Application of the angular momentum projection method to some even mass nuclei

Nuclear Physics A557 (1993) 44-58~ North-Holland. Amsterdam

NUCLEAR PHYSICS A

Application of the Angular Momentum Projection Method to some Even Mass Nuclei

Kenji Hara

Physik-Department, Technische Universitat Miinchen W-8046 Garching bei Miinchen, Germany

Abstract Spin alignment and BE2 values for doubly even nuclei are compared with experimen-

tal data. For doubly odd nuclei, it is demonstrated that two alternative mechanisms exist for signature inversion. Analysis of new data on 138Pr is also presented.

1. Outline of the Theory

Angular momentum projection method is an efficient way of carrying out the shell model configuration mixing calculations for deformed nuclei. It has a great advantage over the standard (spherical) shell model: The dimension of the configuration space can be reduced drastically, so that the shell model calculations become feasible even for heavy systems. We take an appropriate deformed basis (Nilsson) and build upon it a quasiparticle representation (BCS). The hierarchy of the (intrinsic) states is thus characterized by the energy and number of quasiparticles. We project out thus obtained (m&i-) quasiparticle states onto good angular momenta. This procedure corresponds to the transformation from the body-fixed system to a space-fixed system. The Hamil- tonian is then diagonalized in this (many-body) basis for each spin. Interpretation of the results can be done by looking at a diagram in which the rotational energies (i.e. expectation values of the Hamiltonian) of various bands are plotted against spin. Band crossing is often observed in such a diagram. It is a purely quantum mechanical theory and provides us with reliable nuclear wavefunctions.

In all calculations, we use a Q.Q + Pairing + Quadrupole-Pairing force model, which has been quite successful despite its simplicity. The strength of Q.Q force is adjusted to give the right deformation and that of Pairing force to the known energy gap. The strength of Quadrupole Pairing force is assumed for simplicity to be proportional to the (Monopole-) Pairing force with a proportionality constant common for all nuclei.

Expanding eigenstates of the Hamiltonian in the configuration space spanned by the projected quasiparticle states, {PL, 1 @K >}, we obtain the eigenvalue equation:

03759474P3I806.00 0 1993 - Elsevier Science publishers B.V. All rights reserved.

4.5oc K. Hara I Angular momentum projection method

The matrix elements appearing in Eq. (1) are defined by

H’, K< =< @‘c ] HI& ] a,) > , N’,,, =< QK ] I& ] @,I > (2)

We assumed here an axially symmetric system, so that to a given K. there corresponds a unique K. In an axially asymmetric system, however, the “quantum number”K runs independently to IC, so that it is necessary to make the replacement: F’, + FiK and take accordingly the summation over K into account in Eq. (1).

For the interpretation of numerical results, it is quite useful to define the rotational energy and frequency of a band r;:

E,(I) = 2 = < a)‘* 1 tiex, I@, > dE,(I) < @% /6X, 1 !D,$ > ’

w,(I) = -&- = E,(I) - :‘I - AI) (3)

lcI(

If we plot the rotational energy of various bands as a function of spin (Band Diagram), the rotational frequency is the slope of the diagram. Well decoupled band has a smaller rotational frequency (due to a larger moment of inertia) and shows stronger signature dependence. In the band diagram, we often observe that two bands, which have mutually different characteristics (slope and signature dependence), cross each other. This may lead to prominent phenomena such as backbending and signature inversion. Some bands show spontaneous signature inversion by themselves (Self-Inversion). We shall see examples of both types of signature inversion in section 3.

Matrix elements of a tensor operator TX, can be evaluated by using the expression

< Q?&, 1 TX, 1 Q& > < IM, Xp I I’M’ >

= c < IK’ - V, XV 1 I’K’ >< @,I I +Avi)k+,K 1 @K > F;,F’, ” K K’

In section 2, some results of BE2 calculations will be compared with the experimental data by using the quantity

Qt(I -+ I’) = <10,2;lI,0>Jw

which is a measure for the magnitude of the intrinsic electric quadrupole moment of a deformed (axially symmetric) nucleus. It differs of course from the expectation value of the electric quadrupole operator.

A fast algorithm t.o evaluate matrix elements of operators with respect to projected (multi-) quasiparticle states is given in Ref. [l] and its early applications in Refs. [2, 3, 41. More extended applications and systematic analysis of recent data on doubly even, doubly odd and odd mass rare-earth nuclei can be found respectively in Refs. [5, G, 71.

