intershell function of the unperturbed (lp-lh) energy.
For the and three form J = attributed
11.5
coupling,
drawn as a
2: state having a [p$lp$,] structure, there are at least four J: = 3/2+ states J,’ = 5/2+ states in the region of excitation energy concerned in I50 which 2+ together with the lsi/s A particle. In the simplest prescription, they are to the 2p-2h configuration: [{p-‘(sd)‘}(J,
= 3/2+, 5/2+) x ls$,]s+.
They are distributed over excitation energies ranging from 5.24 MeV to 11.76 MeV, three of them lying quite close to the energy of the [p$lp$]s+ state. Using these simplest wavefunctions and the experimental energy input, the energy shift due to the intershell coupling is estimated as shown in Fig. 4. Since the couplings of those states lying above the Ip;,?slp$,],: state are stronger than for the other core-excited states, the resulting effect of the intershell couplings is to push the lp-lh 2+ state down. As the resultant excitation energy of 2: is about 10.3 MeV, and we obtain the energy shift AE(2:) Y -0.7 MeV. intershell As a result, the extended model for ‘20; suggests that the parity-mixing coupling gives rise to a spreading of about 1 MeV, which amounts to 2/3 of AE@“P)(O~ 2:) = 1.56 MeV from the emulsion data of Eq. (2). In other words, we have to modify the conventional result of Eq. (8) when the intershell coupling effect is taken into account, and the intrinsic spin-orbit splitting S,(‘) of the A p-state orbits should be 0.30 MeV 5 I$) 5 0.6 MeV , depending
on the AN interaction
(10) adopted.
l? Motoba/Nuclear
142~
4.
STRUCTURE
Physics A639 (1998) 13.k146~
OF YC DISCLOSED
IN THE KEK (r+, K+) REACTION
4.1. Results of the conventional description of ‘;C Basic aspects of the observed ‘:C structure can be well described with the conventional shell model with an SA or pi coupled to p-shell configurations [21] in a natural extension of the Cohen-Kurath prescription for p-shell nuclei. In fact, this model predicts two minor peaks in addition to the pronounced two peaks of [p-‘ls”](J = l&.) and [p-ll$](J = 2+), as shown in Fig. 5, and actually two minor intruders have been clearly confirmed at E, = 2.6 and 6.8 f 0.7 MeV in a recent KEK (7rr+,K+) experiment [4]. One may refer to Ref. [22] for possible test of the AN potential models in comparison with these new data for “C. At an additional test, we investigated the AN interaction dependence of the (r+,K+) spectra. We found that only the JB potential fails to reproduce the basic character of the experimental spectrum [4]. As shown in Fig. 6, the clear disagreement with the experiment tells us that JB is not suitable for describing the low-lying states of p-shell hypernuclei. This is due to the special property of JB that the spin-triplet interaction is too strong. For a more quantitative expression, we compare the ratios of the ‘E and 3E interaction strengths:
SE1 [3El
=
L(JA) 7.6
: -&(JB)
: &+NF)
: &ND)
: &(Ns)
: A(Ns97d)
: A(NS97e)
.
(11) This extreme unbalance found in JB destroys the basic spin character of the low-lying hypernuclear level structure. However, the (“+,K+) spectra themselves are not able to determine the most appropriate AN interaction, since all available interactions except JB
-I : > 2
‘*C ND
(n*,K’)‘;C
p--l
.04GeV/c
8=5”
Figure 5. The (7r+,K+) strength calculated with ND potential.
I
function
: >
‘*C r
h’,K+l’;C
ps-1
.04GeV/c
0=5”
Figure 6. The (n’+,K+) strength calculated with JB potential.
function
143c
T. Motoba/Nuclear Physics A639 (1998) 135146~
lead to spectra similar to that of ND (Fig. 5). The differences in the spin-spin interaction component of the available AN potentials should be reflected in the ground-state doublet splittings. Some time ago this fact was demonstrated in Fig. 5 of Ref. [23] with the central interaction only. However, the spin-orbit interaction is also influential on these splittings and, in fact, almost degenerate doublets obtained before [23] separate appreciably with the J, partner having the lower energy. The updated comparison is made in Fig. 7, which shows the low-lying doublet energy levels calculated with w, + VT + ‘ULS+ VALS for *iB, ‘iB, and ‘iB(‘zC). The NS interaction is representative for the potentials with a stronger singlet force [22]. It is interesting to note that the recent soft-core versions from Nijmegen (NS97e, NS97d, etc.) [14,15] seem to maintain a reasonable combination of both the spin-singlet/triplet relation and the spin-orbit interaction, although we need experiments with better resolution to confirm this. Next we discuss two peaks observed for the first time at E, = 2.58f0.17 and 6.89hO.42 MeV in the KEK (x+,K+) experiment [4]. These two minor peaks are naively attributed to the following structure based on llC core excited states: [“C(1/2;;
2.00MeV) PJ ls”](J
= 1;)
and
[11C(3/2;; 4.80MeV) @ ls*](J
= 1;).
(12)
In fact, two minor peaks obtained theoretically in Fig. 5 have such structures. The excitation energy shift from 2.00 MeV(llC) to 2.58 MeV(yC) may be possibly explained with an appropriate AN interaction in view of the results shown in Fig. 7. In fact, the
(CTW\)
. .. ... .. ... 2-
...........
.........._ --.*.*..-..'
1-
,.......... 7/z+ -
_lII_
... .. ... .
10 .B
............
--,----
. .. .. ... .. . -
11 -
.
AB
-
5/2+
.. .. ... .. .
