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Submitted on 1 Jan 1971
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SHELL-MODEL CALCULATIONS FOR 28Si
M. Soyeur, A. Zuker
To cite this version:
M. Soyeur, A. Zuker. SHELL-MODEL CALCULATIONS FOR 28Si. Journal de Physique Colloques, 1971, 32 (C6), pp.C6-283-C6-285. �10.1051/jphyscol:1971666�. �jpa-00214884�
JOURNAL DE PHYSIQUE Collogue C6, supplement au n° 11-12, Tome 32, Novembre-Decembre 1971, page C6-283
SHELL-MODEL CALCULATIONS FOR
28Si
M. SOYEUR and A. P. ZUKER
Service de Physique Theorique, C. E. N . Saclay, France
Resume. — Un calcul de modele en couches pour le 28Si utilisant une interaction effective et un grand nombre de configurations, indique qu'une bonne description du spectre et des proprietes electromagnetiques de ce noyau peut etre obtenue dans l'espace conventionnel de la couche (s-d).
Les calculs precedents considerant des espaces plus reduits, sont discutes.
Abstract. — Results of extensive shell-model calculations with realistic interactions in 28Si are described. They indicate that a good description of the spectrum and electromagnetic properties can be obtained within the conventional (s-d) shell space. Remarks are made concerning previous calculations and the use of different truncation schemes.
An extensive shell-model study of 28Si is particularly worthwhile since this nucleus seems to have a rather complicated structure.
The SU3 scheme and the deformed Hartree-Fock method using central forces, which are quite successful at the beginning of the (s-d) shell, lead to degenerate or almost degenerate solutions for the ground state (one prolate and one oblate) [1], in obvious disagree- ment with experiment.
Since 28Si lies in the middle of the (s-d) shell it seems safe to assume that excitations from the p-hole states or into the (fp)-particle states can be disregarded for the lowest lying positive parity levels. Even if we confine ourselves to (s-d)12 configurations we are faced with too large matrices and some truncations have to be accepted.
Our first step was to perform calculations in a restricted space containing jumps of up to 4 particle- 4 holes with respect to the closed (d 5/2)12 subshell, in other words (d 5/2)1 2 -", (s 1/2 d 3/2)", n < 4, confi- gurations. We chose the renormalized Kahana-Lee- Scott interaction [2] and no attempt was made, in this first step or subsequently, to «improve» the matrix elements. The experimental single particle spectrum of 1 70 was used.
The agreement between experimental and calculated spectra turned out to be very good and we thought the 28Si problem had been solved. It was interesting to understand why deformed Hartree-Fock had failed for such a long time and a calculation was undertaken by S. Das Gupta [3], using exactly the same interaction we had used. The results were somewhat surprising (and reassuring) in that they showed a marked impro- vement compared to those obtained previously : a nice gap had developed between the prolate and oblate solutions. Furthermore the (unprojected) ground state energy was slightly lower than the one obtained in the
shell-model calculations, indicating a better variational wave function. What was somewhat disturbing is the fact that the occupation probabilities for the orbits differed substantially from our results.
To remove these discrepancies we carried out the exact diagonalization for the 0+ states (dimensio- nality = 839). It required only trivial changes in the Oak Ridge-Rochester shell-model program [4], a large computer and a lot of time. The ground state was now well below the Hartree-Fock result but, as expected, the wave functions contained dominant components lying outside our original truncated space. Figure 1 gives an idea of the structure of the wave function.
0 1 1 2 2 2 3 3 3 3 4 1 4 4 4 5 5 5 5 5 8 6 5 6 5 7 7 7 7 7 8 8 8 8 8 9 9 9 9 ID 10 U1111 12 i u n t i n t o i n ; i [ l u a i i n n i u B i u a i 2 3 0 1 2 0 1 0 S 8 7 8 7 6 8 T 6 S 8 7 6 5 4 7 6 5 4 3 6 S 4 3 2 5 4 3 2 1 4 3 2 1 0 3 2 I 0 2 1 O 1 0 O
FIG. 1.—-This figure shows the distribution of the different configurations in the ground state wave function of the Si28
nucleus calculated in the complete (s-d) shell space. In abscissa we put the 45 possible configurations. A vector represents a configuration having n\ particles in d 5/2, nj particles in s 1/2 and »3 particles in d 3/2. In ordinate, we put the weight (squared)
of each configuration.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1971666
C6-284 M. SOYEUR AND A. P. ZUKER At this point, it became clear that the difficulties
associated with Hartree-Fock calculations were not due to the method but to the forces used. It also became clear why truncations based on a closed (d 512)'' subshell were insufficient. For a Rosenfeld- Yukawa interaction, (d 512)'' is the lowest unperturbed state and for realistic forces, a large number of configurations come much lower : (d&2 s?,2), (d&
4 2
s112 d3/2), etc
...