2. Doubly Even Nuclei - BE2 Values

The mechanism of (first and second) backbending in doubly even rare-earth nuclei was discussed in the framework of the angular momentum projection theory in Ref. [5] by taking up to two-broken-pair states. On the other hand, the BE2 measurements are

K. Hara I Angular momentum projection method 451c

available mostly up to spin 20 at the moment. Therefore, for the present purpose, it is sufficient to take quasiparticle configurations only up to one-broken-pair states:

I @K >= { IO >, atat, IO >, aLa$ / 0 > } (5)

where IO > is the qua&particle vacuum state and the quantum numbers v and V’ (r and x’) run over the neutron (proton) Nilsson levels around the Fermi energy. Figs. 1 and 2 compare the results (dots) with experiments (open squares and also open circles if more than one measurement exists) for Er [8, 9, lo] and Yb [ll, 121 isotopes, respectively. In each set of figures, the upper one shows the spin alignment and the lower one the ratio Qt(I + I - 2)/Qt(2 + 0). Quality of agreement is about the same for other nuclei.

3. Doubly Odd Nuclei - Signature

Up to a moderate spin, the quasiparticle configurations in doubly odd nuclei are taken to be l-neutron@l-proton quasiparticle states:

1 QK >= { a+a+ IO > } Y A (6)

The problem of signature inversion in doubly odd nuclei was discussed extensively in Ref. [6]. It was shown that signature dependence in the rotational energy of a band may occur if I K, + K, ( is small. Spins of the favored (unfavored) states satisfy normally the signatzlre ruIe I - j, - j, = even (odd) where j, and j, denote the (decoupled) neutron and proton sub-shells, respectively. However, it was shown [6] that violation of this rule (signature anomaly) occurs at lower spins when K, and K, have mutually opposite signs, though it becomes normal at higher spins. Such a spontaneous signature inversion in a given band is called self-inversion. Thus, there are two possible mechanisms [6] for the signature inversion, namely the one due to self-inversion and the other due to band crossing. Either of the two mechanisms may alternatively occur in actual nuclei depending on the Fermi energy (shell filling). W e can in fact see this by studying, for example, a series of isotones.

Figs. 3, 4, 5 and 6 show the AE plots (upper figures) and the band diagrams (lower figures showing some selected bands only) for the negative parity band of isotones ls6Tb, “‘Ho, “jOTrn and 162Lu respectively. Theoretical AE plots (dots) are compared with experiments (open squares) for the first two nuclei [13, 141.

Let us study the nucleus issTb in some detail. The AE plot shows that the signature inversion occurs at spin 12. In this nucleus, low-lying configurations are I/i1si2 @ xhr,,,. In the band diagram, they are plotted in solid lines. Obviously, the signature rule is fulfilled for spins higher than 12. At lower spins, the staggering phase is anomalous. This comes from the self-inversion of the band Kv=1/2i1si2 @ K,=-3/2hr1,2 (the one which starts at spin I=1 in the second low-lying group of bands). In this nucleus, the configurations yi18,2 @J Thgj2 (dashed lines) play no role as they are lying much higher. However, they will come down as proton number increases because the proton Fermi energy moves toward the Nilsson level K,=1/2hs,z. This can be seen in the band diagrams for 15sHo and 16’Tm. In 15sHo, the zig-zag amplitude gets smaller because the proton Fermi energy moves away from the Nilsson level I(X=-3/2h11,2.

452~ K. Hara I Angular momentum projection method

On the other hand, in 16’Trn, the signature inversion will be due to crossing of two bands which have mutually opposite signature dependence (dashed and solid lines). Such cases were already discussed in Ref. [6] in detail. Beyond this nucleus, the proton Nilsson level K,=1/2hs,z becomes more and more important (see “*Lu). In the present series of isotones, proton number 69 (Tm) seems to be the border, below (above) which the proton Nilsson level K,=-3/2hir,z (K,=1/2hs,z) plays an essential role in the mechanism of signature inversion (self-inversion vs. band crossing). Figs. 7 and 8 compare our results with recent data for the nucleus issTm [15] and 166Lu [16].