5/2+
_____.()____m. ____-. --
_.___.
1-
-
___--. -
7
::
. .........2- .. . ......
---...........
......... .
ND
NP
1-
NS
Figure 7. Energy levels of low-lying doublets :B and TB(C).
&w
1:
--
B(C)
y_1;
NS97e
calculated
NS97d
EXP
with o, + VT+ VL.S+ VALs for ‘:B,
144c
T: Motoba/Nuclear
Physics A639 (1998) 13.5-146~
soft-core interactions
have large (p3,*~:,~1~1~,,2~:,,)~=1
matrix
elements,
which give rise
to mixing of the hypernuclear states based on the 3/2; and l/2; core states. Another energy shift from 4.80 MeV in ‘lC to 6.89 MeV in ‘,$ is too large to be explained by the AN two-body interaction. Moreover, it seems hard to reproduce the experimental cross section for the peak at 6.89 MeV with the ‘standard’ wave functions calculated within the conventional configuration of Eq. (12). For reference, we compare the theoretical and observed cross section ratios: W13) WG)
= O.l7(JA),
0.60(JB),
O.l5(ND),
O.l4(NF),
O.O8(NS) vs. 0.28 f O.O8(EXP) (13)
The theoretical cross sections amount to only half (except for JB) of the observed value. This discrepancy is remarkable in view of the fact that the cross sections for the 1, and 2.6-MeV 12 peaks can be reproduced fairly well except for JB. JB explains neither the level energies nor the cross sections. 4.2. An extended model with the intershell couplings for ‘ZC In order to overcome the difficulty in explaining the intruder state at E, 2: 6.9 MeV in ‘;C, we consider that the positive parity core-excited states in “C should be involved [24], since their excitation energies are very close to this energy: J = l/2+(&=6.34 MeV), 5/2+(6.90 MeV), 3/2+(7.50 MeV). The conjecture has been mentioned also by Millener 191, Gal [25] and Majling et al. [7]. These core-excited states can be primarily described by lhw excited configurations of the form {[s4p6(sd)‘] + [s3p8]}, while the natural-parity low-lying states are described by OLJ [s4p7] configurations. Then, the positive-parity excited states in ‘;C should be calculated within the extended model space of I’,$; J+) = [s4p7]_ @ lp” + {[s4p6(sd)‘]+
+ [s”p”]+} C?JIs”.
(14)
In the conventional model used so far, only the first configuration has been taken into account for the positive-parity states. One should note here that both configurations are 1Ww in character. For the low-lying negative-parity states, we confine ourselves here to the OAw configuration [s4p7]_ @ Is”. (Note that the states 5/2-(4.32 MeV) @ls” and 7/2- (6.48 MeV) @ls” at similar excitation energies are not excited appreciably in the (7r+,K+) reaction because a two-step process is needed.) In the extended shell-model calculation, the spurious center-of-mass excitations are removed. The results of the calculation for “C are shown in the left half of Fig. 8. The l/2+ and 5/2+ states are easily obtained at right positions, while the present NN interaction does not adequately explain the 3/2+ state energy. Thus we still need fine tuning to get a reasonable description of the spectrum of “C. In spite of t,his preliminary stage, it is interesting to see what the intershell coupling model of Eq. 14 provides for the energy levels of ‘;C. The calculated results are shown in the right half of Fig. 8. The most remarkable result is that a new 2: state is obtained at EzAL = 7.4 MeV which is well below the [p_lp*](2+) dominant states around 11 MeV. The main character of this 2: state is simply explained with ‘lC(5/2+, 6.9 MeV)@ls$. It is notable that the 2: state carries also several % of the [p-‘p”] component which gives rise to (r+,K+) strength. In other words, the considerable (7r+,K+) strength observed at E, = 6.9 MeV is possibly explained with the mixed contributions from the 1; and
7: Motoba/Nuclear
__._.._.._
-_..__.._. -.._-..-.
3/2+
5/z+
--.___---1/z+
3+ =:::~rr;;~n;n:nr 2+
____.__._.
_______._.
_. . _ _ . . _ _ _ 3/2-
145c
Physics A639 (1998) 13.5-146~
12-
-
-
0+
6. 9
32-
5/z-
0-
%6
l/2-
-
12-
3/2-
-
EXP
CAL “C
Figure 8. Comparison
CAL '-
0. 0
EXP
.:“C
of energy levels of ‘lC (left half) and l,$
(right half).
the new 2: states. A quantitative evaluation of the strength is in progress. It is also interesting to see the 0: state at 6.7 MeV excitation which has a small component of the [p-‘p^] (Of) configuration. In this way, if the (n+,K+) and/or (K-,X-) cross sections become sufficiently wellresolved, we expect to see such intruder states more clearly and they should provide valuable information on the parity-mixing intershell couplings mediated by the A particle.
5. CONCLUDING
REMARKS
On the basis of conventional model descriptions for ‘:O and ‘$2, for which data are available, we have examined the theoretical ability to explore the spectra in terms of specific properties of the AN interaction. We have pointed out the importance of the intershell configuration mixing effect in hypernuclear structure analyses, especially for the ‘lfiw’ excited states. Such coupling, mediated by the A particle, is energetically favored and is a unique feature of hypernuclei. We remark that a similar approach has been reported by Majling[7,8] The calculations for the reaction strengths with extended wave functions are now in progress.
