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Evidence for strong ground-state shape asymmetry in even germanium and selenium isotopes
9 June 1994
PHYSICS LETTERS B ELSEVIER
Physics Letters B 329 (1994) 1-4
Evidence for strong ground-state shape asymmetry in even germanium and selenium isotopes W. Andrejtscheff, R Petkov Bulgarian Academy of Sciences, Institutefor Nuclear, Research and Nuclear Energy, 1784 Sofia, Bulgaria Received 17 February 1994 Editor: C. Mahaux
Abstract
The asymmetry of even-even nuclei with 46< A <82 (22< Z <34)is studied by the sum-rule method applying recently introduced approximations. The uniquely high eccentricities of 72-76Ge and 74-78Se point at the most pronounced (effective) triaxiality of ground states emerging so far from experimental data for 46< A <192.
The high interest in the structure of Germanium and Selenium nuclei has persisted over many years. Their low-lying states offer a variety of"shape phenomena" extensively discussed in the literature: shape change in an isotopic chain (e.g. between N--38 and 40), shape coexistence (i.e. states with different shape: prolate, oblate or spherical in the same nucleus) (cf. e.g. Refs. [ 1-3] ). To some extent in controversy to these considerations of axially symmetric shapes, earlier calculations [4,5] and recent experimental evidence [ 6,7 ] point at the possibility of asymmetric (triaxial) shapes in the even germanium isotopes. In recent years, possible deviations from axial symmetry have been extensively discussed (Ref. [8] and references therein) mainly for high-spin states. In the present paper, we exploit the recently introduced approximations [9] in the sum-rule method [ 10,11 ] to investigate the shape of even-even Ge and Se isotopes in their ground states. For the sake of comparison, we display the earlier [9] data (94< A <192) and for the first time relevant results down to A = 46. Let us briefly recall the basic expressions needed.
Using invariant products of the collective E2 operator, expectation values of (Q~) and (cos 38j) can be derived from expansions over the experimental reduced E2 matrix elements associated with the state J. In the case of the J~---0+ ground state of an even-even nucleus: 2
I(0+1[E2112+)12
(Qg.s.) = E
(1)
r
where r denotes the different 2 + excitations. In practice, the restriction to r=l,2 turns out to be sufficient [9]. The quantity (Q2) is a measure of the symmetric quadrupole deformation firms which includes both static and dynamic contributions: 47/"
#rms = 3 ZR2o~/ (a))
(2)
Furthermore, in the estimate of (cos 388.s.) one can again involve only the first and second 2 + excitations ( r= 1,2) but also additionally neglect [ 9 ] the term (0~- IIE2112~-)2(2~-IIE21[2~-):
0370-2693/94/$07.00 (~) 1994 Elsevier Science B.V. All fights reserved SSDI 0 3 7 0 - 2 6 9 3 ( 94 ) 0 0 4 8 2 - M
W. Andrejtscheff, P. Petkov / Physics Letters B 329 (1994) 1-4
,/~--m= x-3/2 [ <0TllE2112+>= (2cii E21[2D
-- V lO\~g's'/
+ 2<2yIIEeIIo+>] (3) The value of
/N
~eff = 31-arccos((cos38g s))
(4)
corresponds (up to higher order terms) to the collective-model asymmetry angle y. As an example of a case where a check is possible, the neglection of the product <0+IIE2112~>=<2~-IIE2112~ -> in 7=Ge (data taken from Ref. [6] ) results in a reduction of 3eff by less than 0.1 °. The approximation (Eq. ( 3 ) ) turns out to be very efficient. It facilitates the estimate of 6elf for the ground states of many even-even nuclei (nearly seventy in this paper) in which the four matrix elements entering Eq. (3) have been experimentally determined. Let us consider now the eccentricity eK (generally: about axes x= 1,2,3) of the nuclear ellipsoid with semiaxes R,~ (Eqs. (5.74) and (5.75) in Ref. [12]). A comparison of the asymmetry of different ellipsoids (with arbitrary values o f / 3 and y) appears possible using that eccentricity eK which is the lowest one (associated with the smallest moment of inertia) among those with x=l,2 and 3. For 0o<9,<30 °, the lowest eccentricity is that about axis 3 (symmetry axis at y--O°) : e3 = R~ - R22 ~ V ~ - ~ / 3 s i n y
(5)
Accordingly, for 300_<3,<60 ° the lowest eccentricity is that about axis 2 (symmetry axis at 3/--60 °) : e2
R~
~"
.