To improve our truncation scheme, and because of the impossibility of calculating exact eigenvectors for J" # O+, we decided t o choose a space based on the
(( suggestions )) of the exact calculation. Accordingly, we kept all the configurations including states contri- buting more than 0.1 in amplitude to theexact three lowest 0' states : (dij2 s'$~ d32) with n1 +n2 +n3 = 12, n1 2 4 a n d n , < 4 ; n,, n,, n3 = 5, 3 , 4 a n d 6 , 2 , 4 were excluded. The resulting matrices had dimensions ranging up to 1 292 (J" = 4') for the spins considered ( J = 0, 1, 2, 4, 6).
The spectrum obtained is shown in figure 2. It is quite satisfactory although there is definitely too much spreading for J = 0, 2. This may be attributed t o a slightly too attractive interaction due t o the neglect of size effects (we use matrix elements designed for A = 18) or core polarization corrections (again designed for A = 18). It is interesting to mention that results for the bare matrix elements show the 0 + states coming in the right position, but a too contracted spectrum for the other spins.
By far the most satisfying results of our calculations are the B(E 2) transition rates shown in table I. They are obtained with a code constructed following the techniques outlined in ref. [4]. Using the conventional effective charge of 0.5 e for the neutrons and 1.5 e for the protons and tio = 41A-'I3, we obtain agreement within experimental bounds. The experi- mental values quoted have been measured recently in (py) reactions on 27Al by J. P. Gonidec, C . Miehe
0.00 O + OM) Ot
Theory Experiment
FIG. 2. -Energy spectrum obtained by calculating the 0+, l+, 2+, 4+ and 6" energy eigenvalues in a large (sd) shell space.
and G. Walter [5]. It should be noticed that the cal- culation using bare matrix elements gives similarly good agreement in spite of the rather large energy discrepancies mentioned above.
Last, we would like to come back to our original simplistic space which produced nice spectra and apparently bad wave functions. While our work was in progress a paper by J. B. Mc Grory and B. H. Wilden- thal [6] appeared in which a calculation is described
TABLE I
Calculated and experimental
16]
values for the B(E2).Initial state Energy (MeV) calc. exp
.
- -
1.86 1.78
5.00 4.62
6.02 4.98
7.96 6.69
6.84 7.38
6.84 7.38
8.91 7.42
8.91 7.42
8.91 7.42
8.91 7.42
8.39 8.54
Final state Energy (MeV) calc. exp.
- -
0.00 0.00
1.86 1.78
1.86 1.78
1.86 1.78
0.00 0.00
1.86 1.78
0.00 0.00
1.86 1.78
6.02 4.98
7.96 6.69
5.00 4.62
Reduced transition probabilities (e2 fm4)
Calculated Experimental values values [6]
SHELL-MODEL CALCULATIONS FOR 28Si C6-285 that used what amounts to our original space plus
the additional restriction of at most two particles in the d 3/2 orbit. Their interaction contains a number of adjustable parameters and reproduces well the expe- rimental spectrum. In spite of the shortcomings we have pointed out for such a model space, they obtained good agreement for the 2X H—• 0t + transition (others are not mentioned in their paper). This puzzling result made us suspect that there must be something interesting in our first calculations.
However, no confirmation was obtained when transition rates involving / = 0 and 2 were calculated ; some of them are acceptable and the 2x -*• Ot is very good (as found in ref. [6]), but most are off the experimental values by large factors (5 to 10). The
conclusion is that a good energy fit is certainly not sufficient to ensure the validity of a model.
We can summarize our results as follows : an inter- action containing no adjustable parameters gives reasonable spectra and good transition rates for 28Si in a large, but truncated, space. When a more severe truncation and the same interaction are used, the energy agreement remains good but the electromagnetic properties become quite poor.
Acknowledgements. — We are very grateful to G.
Walter for communicating numerous measurements before publication and to Mme N . Tichit for her valuable computational help.
References [1] DAS GUPTA (S.) and HARVEY (M.), Nucl. Phys., 1967,
A 94, 602.
[2] KAHANA (S.), LEE (H. C.) and SCOTT (C. K.), Phys.
Rev., 1969, 185, 1378, and private communica- tions from H. C. Lee.
[3] DAS GUPTA (S.), private communication. At the time of Das Gupta's calculations we were not aware of the existence of a paper by WONG (S. K. M.) et al. ; {Nucl. Phys., 1969, A 137, 318) in which realistic forces are shown to improve markedly
Hartree-Fock results.
[4] FRENCH (J. B.), HALBERT (E.), M C GRORY (J. B.)
and WONG (S. S. M.), Advances in Nuclear Physics, vol. 3, Ed. by M. Baranger and E. Vogt (Plenum Press, New York, 1969).
[5] GONIDEC (J. P.) MIEHE (C.) and WALTER (G.), C. R.
Acad. Sci. Paris, 1971, 272, 1385 and private communications.
[6] MCGRORY (J. B.) and WILDENTHAL (B. H.), Phys.
Letters, 1971, 34B, 373.