146~
T. Motoba /Nuclear Physics A639 (1998) 135c-146~
ACKNOWLEDGEMENTS The analysis for 16’y0 discussed in Section 2 is based on work done in collaboration with R.H. Dalitz, D.H. Davis and D.N. Tovee. The author is grateful to Y. Yamamoto for providing the YNG interactions and for discussion and to D.J. Millener for useful comments. Discussions with A. Gal, 0. Hashimoto, L. Majling, Th.A. Rijken, and M. Sotona are appreciated. REFERENCES 1. R.H. Dalitz, D.H. Davis and D.N. Tovee, Nucl. Phys. A 450 (1986) 311~. 2. R.H. Dalitz, D.H. Davis, T. Motoba and D.N. Tovee, Nucl. Phys. A 625 (1997) 71. 3. C. Milner et al., Phys. Rev. Lett. 54 (1985) 1237; P.H. Pile et al., Phys. Rev. Lett. 66 (1991) 2585. 4. T. Hasegawa et al., Phys. Rev. Lett. 74 (1995) 224; Phys. Rev. C 53 (1996) 1210. 5. 0. Hashimoto, in these proceedings; S. Ajimura et al., Genshikaku Kenkyu 41, No.6 (1997) 35. 6. L. Majling et al., Phys. Lett. 92B (1980) 256. 7. L. Majling et al., Phys. Part. Nucl. 28 (1997) 101. 8. L. Majling, in these proceedings. 9. D.J. Millener, in New Vistas in Physics with High-energy Pion Beams, eds. B.F. Gibson and J.B. McClelland, (World Scientific, 1993), p. 19. 10. Y. Yamamoto, T. Motoba, H. Himeno, K. Ikeda and S. Nagata, Prog. Theor. Phys. Suppl. No. 117 (1994) 361, and references therein. 11. M.M. Nagels, T.A. Rijken and J.J. de Swart, Phys. Rev. D 12 (1975) 744; D 15 (1977) 2547; D 20 (1979) 1633. 12. P.M.M. Maessen, T.A. Rijken and J.J. de Swart, Phys. Rev. C 40 (1989) 2226. 13. B. Holzenkamp, K. Holinde and J. Speth, Nucl. Phys. A 500 (1989) 485. 14. Th.A. Rijken, in these proceedings. 15. Y. Yamamoto, in Proc. first Sino-Japan Symposium on Nuclear and Particle Physics with Strangeness (Beijing, September 1997), to be published. 16. A. Bouyssy, Phys. Lett. B 84 (1979) 41; B 91 (1980) 15. 17. E. H. Auerbach et al., Phys. Rev. Lett. 47 (1981) 1110. 18. M. May et al., Phys. Rev. Lett. 47 (1981) 1106. 19. D.E. Alburger and D.J. Millener, Phys. Rev. C 20 (1979) 1891. 20. D.J. Millener, private communication. 21. K. Itonaga, T. Motoba and H. Bando, Prog. Theor. Phys. 84 (1990) 1321. 22. T. Motoba and Y. Yamamoto, Nucl. Phys. A 585 (1995) 29c. 23. Y. Yamamoto, A. Reuber, H. Himeno, S. Nagata and T. Motoba, Czech. J. Phys. 42 (1992) 1249. 24. T. Motoba, in Nuclear and Particle Physics with Meson Beams in the 1 GeV/c Region, eds. S. Sugimoto and O.Hashimoto (Universal Academy Press, Tokyo, 1995), p. 187. 25. A. Gal, ibid. p. 23.
Parity-mixing interactions
intershell coupling in hypernuclei
and properties
of AN
Toshio Motobaa aLaboratory
of Physics, Osaka Electra-Communication
University,
Neyagawa 572, Japan
Results of the conventional analyses of the ‘!O and ‘,$ structures are briefly summarized together with properties of AN G-matrix interactions. As a new dynamical aspect of hypernuclear spectra, it is pointed out that the A induces coupling of positive- and negative-parity nuclear core states through the .s~-p~ transition. The two components appear in the same energy region, so that this aspect is generally expected in hypernuclei and such effects should be taken into account in spectroscopic calculations. 1. INTRODUCTION More than ten years ago Dalitz et al. [l] showed that the old emulsion data have a remarkable power in identifying the details of hypernuclear energy levels due to the emulsion’s high-resolution characteristics. They assigned particle-hole multiplets of the [lp~lp,‘](J = O+, 2tz) configuration in ‘,$ from such data. Recently, a similar analysis of emulsion data on the K- capture reaction, K- +I6 0 -+liN + p + V, has been made [2], providing two peaks of the [lpnlp;‘] configuration which are proton-emitting A-bound states. On the other hand, after the pioneering experiments done at Brookhaven [3], the energy resolution of the (x+,K+) reaction measurements has been improved by using the SKS spectrometer at KEK [4,5]. In fact, the SKS experiments disclosed the A single-particle energies deep inside heavy hypernuclei up to “:Pb . For typical p-shell hypernuclei such as iBe, ‘:C and ‘:O, these reactions provide interesting detailed information on the coreexcited states and their formation rates. Moreover, the (K-,r-y), (K-,7r”), and the (e,e’K+) reactions will provide unique future sources of information on unnatural-parity hypernuclear states. We emphasize that such high-resolution spectroscopy will enable us, on one hand, to interpret hypernuclear structure to reveal details of the AN interactions, on which there is very limited information from the YN elementary scattering data. On the other hand, high-resolution spectroscopy will disclose the unique aspects of strangeness many-body problems. As one such aspect, we point out the unique role of the A particle in inducing parity-mixing intershell coupling in hypernuclear excited states. Here, we focus our attention on the structure of TO and YC for which we have relatively ‘rich’ experimental data. In each case, first we summarize typical results of the conventional shell-model analysis and then we proceed to an extended model in order to take the intershell coupling into account. Such an extended model has been exploited 0375-94741981% 19.00 0 1998 Elsevier Science B.V. All rights reserved PII SO375-9474(98)00262-O
T. Motoba/Nuclear Physics A63Y (1YY8) 135~~146~
136~
and applied to :Li by Majling et al. [6], who called it the translationally invariant shell model (TISM) - see also [7,8] and references therein. Millener [9] has also pointed out possible fragmentation of the ‘%(7r+, K+) strength if one takes into account theoretically the admixture of the positive-parity states of rlC as well. In the calculations we use several AN G-matrix interactions called ‘YNG’, which are derived by Yamamoto et al. [lo] from the Nijmegen [11,12] and Jiilich [13] meson-theoretical YN potentials. The YNG’s are expressed in the three-range Gaussian form, VAN
=
ca ‘h(r)
with
va(r) = c(ui
(1)
+ bikr + c&)exp[-(r/Pi)‘].