.
sin(B0° - 3/)
(6)
Combining Eqs. (5) and (6), we introduce as a measure of the deviation from axial symmetry in the sector 00_<7_<60 ° the eccentricity e =
e3(/3, y) e2(/3, y )
0°_
(7)
The significance of this quantity (Eq. ( 7 ) ) appears plausible from some examples; e.g. for any/3: e ( / 3 , y = O°)=e(/3, y = 60 °) = 0 (axial symmetry); for a given/3=/31:e(/31 , y = 30 °) =emax(/31) etc.
o.6[;., , ~ / t 0 ~°~o
°
0"2Ol JtJ"j" e. 's~'S~Ba'~tm{'TeC}J d -o2 ~tV m'~m -o.4 F ~
~'~
IT~""teFa"0.6 " 0.0
0.05
.]40 ° 0.1
0.15
0.2
0.25
0.3
0.35
0.4
Fig. 1. Values of (cos 38) (Eq. (3)) versus the quadrupole deformation firms (Eq. (2)). On the r.h. axis, ~eff (Eq. (4)) is given. Arrows indicate that in the corresponding isotopic chain the mass number increases (T) or decreases (J.) with increasing quadrupole deformation. Data are taken from Refs. [13,14] and from current issues of Nucl. Data Sheets. For 58,60.62Ni, the values of firms and ~eff are 0.183 and 23.9°, 0.207 and 35.5° as well as 0.202 and 38.6° , respectively. In Fig. 1, the values of (cos 36) (from now on we omit the index g.s.) for nuclei with available data in the range 4 6 < A <192 derived according to Eq. (3) are presented versus the symmetric quadrupole deformation firms (Eq. ( 2 ) ) . The asymmetry angle 8efe is indicated on the r.h.s. The data points of some nuclei constitute a separate track (framed) parallel to the main group. The tendency of the latter indicating a firms-Beff correlation was already discussed in Ref. [ 9 ]. The "new" track includes 60'62Ni, 7°-76Ge and 74-785e. Among them, positive quadrupole moments Q (2~-) were measured [ 14] for 6°'62Ni and 7°Ge resulting in values of 6eff well above 30 °. Of particular interest are the eccentricities e(flrms, Beef) (Eq. ( 7 ) ) displayed versus mass number A in Fig. 2. The overhelming majority of these values belongs to the surprisingly narrow interval 0.14< e <0.21 as already noticed in Ref. [9] for A > 92. Below that are only values for some "classical" axial rotors ( 1 5 0 < A <190) as well as for 138Ba. (Note that 6 = 30 ° may indicate triaxiality but also sphericity, cf. e.g. Ref. [ 10]. The eccentricity (Eq. ( 7 ) ) sharply distinguishes between them.) For several of the nuclei with eccentricities within the above interval, (soft) triaxiality has been considered in the literature. For instance, 98-1°4Ru (Z = 44) contribute to the small bump around A = 100 (Fig. 2).
W. Andrejtscheff, P. Petkov / Physics Letters B 329 (1994) 1-4
n'Ta'~Ge~'~4'~'r%e
0,3
0.0 . -0.1 40
60
80
I00
129
140
160
180
200
MASS NUMBER A Fig. 2. The eccentricity (Eq. (7)) derived from the data in Fig. I. In the insert, the region 68< A _<82 is expanded. For 7°Ge (Ref. [15]), CE data for TI/2(2~- ) are used producing 8eft = 35.8 ° and e = 0.20 rather than DSAM data (Self = 50°, e = 0.08) due to the better fitting into this systematics.
