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SHELL MODEL, QUARTETS AND ROTONS IN THE s-d SHELL
A. Arima
To cite this version:
A. Arima. SHELL MODEL, QUARTETS AND ROTONS IN THE s-d SHELL. Journal de Physique
Colloques, 1971, 32 (C6), pp.C6-33-C6-37. �10.1051/jphyscol:1971605�. �jpa-00214823�
JOURNAL DE PHYSIQUE Colloque C6, supplkment au no 11-12, Tome 32, Novembre-Dhcembre 1971, page C6-33
SHELL MODEL, QUARTETS AND ROTONS IN THE S-d SHELL
A. ARTMA
Department of Physics, State University of New York, Stony Brook, U. S. A.
Rbum6. - On definit la notion de quartet dans la limite du modelc a. Les structures du 2oNe, du 44Ti et du 5 sNi sont discutees dans le cadre du modele des quartets.
Abstract. - The definition of quartet is given in connestion with alpha clusters. Thc structures of zQNe, 44Ti and SYNi arc discussed from the quartet structure point of view.
I . Quartets and space symmetry. - To begin with, I wish to show binding energies of the s-d shell nuclei relative to that of 160. We can see in Table I how stable 'ONe is.
Interaction energies
ABE = BE (8
+
Z , 8+
N) - BE(160)-
4.2 N - 0.6 ZThe easiest way to explain the stability of four body system is givcn by a consideration on the spatial symmetry of wave functions as discussed by Dr.
Messiah in talk of this conference. A short ranged nuclear interaction favours the spatial symmetric states. Because a nucleon has four internal freedoms -
spin up and down, isospin up and down -, the wave function in the isospin-spin space can be fully anti-symmetric and the spatial wave function can be fully symmetric only for four nucleons or less. Tinpor- tance of this aspect of spatial symmetry was recognized by Wigner who introduced the super~nultiplet theory [I].
Table I1 shows approximate intensities in percent for the space symmetries dominant in the ground state domain in the s-d shell region [2]. The percen-
Breaking of spatial symmetry a Nurn. of
nucleons Rosenfeld Kuo
-
--
4 92-8 70-27-3
5 90-9-1 75-20-4- 1
6 87-9-4- 1 73-17-5-5
7 80-19-1 50-45-3- 1-2
8 79- 1 9- 1 36-49-1 1-1-2
Representations arc ordered according to eigenvalue of the two-body Casimir (Majorana) operator.
tages of the highest spatial symmetric states are always very large for a Rosenfeld interaction, but Kuo-Brown's interaction gives rise to large symmetry admixing. Even so at least up to 20Ne, the highest spatial symmetry dominates. Table 111 shows the
Interaction energies ABE
.-
1.9 2.5 6.1 5.7 0.1 5.6 12.6 5.4 12.9 12.4
ABE = B E ( ~ O C ~ )
-
BE(20 - Z 20-
N)+
+
8.33 Z+
15.62 N .Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1971605
C6-34 A. ARIMA
binding energies of several nuclei relative to 40Ca where the spatial symmetry is badly broken because of the strong spin-orbit interaction. We can again clearly see that 36Ar is more stable than other neighbouring nuclei. This is an indication of cr persistence )) of the spatial symmetry. In other words, two neutrons and two protons interact strongly to form a stable structure even when the spatial symmetry is broken rather badly. This structure is called quartet )) following Gillet and Danos [3]. The quartet nature can be found in many physical systems irrespective of coupl- ing schemes, for example the Is coupling shell model state, the H. F. intrinsic state, and the stretched state [3].
If non central, Bartlet and Heisenberg interactions are neglected, the spatial part of all quartet wave functions has the full symmetry [4]. Quartets with the full spatial symmetry [4] can be called the [4] symme- tric quartets. Take 20Ne as a n example. The shell model gives four basis vectors for the T = 0 J = 0, fully symmetric state ;
Namely we can have a certain linear combination of those four states after diagonalizing a Hamiltonian with a residual interaction. This state is favoured by a short ranged interaction and should be called a quartet with the symmetry [4].
Let us take the quadrupole-quadrupole interaction as the residual interaction to the shcll model Hamil- tonian. Diagonalizing this shell model Hamiltonian we can find a special linear combination of these basis vectors as the ground state wave function which is expressed as :
This is nothing but the SU, wave function [4].