i
Here Q distinguishes the central, tensor, LS, and antisymmetric-LS components (vc, Q-, TJLS,UALS). Corresponding to the original potential models, these effective interactions are denoted symbolically as JA, JB, ND, NF and NS, respectively, for which one may refer to Ref. [lo]. In addition, we also use modified versions of the Nijmegen soft-core model which are called NSC97d, NSC97e, etc. [14,15]. 2.
A CONVENTIONAL
DESCRIPTION
OF ‘li”O EXCITED
STATES
A reanalysis of the emulsion data on the Kitomic +160+‘,5N + p + x- reaction yielded two peaks corresponding to ‘:O excited states [2]; their binding energies and the intensity ratios are Bjzup) = 1.54 f 0.09MeV
and
By)
= 3.10 f 0.08MeV
with
+
= 0.29 & 0.14 1OW
These excited states should be described naively figurations, so that their energies reflect both the residual AN p-h interactions. It is interesting to with the various potentials adopted. Therefore, let has no p-h interaction.
(2)
as corresponding to [lp$,lpn] conA p-state spin-orbit splitting and the see the spin-orbit splitting calculated us start with the i60+A system which
2.1. A spin-orbit splittings for 160+A calculated with YNG interactions Since all the nucleons in 160 are bound quite strongly (B, = 12.1 MeV, B, = 15.7 MeV for lpi/z), the I60 core itself can be well described with the single configuration of the harmonic oscillator (HO) double-closed shell [(l~$~)(lp~,,lp‘$]. The HO size parameter of bN = 1.76 fm is adopted so as to be consistent with the charge distribution from electron scattering data. In order to describe the weakly-bound lpn states, we allow the A to occupy higher nodal HO orbits (nej,) until we get stable solutions under the restriction kfNb& = p*bi, where PA denotes the reduced mass. We diagonalize a Hamiltonian consisting of the A kinetic energy th and the AN two-body interaction from . + 16~) YNG: H = TV+ &N VNA. We find that the HO model space of (1~) + 12~) + is large enough to describe the A radial wave functions at large distances (we obtain < ri >li2= 3.29 fm for 1p3/2 and 3.39 fm for @l/z). We also remark that the radial wave functions thus obtained within the HO basis are quite similar to those of an appropriate Woods-Saxon potential, 4(rA) ws [21. The ke parameter in each YNG(AN) interaction is determined so as to reproduce Bpwer) for the state which will be assigned 2+ in YO.
7: Motoba/Nuclear
‘60 + A -2.
s
0
____
__...__...___.____..._
. . . .._.
splitting
__.._
___
‘. ‘.-
-.
.. I.34
I. 48
0. 82
w -3.5z
,-4. 0 - -I’
_.___._
-..‘_3.
+A
ND
CTL
+A
NF
+A
0. 77
;-
CTL
NS
t/2 ;
'-
0. 78 -’
CTL
I
1. 19
I. 2I
-..,G
. ..____....__
-.
1. II
I. 06
-3. 85
: CTL
___
~-
-3.0 1 ~,i_ : I.22 w is
___...____
-.
-.
3
d
spin-orbit . .
.\
-2.5 -
137c
Physics A639 (1998) 135c-146~
-3.74
+A
NS 97e
,-3. 76
-”
CTL
+A
j ““<
/
NS 97d
Figure 1. Spin-orbit splittings calculated for the A p-states in ‘;O. The ‘CTL’ denotes the result with 21,+ ?.@+ VLS, while ‘+A’ adds ‘u&S. The italic numbers denote the splittings.
The calculated results for the spin-orbit Nijmegen potentials. The following relations I’ULSI: (VALSI
:
splittings are shown in Fig. 1 for several are obtained for the LS and ALS strengths:
NS97d N NS97e 21 ND < NF < NS
(3)
NS97d N NS97e N ND > NS > NF
(4)
The splitting due to VLS is largest for NS (1.48MeV), while the ND and NS97 versions have values around 1.2 MeV. It is notable that the reduction of the splitting due to VALS amounts to about 30% (NF) through 40% (ND and NS97’s). Thus the use of the Nijmegen potentials leads to the theoretical splittings 6(YNG) = EA(P~,~) - EA(P~,~) = 0.8 MeV (ND,NS97’s); A
-1.1
MeV (NF,NS)
.