Investigations [ 16] with the general collective model (GCM) found for l°°-l°SRu triaxial minima in the potential-energy surfaces (PES). A striking feature of Fig. 2 are the eccentricities of 72-76Ge and 74-78Se (N = 40, 42, 44) lying appreciably higher than all other values. Apparently, in these isotopes the shapes of the ground states reveal the most pronounced asymmetry among the nearby seventy nuclei in this systematics arising from available data for 46<_ A _<192 (22< Z <76). It should be stressed that the present analysis based on third-order products of matrix elements (Eq. (3)) can not determine whether the triaxiality is soft (dynamic) or rigid (static). Triaxiality of the nuclei under investigation has been discussed in earlier studies. Hartree-Fock-Bogolyubov (HFB) calculations [4] predict dynarrfic triaxial deformation for 74Ge (single potential minimum at ~, ..~ 30 °) and other neighbours. PES derived [5] with the Woods-Saxon potential indicate 3~-softness in several even Se,Ge and Zn isotopes and triaxial deformation in 72'74'76Ge. After comparing experimental E2 transition matrix elements to predictions of several models, the structure of 72Ge is found [6] to be a triaxial (~,=28.5 °) quadrupole-deformed collective rotational pattern perturbed by an "intruder" 0 + state. Ratios of level energies and of B(E2) values in 72-78Ge were compared [7] to predictions of several geometric models. On this basis, it was suggested [7] that 74-78Ge show the characteristics of an asymmetric ro-
tor with "y~30°. In Table 1, we display an excerpt from such a comparison indicating that similar interpretation could also apply to 74-785e but not e.g. to 82Se. Indeed, the shape parameters of S°Se and especially of S2Se (Figs. 1, 2) clearly differ from those of 74-78Se (including the sign of the hexadecapole deformation [ 17] ). The inspection of Table 1 suggest somewhat higher values of ~, (closer to 30 °) in the Z = 32 (Ge) isotopes than in those with Z --- 74 (Se). This suggestion is convincingly confirmed (Fig. 1) by the derived values of Seff (28.50,26.40,29.0 ° for 72'74'76Ge and 22.6 °,24.4 °,26.4 ° for 74'76'788e, respectively). In this way, the high eccentricities of 72-76Ge and 74-78Se could be associated with a triaxiality in their ground states, i.e. with a shape corresponding to a single minimum in the PES at 3' ~ 25°. The origin of asymmetric shapes is associated with orbitals strongly coupled by the }~2 operator as outlined in Ch.5.3 of Ref. [ 18] for 24Mg. In a Nilsson scheme [ 19] or in a Woods-Saxon scheme [20] both generated after a minimization with respect to f14 such closely lying orbitals at f12 = 0.2+0.3 are 1/2-[310] and 3/2-[312] active for Z = 32,34 as well as 1/2 - [ 301 ] and 5 / 2 - [ 303 ] active for N = 40, 42, 44. The corresponding Y22 matrix elements are asymptotically allowed [21] and thus expected to be large. In principle, the observed phenomenon of high eccentricities might be due to an effective triaxiality arising from a prolate-oblate shape mixing (e.g. two minima in the PES at ~,=0° and ~,=60°, respectively, resulting in an effective value of e.g. ~, ~ 30°). Such a mechanism is invoked in the large-scale microscopic (projected HFB) calculations (real and complex VAMPIR family) performed for some of the nuclei studied (Ref. [22] and references therein). The restriction to axial symmetry is made thereby for numerical reasons. In the complex FED VAMPIR calculation, the main component in the ground state of 72Ge is spherical correlated by moderately deformed prolate and oblate determinants. Concerning the matrix elements of importance for the present work (i.e. those involved in Eq. (3)), the quadrupole moment Q(2 +) is approximately reproduced by the calculation [22] but the B(E2,2 + --~0+) value comes out by ..~ 40% too small. In conclusion, uniquely high eccentricities were found for the ground-states of 72-76Ge and 74-78Se among those of nearly seventy nuclei accessible for this study in the region 46< A <192. This feature
4
W. Andrejtscheff, P. Petkov/Physics Letters B 329 (1994) 1-4
Table 1 Comparison of B(E2) ratios to predictions of different geometrical models [7]. The experimental matrix elements used (cols. 7-11 ) are the same employed to derive the data in Figs. 1, 2. B(E2) ratio
Sym. rotor
Vibrator
Asym. rotor = 20 °
22---.2 21---,0:I
<< 1
2.0
0.38
22"21 22"-,01
1.43
cx~
5.33
p o i n t s at t h e m o s t p r o n o u n c e d ( e f f e c t i v e ) t r i a x i a l i t y e m e r g i n g f r o m t h e s e e x p e r i m e n t a l data.