At this point, we comment on the relation between the y-deformation and signature dependence. We notice that our theoretical predictions for the staggering amplitudes are in general much smaller than what is observed at lower spins where anomalous signature splitting occurs. The reason for this is as follows. Due to assumed axial symmetry, two bands (K*=K,fK,) which have mutually opposite phases at lower spins are nearly degenerate. In fact, we can see two such closely lying bands appearing pairwise in the band diagrm. As a result of this degeneracy, the net signature effect is almost canceled out. We expect that introduction of y-deformation removes such a degeneracy and makes zig-zag amplitudes larger. This is our understanding of the reason why the presence of tri-axiality can enhance the signature dependence at lower spins.

Finally, we show the analysis on the positive (vhlliz @rhrr,z) and negative (Vhir,z @ 7rg7j2) parity bands of 138Pr [17] in Figs. 9 and 10. The spectra of both bands can be consistently described only if we assume an oblate shape (E M - 0.16) in the calculation. We can therefore state with confidence that this nucleus is oblate. Our theory predicts that pairs of negative parity states (8-, 9-), (lo-, ll-), (12-, 13-), ... etc. are almost degenerate. This explains the reason why only one of the partners is measured in the experiment through the stretched E2 transitions (AI=2 sequence). How one can detect the missing partner-states poses an interesting question. We have also studied a number of neighboring doubly odd and odd mass nuclei systematically. The result indicates that a transition from prolate to oblate shape takes place across the neutron number 77. Nuclei around this neutron number (N = 76, 77, 78) are most likely y-deformed. This conclusion has been deduced from the fact that calculations using axially symmetric basis (Nilsson + BCS) could not reproduce the spectra of these nuclei. The following table summarizes the deformation parameters (E) compiled by the analysis.

Table

N 73 74

soNdN + 0.22 + 0.20 * - 0.16 5&N + 0.22 + 0.21 + 0.20 * * * - 0.16 58CeN + 0.20 * - 0.16 57LaN + 0.22 + 0.21 + 0.20 * *

+ : prolate shape + : presumably y-deformed - : oblate shape


There is no doubt that axial symmetry is broken in some nuclei. In this connection, we point out that most of prominent phenomena is caused by spin alignment and takes place in nuclei whose deformation is not very large. Small deformation is in fact a necessary condition for spin alignment to be substantial. However, if the deformation is not large, we may generally expect the presence of T-deformation [18]. As mentioned before, our present computer code assumes axial symmetry. We shall remove this restriction and extend the program to include tri-axiality in the nearest future.

A number of people contributed to the present article which is a combination of two independent works. The calculations for doubly even and doubly odd rare-earth nuclei were carried out in Munich by Y. Sun (University of Madrid) during the summer

vacation, while those for 13sPr and neighboring nuclei were done in collaboration with Gamma-Spectroscopy group of the Pelletron Laboratory at University of S%o Paulo,

where a number of doubly odd nuclei in the mass region A=130-140 has been measured. Those who were involved in the present analysis are E. W. Cybulska, M. A. Rizzutto, L. G. R. Emediato, R. V. Ribas and C. L. Lima. Detailed account of individual work is in preparation and will be published elsewhere.

References

1 K. Hara and S. Iwasaki, Nucl. Phys. A332 (1979) 61

2 K. Hara and S. Iwasaki, Nucl. Phys. A348 (1980) 200 3 S. Iwasaki and K. Hara, Prog. Theor. Phys. 68 (1982) 1782 4 K. Hara and S. Iwasaki, Nucl. Phys. A430 (1984) 175 5 K. Hara and Y. Sun, Nucl. Phys. A529 (1991) 445 6 K. Hara and Y. Sun, Nucl. Phys. A531 (1991) 221 7 K. Hara and Y. Sun, Nucl. Phys. A537 (1992) 77 8 M. Ohshima et. al., Phys. Rev. C33 (1986) 1988 9 M. de Voit et. al., Rev. Mod. Phys. 55 (1983) 949

10 B. Bother et. al., Sov. 3. Nucl. Phys. 30 (1979) 305 11 B. Bother et. al., Nucl. Phys. A267 (1976) 344

12 J. C. Becker et. al., Phys. Rev. C35 (1987) 1170 13 R. Bengtson et. al., Nucl. Phys. A389 (1982) 158 14 M. A. Lee (ed.), Nucl. Data Sheets 56 (1989) 219 15 D. Hojman et. al., Phys. Rev. C45 (1992) 90 16 S. Drissi et. al., Nucl. Phys. A543 (1992) 495 17 M. A. Rizzutto et. al., Z. f. Physik to be published 18 A. Hayashi, K. Hara and P. Ring, Phys. Rev. Lett. 53 (1984) 337