(5)
It is hard to expect data from the direct production of YO for comparison. These values, however, are incorporated when the potentials are applied to the theoretical analysis of ‘i0*. 2.2. Spin-orbit splitting necessary to explain the excited states in ‘iO* The above-mentioned two peaks in ‘jO* lie between the p+‘iN and A+150 thresholds and they should be attributed to the lp-lh configuration [(1p;;2)N(1p1,21p3,2)f-\l (J” = Ot, 2:). Here, both the A spin-orbit splitting 6~ and the p-h interactions are relevant to these energies. In order to describe the weakly-bound A p-states and the positions of 0; and 2: states as well, we naturally adopt the model space KW,/$W(lP
+ 2P + *..+6p);]
@ [(lp;;J~(lp+2p+.-.+6p);].
(6)
7: Motoba/Nuclear Physics A639 (1998) 13.S146~
13%
The Hamiltonian ?I! = tl\ +
now contains
c2)NA(r)+ iEN
c
the nucleon-hole
energies,
fN>
N-hole
with the empirical
nucleon energy difference from 150, 6~ = eN(lp$)
- e~(lp;$
= 6.18
As we do not know the SA value which MeV (we put eN( lp$) = 0 for simplicity). is necessary to reproduce the BA’s from the emulsion data, given in Eq. (2), we treat 6, = @No) + z as an adjustable parameter by changing z in every diagonalization. Before going to the results of the diagonalization, we report the spin-parity assignment for the two peaks found in the emulsion data. On the basis of the analysis for the stopped K- capture rates, the larger peak at Bz”’ = 3.10 MeV should be assigned as 2: and the smaller peak at Bip = 1.54 MeV as O+. The assignment is confirmed by the comparison shown in Fig. 2, where the theoretical formation rates for the 2: and 0:: states are displayed and the relative ratio is compared with N,,/Nr,, given by Eq. (2). The 2: state is found to be kinematically favored [2]. Thus, the emulsion data lead to the experimental energy difference LLL?(~~P) = E(Ot) - E(2t) = 1.56 & 0.12 MeV.
K-: 3d (82%)+2p(18%)
2.00 ,
160 (stopped K-, x-1 ‘20 .,9
1.50 m._.-m c3 *. 1.00
0. 50
*./
*’
,_.-m’.*
!\
*\ IJ
R(2+)
-
I
o.oo’.,,
,““N1”lll’,‘l”‘, JA
JB
“’ ND
NF
NS
,“,,I”’
(BMZ) (GK)
EXP
Figure 2. Formation rates, R(J+)/A m units of 10e3, and the ratio R(O+)/R(2+) line) calculated for the 160(stopped K-, r-)‘i 0* reaction.
(solid
The diagonalization of ?L, Eq. (7), to obtain the O+ and 2+ energies has been repeated by changing x for each of the YNG’s. All the YNG interactions lead to the right order for these two energy levels. It is notable that the 2: state hardly changes with 2, since the dominant component is [lp$lp$,]. On the contrary, the the Of state energy is sensitive to 6~ or the position of the lp$, orbit. The obtained results are summarized in Fig. 3 as a function of SA for each YNG interaction.
l? Motoba /Nuclear Physics A639 (1998) 135c-146~
139c
_,,..._,....._........... ,,.........._.....__.............~
6
B
t
Sd _______________
r,
---__---___-
F
Figure 3. Energies of O+ and 2+ states calculated as a function of 6~ = eh(pi,s) - c~(ps,2). The JA, JB,... NS97d interactions are denoted here by A, B, . . . Sd, respectively.
Here, we have an ambiguity due to the two variables, the AN p-h interaction and 6~. Nevertheless, it is quite interesting to find that four potentials (ND, NF, JA, JB) lead to very similar curves for the 0; state energy. Only the NS curve deviates considerably from the others because of its strong spin-singlet interaction, but the deviation is remedied by the recent improvement in the spin-singlet/triplet balance [14,15] as shown by the Sd curve. It is also interesting to point out that the position of 0; is drastically improved in NS97d (E = 4 MeV, shown by Sd in Fig. 3) in comparison with the NS result (E = 2.9 MeV). Thus we conclude that, in order to reproduce the empirical energy difference LL?Z(~~P) = 1.56 MeV from the emulsion data, the acceptable A spin-orbit splittings are 1.30 5 6~ 5 1.45 in MeV
(for ND, NF, JA, JB) .
For NS97’s, a slightly larger value S,, = 1.64 MeV is necessary
(8) to reproduce
AEe”P
7: Motoba/Nuclear
14oc
Ph,vsics A439 (1998) 135~~146~
An early estimate 6~ = 0.8 f 0.7 MeV was made by Bouyssy [16] from the analysis of recoilless (K-,T-) reaction data. Another estimate 0.36 f 0.3 MeV has been given by Auerbach et al. [17] from their analysis of the liC* data [18]. The bh values deduced here are substantially larger than these estimates. 3. EFFECT
OF INTERSHELL
COUPLING
IN yO*
In the preceding section, only two nuclear core states (J; = l/2- and 3/2-) are considered to describe the low-lying states of fi”O by the [‘“O(J~)@ls”] configuration and the positive-parity excited states by the [150(J;) @lp*] one. The latter configuration has lfiw excitation energy. There is another configuration with the same excitation energy in which the positive-parity nuclear core states are excited by 1Aw with the A particle being kept in the 1s1/2 orbit. Therefore the hypernuclear excited states such as 0: and 2: discussed in the preceding section should be reconsidered within the extended model space of intershell configuration mixings: [lso(J,-)
x lp”]
@
[‘50(5,+) x l.@]
(9)
One should note that it is obviously not sufficient to describe the negative- and positiveparity T=1/2 excited states of 150 by the lh and (lp-2h) configurations, because any negative-parity states other than the lps,z hole state must be due to 2fiw (or higher) configurations and 3p-4h positive-parity configurations are also known be involved at low are important for the excitation energy [19]. H owever, only the simple configurations IL hypernuclear states and it is interesting to examine the effects coming from the new couplings for just the lowest few configurations of this type. For A%, we will show the results of a full lfiw calculation for the energy levels. Here, we assume only the lpi/s nucleon holes for simplicity. Then, we have three lp2h J,’ = l/2+ states in I50 which couple with I$‘,, to form J = O+ of ?O*. Their structures may be characterized by the following wave functions: [{(pd2(1+) .2sl,z}(J, = > [{(P-2(o+). h/2)(& In the evaluation
l/2:)
x ls$,]o+
1/m
x 1s:,210+ .