This work has been supported by the Bulgarian National Research Foundation under contracts PH14 and PH31.
References [I] J.L. Wood et al., Phys. Rep. 215 (1992) 101. [2] J.H. Hamilton, in: Treatise on heavy-ion science, vol. 8, ed. D.A. Bromley (Plenum, New York, 1989) p. 3. [ 3 ] M. Vergnes, in: Proc. EPS Conf. Structure of Medium-Heavy Nuclei (Rhodes, 1979), eds. G.S. Anagnostatos et al., IOP Conf. ser. No. 49 (Bristol, 1980) p. 25. [4] M. Girod and B. Grammaticos, Phys. Rev. C 27 (1983) 2317. [51 R. Bengtsson, in: Proc. Intern. Workshop Nuclear Structure of the Zirconium Region (Bad Honnef, 1988 ), eds. J. Eberth et al. (Springer, Berlin, 1988) p. 17.
Z
y = 30 ° 1.43
[6] [7] [8] [9] [10] [11] [ 12] [13] [14] [15] [16] [17] [18] [19]
[20] [21 ] [22]
Neutron number N 40
42
44
46
48
32 34
2.8 1.6
1.4 1.0
1.4 1.0
1.1 0.8
0.3
32 34
526 69
60 34
44 31
37 14
3.7
B. Kotlinski et al., Nucl. Phys. A 519 (1990) 646. W.T. Chou et al., Phys. Rev. C 47 (1993) t57. I. Hamamoto, Nucl. Phys. A 520 (1990) 2976. W. Andrejtscheff and P. Petkov, Phys. Rev. C 48 (1993) 2531. D. Cline, Ann. Rev. Nucl. Part. Sci. 36 (1986) 683. K. Kumar, Phys. Rev. Lett. 28 (1972) 249. J.M. Eisenberg and W. Greiner, Nuclear Theory, Vol. 1 (North-Holland, Amsterdam, 1970). S. Raman et al., At. Data Nucl. Data Tables 36 (1987) 1. P. Raghavan, At. Data Nucl. Data Tables 42 (1989) 189, M.R. Bhat, Nucl. Data Sheets 68 (1993) 117. D. Troltenier et al., Z. Phys. A 338 (1991) 261. S. Matsuki et al., Phys. Rev. Lett. 51 (1983) 1741. A. Bohr and B.R. Mottelson, Nuclear Structure, Vol. 2 (Benjamin, Reading, MA, 1975). K. Heyde, in: Proc. Intern. Workshop Nuclear Structure of the Zirconium Region (Bad Honnef, 1988), eds. J. Eberth et al. (Springer, Berlin, 1988) p. 3. W. Nazarewicz et al., Nucl. Phys. A 435 (1985) 397. J.P. Boisson and R. Piepenbring, Nucl. Phys. A 168 (1971) 385. A. Petrovici et al., Nucl. Phys. A 549 (1992) 352.