454c K. Hara I hng~iar morn~ni~ ~roj~~~io~ meFho~

w = [ E(l) - E(I-2) ] / 2 o = [ E(l) - E(I-2) ] I2

o.5’ o.5 1

Q,(I -> l-2) I Q{(2 -> 0) Q,(l -> t-2) / C&(2 -> 0)

1.50. I I / 4 / ’ 1.50 I I 6 , ,

‘6oEr 1.25

0.50. ! , ) l , 0.50 4 I I , *, I

0 4 8 12 16

2cJ

0 4 8 12 16 20

Spin spin

7

Fig. 1: Alignment (upper fig.) and BE2 amplitude (lower fig.) for Er isotopes

o = [ E(l) - E(L2) ] / 2 w = [ E(I). E(I-2) ] / 2

o.51 o.51

0.4

t ‘64Yb

t

0.4

t

“=Yb

t

0.0 -

Q,(l -> I-2) / Q,(2 -> 0)

1 so 1 I / , I

‘Vb 1.25 .-

0.0 1 I I I / I

Q,(l -> l-2) / Qt(2 -> 0)

1.50, / / / 1 I

‘=Yb 1.25 -

f.OO.- * :' '9 '

0.75 '- . I 1 / _. Ill

/J ,, II - J.OO.- c 2 j l/l 11: -~ : . 'I z

.

II-

,,

$1 i,

,I

. , . 0.75 - .

. .

0.50 .I 0.50 < I I ! I I _

0 4 8 12 16 20 0 4 8 12 16 20

Spin Spin

Fig. 2: Alignment (upper fig.) and BE2 amplitude (lower fig.) for Yb isotopes


0 10 14 18 ;

Spin

5.0

4.5

4.0

3.5

3.0

2.5

2.0

1.5

1.0

0.5

E(I) - E(f-1) (MeV) E(l) - E(l-1) (MeV)

Energy (MeV) Energy (MeV)

0.01 ’ ’ 1 ! ’ ’ ’ + ! ! 1 I 0 2 4 6 8 10 12 14 16 18 20 22

Spin

Fig. 3: AE and Band diagram for ‘?C’b Fig. 4: AE and Band diagram for 158H~

Spin

5.0

4.5

4.0

3.5

3.0

2.5

2.0

1.5

1.0

0.5

0.0 1-l 0 2 4 6 8 10 12 14 16 18 20 22

Spin

456c K. Hara I Angular momentum projection method

0.5

0.4

0.3

0.2

0.1

0.0

E(l) - E(I-1) (MeV) E(l) - E(I-I) (MeV)

6

16oTm

Spin

L I , I

6 10 14 18

Spin


0.01 ’ ’ ’ ’ ’ ! ! ’ ’ ’ 0 2 4 6 8 10 12 14 16 18 20 22

Spin

Fig. 5: AE and Band diagram for lsOTm Fig. 6: AE and Band diagram for ‘%u

0.5

0.4

0.3

0.2

0.1

0.0

0.5 -

0 2 4 6 8 10 12 14 16 18 20 22

Spin


E(l) - E(I-1) (MeV) E(l) - E(I-1) (MeV)

0.5

0.4

0.3

0.2

0.1

0.0

166Tm

‘rl I

4’ ,

>-

, ,

.;y/y ;

,iA’ If

P’ d

u’

6 10 14 16 22


0.01 ! ! ’ ! ’ ! ! ’ ’ ! ! ’ 0 2 4 6 6 10 12 14 16 18 20 22

Spin

Fig. 7: AE and Band diagram for lssTm Fig. 8: AE and Band diagram for 166Lu

2.5

2.0

Spin

0 2 4 6 8 10 12 14 16 18 20 22 Spin

458c K. Hara I angular ~rnent~ project~o~ method

8.0

6.0

5:

2 4.0 E W

2.0

00

IrH(Thcory +-O.IO) & dlluperinm~l

2 2 6 8 10 12 14 16 18 20

8.0

6.0

u

Fig. 9: Positive parity band fuhtl,2 @I rh11,2] of *38Pr

8 IO 12 14 16 18

Fig. 10: Negative parity band [vhl,p @ xg7i2j of 138Pr

20

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Application of the angular momentum projection method to some even mass nuclei