Bohr and Bayman [S] proved that this SU, wave function is identical with Wildermuth's wave func- tion [6] ;
1
O g n>
= N ~ x ( ' ~ ~ ) . x ( ~ ) . J ~ ~ ~ ( J ~ ~ - ~ 1 6 ) 'p:?(R20) (2) where X'S are the intrinsic wave functions of clusters,R,,
is the relative wave function between two clusters and @CM is the wave function of the centre of mass motion.We can easily calculate the spectroscopic factor of alpha particle. Here we assumed that any relative
motions between any two nucleons in a n alpha cluster are in the 0 s state and ho of the cluster is assumed to be the same as that of the target nucleus.
It is extremely interesting to observe that only this particular linear combination given by (1) can have non vanishing alpha spectroscopic factor 0.24 and three other states which are orthogonal t o the state (1) havc zero spectroscopic factor. In other words, the state (1) can be considered a s the state in which the alpha cluster moves around the 160 cluster. This kind of special state can be called a n cc alpha cluster )) state. We must, however, bear in mind that the Pauli principle plays a very important role. Because of the Pauli principle, the spectroscopic factor is reduced to 0.24 instead of 1.
Exciting the relative wave function in (2) to 3 L),
2 G , I I and 0 L, we obtain the wavc functions for In = 2+, 4'", 6' and 8' which again coincide with those given by the SU, model. They belong to the irreducible representation (80) of the SU, group.
Among many stateSbelonging to the(sd), configuration, only those states with the (80) label have non-vanishing spectroscopic factor of alpha cluster. All (80) states have the samc value 0.24, which is the probability to find simultaneously the alpha cluster and the 160 cluster in 'ONe. Table I V shows the wave functions obtained by the shell-model calculation. From this
Percentage ana!,~sis of' wave functions of 20Ne
table, we can confirm that the (80) states dominate in the shell model wave functions for the ground state band. Thus we can expect that only the ground state band among many positive parity levels should be strongly excited. This is proved experimentally by Middleton and his colleagues 171. Their results on 160(7Li, t)20Ne are given in Table V which shows that the members of the ground state band are strongly excited, but none of 0' and 2' states around 7 MeV are excited. This table, however, shows that the 1- and 3- states are also strongly excited by the alpha transfer reaction. It is very reasonable because we can easily construct the negative parity states with large alpha spectroscopic factors, exciting the relative
SHELL MODEL, QUARTETS AND ROTONS IN T H E S-d SHELL C6-35
Results for reaction '60(7Li, t)20Ne
Exc. Energy Relative
(MeV) J n (da/dQ) max
motion in (2) from 4 S to 4 P, 3 F, 2 H, 1 K and OM.
They belong to the (90) representation of the SU, group.
In theabove discussion, I have shown that the shell model wave functions of the ground band members in 20Ne are approximated by the SU, (80) wave functions which turn out to have a clustering naturc. The mean square distance between two nucleons, however, is not changed very much by the configuration mixing like as in (1). This fact can be proved as follows.
Take 180 which has two neutrons outsidc 160. The mean square distance between these two nucleons is the expectation value of the opcrator (r, - r2)2 ;
If the harmonic oscillator shell-model is assumed, the first term in the right hand side is constant irres- pectively to configuration mixing. The second term in the right hand side is always zero. Thus this quantity does not depend on a mixture of basis states.
The same argument can be applied to 20Ne. Therefore if we want to shrink the alpha cluster found in the shell model wave function, we have to mix more shell-model states with higher excitations 2 hw 4 hw and so on. If this happens, the shrinked alpha cluster is localized and should be called the localized alpha cluster. At moment, we do not haveany direct evidence that such a localized alpha cluster exists in 20Ne.
Thus we have scveral different steps of quartets ; Quartet 3 Quartet with [4] symmetry 3
alpha cluster 3 localized alpha cluster.
In 'ONe, thc ground state band consists of alpha cluster. 44Ti, however, does not seem to have large amplitude of alpha cluster. According to the shell- model calculation, the probability of (12,O) state which has the sum rule limit of alpha spectroscopic factor is only 20
%.