PHYSICS LETTERS B ELSEVIER
Physics Letters B 329 (1994) 1-4
Evidence for strong ground-state shape asymmetry in even germanium and selenium isotopes W. Andrejtscheff, R Petkov Bulgarian Academy of Sciences, Institutefor Nuclear, Research and Nuclear Energy, 1784 Sofia, Bulgaria Received 17 February 1994 Editor: C. Mahaux
Abstract
The asymmetry of even-even nuclei with 46< A <82 (22< Z <34)is studied by the sum-rule method applying recently introduced approximations. The uniquely high eccentricities of 72-76Ge and 74-78Se point at the most pronounced (effective) triaxiality of ground states emerging so far from experimental data for 46< A <192.
The high interest in the structure of Germanium and Selenium nuclei has persisted over many years. Their low-lying states offer a variety of"shape phenomena" extensively discussed in the literature: shape change in an isotopic chain (e.g. between N--38 and 40), shape coexistence (i.e. states with different shape: prolate, oblate or spherical in the same nucleus) (cf. e.g. Refs. [ 1-3] ). To some extent in controversy to these considerations of axially symmetric shapes, earlier calculations [4,5] and recent experimental evidence [ 6,7 ] point at the possibility of asymmetric (triaxial) shapes in the even germanium isotopes. In recent years, possible deviations from axial symmetry have been extensively discussed (Ref. [8] and references therein) mainly for high-spin states. In the present paper, we exploit the recently introduced approximations [9] in the sum-rule method [ 10,11 ] to investigate the shape of even-even Ge and Se isotopes in their ground states. For the sake of comparison, we display the earlier [9] data (94< A <192) and for the first time relevant results down to A = 46. Let us briefly recall the basic expressions needed.
Using invariant products of the collective E2 operator, expectation values of (Q~) and (cos 38j) can be derived from expansions over the experimental reduced E2 matrix elements associated with the state J. In the case of the J~---0+ ground state of an even-even nucleus: 2
I(0+1[E2112+)12
(Qg.s.) = E
(1)
r
where r denotes the different 2 + excitations. In practice, the restriction to r=l,2 turns out to be sufficient [9]. The quantity (Q2) is a measure of the symmetric quadrupole deformation firms which includes both static and dynamic contributions: 47/"
#rms = 3 ZR2o~/ (a))
(2)
Furthermore, in the estimate of (cos 388.s.) one can again involve only the first and second 2 + excitations ( r= 1,2) but also additionally neglect [ 9 ] the term (0~- IIE2112~-)2(2~-IIE21[2~-):
0370-2693/94/$07.00 (~) 1994 Elsevier Science B.V. All fights reserved SSDI 0 3 7 0 - 2 6 9 3 ( 94 ) 0 0 4 8 2 - M
W. Andrejtscheff, P. Petkov / Physics Letters B 329 (1994) 1-4
,/~--m= x-3/2 [ <0TllE2112+>= (2cii E21[2D
-- V lO\~g's'/
+ 2
/N
~eff = 31-arccos((cos38g s))
(4)
corresponds (up to higher order terms) to the collective-model asymmetry angle y. As an example of a case where a check is possible, the neglection of the product <0+IIE2112~>=<2~-IIE2112~ -> in 7=Ge (data taken from Ref. [6] ) results in a reduction of 3eff by less than 0.1 °. The approximation (Eq. ( 3 ) ) turns out to be very efficient. It facilitates the estimate of 6elf for the ground states of many even-even nuclei (nearly seventy in this paper) in which the four matrix elements entering Eq. (3) have been experimentally determined. Let us consider now the eccentricity eK (generally: about axes x= 1,2,3) of the nuclear ellipsoid with semiaxes R,~ (Eqs. (5.74) and (5.75) in Ref. [12]). A comparison of the asymmetry of different ellipsoids (with arbitrary values o f / 3 and y) appears possible using that eccentricity eK which is the lowest one (associated with the smallest moment of inertia) among those with x=l,2 and 3. For 0o<9,<30 °, the lowest eccentricity is that about axis 3 (symmetry axis at y--O°) : e3 = R~ - R22 ~ V ~ - ~ / 3 s i n y
(5)
Accordingly, for 300_<3,<60 ° the lowest eccentricity is that about axis 2 (symmetry axis at 3/--60 °) : e2
R~
~"
.