[p;,\lp$&,+
= l/2$) x $,]o+
of the coupling
state, we use the experimental
excitation
> and
[{b-“(I+) . ld3/2}(Jc =
effect on the energy shift of the
energies as their unperturbed
energy
positions: EC = 5.18 (Jc = l/2?), 7.56 (l/2:) and 9.53 (1/2z)t MeV. The calculated energy shift obtained in the diagonalization within the present prescription is displayed in Fig. 4 as a function of the unperturbed position of the main term. The excitation energy of 0: is about 11.7 MeV (BA(gs) = 13.38 MeV and B*(Ot) = 1.54 MeV), the corresponding energy shift coming from the intershell coupling is about &?Z(Ot) N +0.3 MeV (ND,NF,NS) and in this case the unperturbed energy position of + should be at 11.4 MeV. Note that all of the core-excited intruder states lie the K/M,210 below this ‘main term’ having the [p~~l~p$,]~ + character. There are no other l/2+ states near the energy region, so that the coupling pushes its energy up. In order to get more quantitative energy shift, we should take into account of p;/2p$\(sd)1 @ SA configurations. tThe assignment was based on old data sheets, but this state is 3/2+ according to the recent data [20]. Thus we should use 10.94 MeV [19,20] instead of 9.53 MeV. However the essence of the present story persists when the [p~\lp~,,]o+ state lies above 11 MeV.
141c
C Motoba/Nuclear Physics A639 (1998) 135c-14612
10. 5
10
11
Figure 4. Energy shift of the O+ and 2+ states due to the intershell function of the unperturbed (lp-lh) energy.
For the and three form J = attributed
11.5
coupling,
drawn as a
2: state having a [p$lp$,] structure, there are at least four J: = 3/2+ states J,’ = 5/2+ states in the region of excitation energy concerned in I50 which 2+ together with the lsi/s A particle. In the simplest prescription, they are to the 2p-2h configuration: [{p-‘(sd)‘}(J,
= 3/2+, 5/2+) x ls$,]s+.
They are distributed over excitation energies ranging from 5.24 MeV to 11.76 MeV, three of them lying quite close to the energy of the [p$lp$]s+ state. Using these simplest wavefunctions and the experimental energy input, the energy shift due to the intershell coupling is estimated as shown in Fig. 4. Since the couplings of those states lying above the Ip;,?slp$,],: state are stronger than for the other core-excited states, the resulting effect of the intershell couplings is to push the lp-lh 2+ state down. As the resultant excitation energy of 2: is about 10.3 MeV, and we obtain the energy shift AE(2:) Y -0.7 MeV. intershell As a result, the extended model for ‘20; suggests that the parity-mixing coupling gives rise to a spreading of about 1 MeV, which amounts to 2/3 of AE@“P)(O~ 2:) = 1.56 MeV from the emulsion data of Eq. (2). In other words, we have to modify the conventional result of Eq. (8) when the intershell coupling effect is taken into account, and the intrinsic spin-orbit splitting S,(‘) of the A p-state orbits should be 0.30 MeV 5 I$) 5 0.6 MeV , depending
on the AN interaction
(10) adopted.
l? Motoba/Nuclear
142~
4.
STRUCTURE
Physics A639 (1998) 13.k146~
OF YC DISCLOSED
IN THE KEK (r+, K+) REACTION
4.1. Results of the conventional description of ‘;C Basic aspects of the observed ‘:C structure can be well described with the conventional shell model with an SA or pi coupled to p-shell configurations [21] in a natural extension of the Cohen-Kurath prescription for p-shell nuclei. In fact, this model predicts two minor peaks in addition to the pronounced two peaks of [p-‘ls”](J = l&.) and [p-ll$](J = 2+), as shown in Fig. 5, and actually two minor intruders have been clearly confirmed at E, = 2.6 and 6.8 f 0.7 MeV in a recent KEK (7rr+,K+) experiment [4]. One may refer to Ref. [22] for possible test of the AN potential models in comparison with these new data for “C. At an additional test, we investigated the AN interaction dependence of the (r+,K+) spectra. We found that only the JB potential fails to reproduce the basic character of the experimental spectrum [4]. As shown in Fig. 6, the clear disagreement with the experiment tells us that JB is not suitable for describing the low-lying states of p-shell hypernuclei. This is due to the special property of JB that the spin-triplet interaction is too strong. For a more quantitative expression, we compare the ratios of the ‘E and 3E interaction strengths:
SE1 [3El
=
L(JA) 7.6
: -&(JB)
: &+NF)
: &ND)
: &(Ns)
: A(Ns97d)
: A(NS97e)
.
(11) This extreme unbalance found in JB destroys the basic spin character of the low-lying hypernuclear level structure. However, the (“+,K+) spectra themselves are not able to determine the most appropriate AN interaction, since all available interactions except JB
-I : > 2
‘*C ND
(n*,K’)‘;C
p--l
.04GeV/c
8=5”
Figure 5. The (7r+,K+) strength calculated with ND potential.