The O f state predicted around 5 MeVshould carry 45
%
probability of (12,O) state [16].This fact seems to explain why 40Ca(160, 12C)44Ti reaction excited fairly strongly the ground band but much more strongly the levels above 5 MeV excitation, although the difference of Q value should be taken into consideration.
2. Multi-particle multi-hole excitation. - Gillet, Ginocchio and myself proposed a model based on the quartet hypothesis to predict energies of multi- particle multi-hole states [8]. An essential point of this model is that the strong internal binding of a quartet system and the relatively weak interactions between diffcrent quartets give rise to low-lying excitations in which an entire quartet is excited. One of typical example is the 6 MeV, Of state and one of the two Of levels around 7 MeV. Theoretically the 6 MeV, O f state is dominated by most probably 4 p-4 h state.
The levels based on this Of band are strongly excited by the 12C(Li, t ) I 6 0 reaction. In order to eliminate a possibility of two-particle two-hole states, we calculate the spectroscopic factor of alpha cluster. Because we must avoid the spurious center of mass motion, we take the strong coupling model between holes and particles. The (42) representation is assumed for the 2 p-2 h state and the (84) representation is assumed for the 4 p-4 h state. These wave functions are givcn as follows :
and
I
p-4(sd)4 [I4441 (84) ' I S>
=Ichimura and I calculated spectroscopic factors using these wave functions. The calculated factors are 0.296 for the ground state (0 p-0 h), 0.178 for the (42) state and 0.244 for the (84) state.
Since the (42) state has smaller spectroscopic factor and the relative motion of the alpha cluster to the 12C cluster has less quanta by 2 h o than (84) state, we can expect that the (42) state will be weakly excited than the (84). This seems to be consistent with available experiments. From the argument described above, wc can predict that the 6 p-2 h states in 2 0 ~ e should be more
weakly excited than the 8p-4h states in the two alpha transfer experiment on 12C, for example 12C(12C, a) 'ONe and 12C(14N, 6Li)20Ne. The former experiment has been recently carried out by Middleton and his colleagues [9]. The later was done by Marquardt and Von Oertzen and his colleagues [lo] and by Nagatani et al. [ll]. Combinding these experiments, we can conclude that the 7.20 MeV, O', 7.84 MeV, 2+, 9.08 MeV, 4' and 12.1 9 MeV, 6' states are members of the 8p-4h quartet states. On the other hand, the 6.72 MeV, Of, 7.32 MeV, 2' and 9.89 MeV, 4' are levels of the normal ( ~ d ) ~ configuration. According to the recent experiments done by Dixon et al. [I21 the first excited 0' state in 44Ti is a t 1.9 1 MeV which is not predicted by the normal shell model calculation which assumes the (p, f)4 configuration. This level is not excited by the alpha transfer reaction a t all.
Arima, Gillet and Ginocchio predicted the quartet excited level a t 2 MeV. If this prediction is correct, this level should be excited by the 36A(12C, c ( ) ~ ~ T ~ reaction.
Beyond Ca region, because of the neutron excess, protons occupy the different levels than the neutrons for low energy excitations. This results in a very small alpha transfer cross section. In order to see this effect clearly, let us study the structure of 5 8 ~ i . Jaffrin took the stretch scheme to approximate 4 par- ticle states in thc (p3/, fs12 pl12) shell and coupled these 4 particle states with 2 f7/2 proton holes.
Shimizu [13] has taken the pseudo SU3 scheme [I41 instead of the stretch scheme. He has taken into account 2p, 3p-lh and 4p-2h states. The Kuo- Brown interaction together with a phcnomenological interaction are assumed. His wave functions are shown in Table VI for the 0' states. The third Of is mainly 4p-2h state. He calculated the spectroscopic factors of alpha cluster too. Table VII the pseudo SU, (80) state has the largest value which is three times as large as that of the 2 particle state. Thus we can expect that the alpha transfer reaction on 54Fe has very small cross section because of the Q-value mis- matching and because of the small spectroscopic
Wave functions of low-lying 0' states iin 58Ni
(pfI2 [21 f7;22(~1) ( p 0 4 [41 (80) (52) f7;: (pf), Observed
J, = 0 Excitation
(40) (02) (21) J2 = 0 22 44 MeV
(2, p ) stands for the irreducible representation of the pseudo SU, coupling.
factor, but the same reaction can excite strongly SimiIarly the calculation predicts that the 4.10 MeV, the 3.59 MeV, third 0' state. 2+ state has a large alpha spectroscopic factor. This is consistent with the experiment done in Saclay [15].