.
sin(B0° - 3/)
(6)
Combining Eqs. (5) and (6), we introduce as a measure of the deviation from axial symmetry in the sector 00_<7_<60 ° the eccentricity e =
e3(/3, y) e2(/3, y )
0°_
(7)
The significance of this quantity (Eq. ( 7 ) ) appears plausible from some examples; e.g. for any/3: e ( / 3 , y = O°)=e(/3, y = 60 °) = 0 (axial symmetry); for a given/3=/31:e(/31 , y = 30 °) =emax(/31) etc.
o.6[;., , ~ / t 0 ~°~o
°
0"2Ol JtJ"j" e. 's~'S~Ba'~tm{'TeC}J d -o2 ~tV m'~m -o.4 F ~
~'~
IT~""teFa"0.6 " 0.0
0.05
.]40 ° 0.1
0.15
0.2
0.25
0.3
0.35
0.4
Fig. 1. Values of (cos 38) (Eq. (3)) versus the quadrupole deformation firms (Eq. (2)). On the r.h. axis, ~eff (Eq. (4)) is given. Arrows indicate that in the corresponding isotopic chain the mass number increases (T) or decreases (J.) with increasing quadrupole deformation. Data are taken from Refs. [13,14] and from current issues of Nucl. Data Sheets. For 58,60.62Ni, the values of firms and ~eff are 0.183 and 23.9°, 0.207 and 35.5° as well as 0.202 and 38.6° , respectively. In Fig. 1, the values of (cos 36) (from now on we omit the index g.s.) for nuclei with available data in the range 4 6 < A <192 derived according to Eq. (3) are presented versus the symmetric quadrupole deformation firms (Eq. ( 2 ) ) . The asymmetry angle 8efe is indicated on the r.h.s. The data points of some nuclei constitute a separate track (framed) parallel to the main group. The tendency of the latter indicating a firms-Beff correlation was already discussed in Ref. [ 9 ]. The "new" track includes 60'62Ni, 7°-76Ge and 74-785e. Among them, positive quadrupole moments Q (2~-) were measured [ 14] for 6°'62Ni and 7°Ge resulting in values of 6eff well above 30 °. Of particular interest are the eccentricities e(flrms, Beef) (Eq. ( 7 ) ) displayed versus mass number A in Fig. 2. The overhelming majority of these values belongs to the surprisingly narrow interval 0.14< e <0.21 as already noticed in Ref. [9] for A > 92. Below that are only values for some "classical" axial rotors ( 1 5 0 < A <190) as well as for 138Ba. (Note that 6 = 30 ° may indicate triaxiality but also sphericity, cf. e.g. Ref. [ 10]. The eccentricity (Eq. ( 7 ) ) sharply distinguishes between them.) For several of the nuclei with eccentricities within the above interval, (soft) triaxiality has been considered in the literature. For instance, 98-1°4Ru (Z = 44) contribute to the small bump around A = 100 (Fig. 2).
W. Andrejtscheff, P. Petkov / Physics Letters B 329 (1994) 1-4
n'Ta'~Ge~'~4'~'r%e
0,3
0.0 . -0.1 40
60
80
I00
129
140
160
180
200
MASS NUMBER A Fig. 2. The eccentricity (Eq. (7)) derived from the data in Fig. I. In the insert, the region 68< A _<82 is expanded. For 7°Ge (Ref. [15]), CE data for TI/2(2~- ) are used producing 8eft = 35.8 ° and e = 0.20 rather than DSAM data (Self = 50°, e = 0.08) due to the better fitting into this systematics.