I
function
: >
‘*C r
h’,K+l’;C
ps-1
.04GeV/c
0=5”
Figure 6. The (n’+,K+) strength calculated with JB potential.
function
143c
T. Motoba/Nuclear Physics A639 (1998) 135146~
lead to spectra similar to that of ND (Fig. 5). The differences in the spin-spin interaction component of the available AN potentials should be reflected in the ground-state doublet splittings. Some time ago this fact was demonstrated in Fig. 5 of Ref. [23] with the central interaction only. However, the spin-orbit interaction is also influential on these splittings and, in fact, almost degenerate doublets obtained before [23] separate appreciably with the J, partner having the lower energy. The updated comparison is made in Fig. 7, which shows the low-lying doublet energy levels calculated with w, + VT + ‘ULS+ VALS for *iB, ‘iB, and ‘iB(‘zC). The NS interaction is representative for the potentials with a stronger singlet force [22]. It is interesting to note that the recent soft-core versions from Nijmegen (NS97e, NS97d, etc.) [14,15] seem to maintain a reasonable combination of both the spin-singlet/triplet relation and the spin-orbit interaction, although we need experiments with better resolution to confirm this. Next we discuss two peaks observed for the first time at E, = 2.58f0.17 and 6.89hO.42 MeV in the KEK (x+,K+) experiment [4]. These two minor peaks are naively attributed to the following structure based on llC core excited states: [“C(1/2;;
2.00MeV) PJ ls”](J
= 1;)
and
[11C(3/2;; 4.80MeV) @ ls*](J
= 1;).
(12)
In fact, two minor peaks obtained theoretically in Fig. 5 have such structures. The excitation energy shift from 2.00 MeV(llC) to 2.58 MeV(yC) may be possibly explained with an appropriate AN interaction in view of the results shown in Fig. 7. In fact, the
(CTW\)
. .. ... .. ... 2-
...........
.........._ --.*.*..-..'
1-
,.......... 7/z+ -
_lII_
... .. ... .
10 .B
............
--,----
. .. .. ... .. . -
11 -
.
AB
-
5/2+
.. .. ... .. .
5/2+
_____.()____m. ____-. --
_.___.
1-
-
___--. -
7
::
. .........2- .. . ......
---...........
......... .
ND
NP
1-
NS
Figure 7. Energy levels of low-lying doublets :B and TB(C).
&w
1:
--
B(C)
y_1;
NS97e
calculated
NS97d
EXP
with o, + VT+ VL.S+ VALs for ‘:B,
144c
T: Motoba/Nuclear
Physics A639 (1998) 13.5-146~
soft-core interactions
have large (p3,*~:,~1~1~,,2~:,,)~=1
matrix
elements,
which give rise
to mixing of the hypernuclear states based on the 3/2; and l/2; core states. Another energy shift from 4.80 MeV in ‘lC to 6.89 MeV in ‘,$ is too large to be explained by the AN two-body interaction. Moreover, it seems hard to reproduce the experimental cross section for the peak at 6.89 MeV with the ‘standard’ wave functions calculated within the conventional configuration of Eq. (12). For reference, we compare the theoretical and observed cross section ratios: W13) WG)
= O.l7(JA),
0.60(JB),
O.l5(ND),
O.l4(NF),
O.O8(NS) vs. 0.28 f O.O8(EXP) (13)
The theoretical cross sections amount to only half (except for JB) of the observed value. This discrepancy is remarkable in view of the fact that the cross sections for the 1, and 2.6-MeV 12 peaks can be reproduced fairly well except for JB. JB explains neither the level energies nor the cross sections. 4.2. An extended model with the intershell couplings for ‘ZC In order to overcome the difficulty in explaining the intruder state at E, 2: 6.9 MeV in ‘;C, we consider that the positive parity core-excited states in “C should be involved [24], since their excitation energies are very close to this energy: J = l/2+(&=6.34 MeV), 5/2+(6.90 MeV), 3/2+(7.50 MeV). The conjecture has been mentioned also by Millener 191, Gal [25] and Majling et al. [7]. These core-excited states can be primarily described by lhw excited configurations of the form {[s4p6(sd)‘] + [s3p8]}, while the natural-parity low-lying states are described by OLJ [s4p7] configurations. Then, the positive-parity excited states in ‘;C should be calculated within the extended model space of I’,$; J+) = [s4p7]_ @ lp” + {[s4p6(sd)‘]+
+ [s”p”]+} C?JIs”.
(14)
In the conventional model used so far, only the first configuration has been taken into account for the positive-parity states. One should note here that both configurations are 1Ww in character. For the low-lying negative-parity states, we confine ourselves here to the OAw configuration [s4p7]_ @ Is”. (Note that the states 5/2-(4.32 MeV) @ls” and 7/2- (6.48 MeV) @ls” at similar excitation energies are not excited appreciably in the (7r+,K+) reaction because a two-step process is needed.) In the extended shell-model calculation, the spurious center-of-mass excitations are removed. The results of the calculation for “C are shown in the left half of Fig. 8. The l/2+ and 5/2+ states are easily obtained at right positions, while the present NN interaction does not adequately explain the 3/2+ state energy. Thus we still need fine tuning to get a reasonable description of the spectrum of “C. In spite of t,his preliminary stage, it is interesting to see what the intershell coupling model of Eq. 14 provides for the energy levels of ‘;C. The calculated results are shown in the right half of Fig. 8. The most remarkable result is that a new 2: state is obtained at EzAL = 7.4 MeV which is well below the [p_lp*](2+) dominant states around 11 MeV. The main character of this 2: state is simply explained with ‘lC(5/2+, 6.9 MeV)@ls$. It is notable that the 2: state carries also several % of the [p-‘p”] component which gives rise to (r+,K+) strength. In other words, the considerable (7r+,K+) strength observed at E, = 6.9 MeV is possibly explained with the mixed contributions from the 1; and
7: Motoba/Nuclear
__._.._.._
-_..__.._. -.._-..-.