Higher spin states of 4 p-2 h are expected to have fairly
TABLE VII large spectroscopic factors.
Specrroscopic factors of alpha cluster to the ground state of 54Fe
J n States S
*
f,;
pseudo (80) S 0.008 5 o* " - 2 4j 7 / 2 P3/2 0.002 2
fi,'f,"!2 0.000 0
2 4
f G 2 P1/2 0.000 5
P3/2 2 0.000 4
J R = 2+ j'; pseudo (80) L) 0.007 7
(*) Without the correction due to the centre of mass motion.
3. Rotons. - My final remark should concern the rotons. Suppose that we are given a Hamiltonian.
Because of four body correlations or quarteting, a system with two neutrons and two protons have a very stable states for each spin, for example the members of the ground state band in ''Ne. Because our Harniltonian is general, those states might be very complicated. We can add onc particle, two par- ticles or three particles on these states. The structure of those states may not be changed too much because of their stability. Furthermore, if we have four protons and four neutrons, the low-lying states can be approxi- mated by
A factor
(:)I2
must be multiplied.SHELL MODEL, QUARTETS A N D ROTONS I N THE S-d SHELL C6-37 where A is the antisymmetrizen. Gillet a n d I [17] T h e antisymmetrizen is not easy t o handle. This is called Y, rotons, which a r e realizations of quartets. a drawback of the roton approximation. A very If nucleons in YI occupy different shells than those interesting attempt has been carried o u t by A. Zuker which a r e occupied by nucleons in Y,,, the anti- a n d M. Wong, who will tell us their story in the symmetrizen does not cause any trouble. In general, following session.
nucleons in Y I , and Y,, occupy the same orbits.
References [I] WIGNEK (E. P.), Phys. Rev., 1937, 51, 106.
[2] FRENCH (J. B.) and PARIKH (J. C.), Phys. Letters, 1971, 35B, 1.
[3] GILLET (V.), Proceedings of the International Confe- rence on Proprties of Nuclear states, Montreal Canada 1969, cdited by M. Harvey and al.
Presses de I'universite de Montreal (Montreal, Canada 1969), p. 483.
[4] ELLIO-rr (J. P.), Proc. Roy. Soc., 1958, A 245, 128, 562.
EI.LIOTT (J. P.) and H A R V ~ Y (M.), Proc. Roy. Soc., 1963, A 272, 557.
[5] BAYMAN (B. F.) and BOHR (A.), Nlicl. PIzys., 1958159, 9, 596.
[h] WII.DERMUTH (K.) and M c CLURE (W.), Cluster representations of nuclei (Springer, Berlin, 1966).
[7] MIDDLETON (R.), ROSNER (B.), PULLEN (D. J.) and POI-SKY (L.), Phys. Rev. Letters, 1968, 20, 118.
[8] ARIMA (A.). GILLET (V.) and GINOCCHIO (J.), Phys.
R E V . Letters, 1970, 25, 1043.
[9] MIDDLETON (R.), private communication.
1101 MARQUAKDT (N.), VON OERTZEN (W.) and WAL-
TER (R. L.), Ph.ys. Letfers, 1971, 35B, 37.
[ l l ] NAGATANI (K.), LL VINE (M. J.), VELOTE (T. A.) and ARIMA (A.), preprint.
[I21 DIXON (W.), private communication.
[I31 SHIMIZU (K.) and ARIMA (A.), to be published.
[14] ARIMA (A.), HARVEY (M.) and SHIMIZU (K.), f'hys.
Letters, 1969, 30B, 51 7.
[15] FAIVRE (J. C.), FARAGGI (H.), GASTEROIS (J.), HAR-
VEY (B. G.), LEMAIRE (M. C . ) , LOISEAUX (J. M.), MERMAZ (M. C.) and PAPINEAU (A.), Phys. Letters, 1970, 24. 1188.
[I61 MC GROKY (J.), Private communication.
[I71 AKIMA (A.) and GILLET (V.), Antr. of Pliys., 1971, 66, 117.