3/2+
5/z+
--.___---1/z+
3+ =:::~rr;;~n;n:nr 2+
____.__._.
_______._.
_. . _ _ . . _ _ _ 3/2-
145c
Physics A639 (1998) 13.5-146~
12-
-
-
0+
6. 9
32-
5/z-
0-
%6
l/2-
-
12-
3/2-
-
EXP
CAL “C
Figure 8. Comparison
CAL '-
0. 0
EXP
.:“C
of energy levels of ‘lC (left half) and l,$
(right half).
the new 2: states. A quantitative evaluation of the strength is in progress. It is also interesting to see the 0: state at 6.7 MeV excitation which has a small component of the [p-‘p^] (Of) configuration. In this way, if the (n+,K+) and/or (K-,X-) cross sections become sufficiently wellresolved, we expect to see such intruder states more clearly and they should provide valuable information on the parity-mixing intershell couplings mediated by the A particle.
5. CONCLUDING
REMARKS
On the basis of conventional model descriptions for ‘:O and ‘$2, for which data are available, we have examined the theoretical ability to explore the spectra in terms of specific properties of the AN interaction. We have pointed out the importance of the intershell configuration mixing effect in hypernuclear structure analyses, especially for the ‘lfiw’ excited states. Such coupling, mediated by the A particle, is energetically favored and is a unique feature of hypernuclei. We remark that a similar approach has been reported by Majling[7,8] The calculations for the reaction strengths with extended wave functions are now in progress.
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T. Motoba /Nuclear Physics A639 (1998) 135c-146~
ACKNOWLEDGEMENTS The analysis for 16’y0 discussed in Section 2 is based on work done in collaboration with R.H. Dalitz, D.H. Davis and D.N. Tovee. The author is grateful to Y. Yamamoto for providing the YNG interactions and for discussion and to D.J. Millener for useful comments. Discussions with A. Gal, 0. Hashimoto, L. Majling, Th.A. Rijken, and M. Sotona are appreciated. REFERENCES 1. R.H. Dalitz, D.H. Davis and D.N. Tovee, Nucl. Phys. A 450 (1986) 311~. 2. R.H. Dalitz, D.H. Davis, T. Motoba and D.N. Tovee, Nucl. Phys. A 625 (1997) 71. 3. C. Milner et al., Phys. Rev. Lett. 54 (1985) 1237; P.H. Pile et al., Phys. Rev. Lett. 66 (1991) 2585. 4. T. Hasegawa et al., Phys. Rev. Lett. 74 (1995) 224; Phys. Rev. C 53 (1996) 1210. 5. 0. Hashimoto, in these proceedings; S. Ajimura et al., Genshikaku Kenkyu 41, No.6 (1997) 35. 6. L. Majling et al., Phys. Lett. 92B (1980) 256. 7. L. Majling et al., Phys. Part. Nucl. 28 (1997) 101. 8. L. Majling, in these proceedings. 9. D.J. Millener, in New Vistas in Physics with High-energy Pion Beams, eds. B.F. Gibson and J.B. McClelland, (World Scientific, 1993), p. 19. 10. Y. Yamamoto, T. Motoba, H. Himeno, K. Ikeda and S. Nagata, Prog. Theor. Phys. Suppl. No. 117 (1994) 361, and references therein. 11. M.M. Nagels, T.A. Rijken and J.J. de Swart, Phys. Rev. D 12 (1975) 744; D 15 (1977) 2547; D 20 (1979) 1633. 12. P.M.M. Maessen, T.A. Rijken and J.J. de Swart, Phys. Rev. C 40 (1989) 2226. 13. B. Holzenkamp, K. Holinde and J. Speth, Nucl. Phys. A 500 (1989) 485. 14. Th.A. Rijken, in these proceedings. 15. Y. Yamamoto, in Proc. first Sino-Japan Symposium on Nuclear and Particle Physics with Strangeness (Beijing, September 1997), to be published. 16. A. Bouyssy, Phys. Lett. B 84 (1979) 41; B 91 (1980) 15. 17. E. H. Auerbach et al., Phys. Rev. Lett. 47 (1981) 1110. 18. M. May et al., Phys. Rev. Lett. 47 (1981) 1106. 19. D.E. Alburger and D.J. Millener, Phys. Rev. C 20 (1979) 1891. 20. D.J. Millener, private communication. 21. K. Itonaga, T. Motoba and H. Bando, Prog. Theor. Phys. 84 (1990) 1321. 22. T. Motoba and Y. Yamamoto, Nucl. Phys. A 585 (1995) 29c. 23. Y. Yamamoto, A. Reuber, H. Himeno, S. Nagata and T. Motoba, Czech. J. Phys. 42 (1992) 1249. 24. T. Motoba, in Nuclear and Particle Physics with Meson Beams in the 1 GeV/c Region, eds. S. Sugimoto and O.Hashimoto (Universal Academy Press, Tokyo, 1995), p. 187. 25. A. Gal, ibid. p. 23.