Investigations [ 16] with the general collective model (GCM) found for l°°-l°SRu triaxial minima in the potential-energy surfaces (PES). A striking feature of Fig. 2 are the eccentricities of 72-76Ge and 74-78Se (N = 40, 42, 44) lying appreciably higher than all other values. Apparently, in these isotopes the shapes of the ground states reveal the most pronounced asymmetry among the nearby seventy nuclei in this systematics arising from available data for 46<_ A _<192 (22< Z <76). It should be stressed that the present analysis based on third-order products of matrix elements (Eq. (3)) can not determine whether the triaxiality is soft (dynamic) or rigid (static). Triaxiality of the nuclei under investigation has been discussed in earlier studies. Hartree-Fock-Bogolyubov (HFB) calculations [4] predict dynarrfic triaxial deformation for 74Ge (single potential minimum at ~, ..~ 30 °) and other neighbours. PES derived [5] with the Woods-Saxon potential indicate 3~-softness in several even Se,Ge and Zn isotopes and triaxial deformation in 72'74'76Ge. After comparing experimental E2 transition matrix elements to predictions of several models, the structure of 72Ge is found [6] to be a triaxial (~,=28.5 °) quadrupole-deformed collective rotational pattern perturbed by an "intruder" 0 + state. Ratios of level energies and of B(E2) values in 72-78Ge were compared [7] to predictions of several geometric models. On this basis, it was suggested [7] that 74-78Ge show the characteristics of an asymmetric ro-
tor with "y~30°. In Table 1, we display an excerpt from such a comparison indicating that similar interpretation could also apply to 74-785e but not e.g. to 82Se. Indeed, the shape parameters of S°Se and especially of S2Se (Figs. 1, 2) clearly differ from those of 74-78Se (including the sign of the hexadecapole deformation [ 17] ). The inspection of Table 1 suggest somewhat higher values of ~, (closer to 30 °) in the Z = 32 (Ge) isotopes than in those with Z --- 74 (Se). This suggestion is convincingly confirmed (Fig. 1) by the derived values of Seff (28.50,26.40,29.0 ° for 72'74'76Ge and 22.6 °,24.4 °,26.4 ° for 74'76'788e, respectively). In this way, the high eccentricities of 72-76Ge and 74-78Se could be associated with a triaxiality in their ground states, i.e. with a shape corresponding to a single minimum in the PES at 3' ~ 25°. The origin of asymmetric shapes is associated with orbitals strongly coupled by the }~2 operator as outlined in Ch.5.3 of Ref. [ 18] for 24Mg. In a Nilsson scheme [ 19] or in a Woods-Saxon scheme [20] both generated after a minimization with respect to f14 such closely lying orbitals at f12 = 0.2+0.3 are 1/2-[310] and 3/2-[312] active for Z = 32,34 as well as 1/2 - [ 301 ] and 5 / 2 - [ 303 ] active for N = 40, 42, 44. The corresponding Y22 matrix elements are asymptotically allowed [21] and thus expected to be large. In principle, the observed phenomenon of high eccentricities might be due to an effective triaxiality arising from a prolate-oblate shape mixing (e.g. two minima in the PES at ~,=0° and ~,=60°, respectively, resulting in an effective value of e.g. ~, ~ 30°). Such a mechanism is invoked in the large-scale microscopic (projected HFB) calculations (real and complex VAMPIR family) performed for some of the nuclei studied (Ref. [22] and references therein). The restriction to axial symmetry is made thereby for numerical reasons. In the complex FED VAMPIR calculation, the main component in the ground state of 72Ge is spherical correlated by moderately deformed prolate and oblate determinants. Concerning the matrix elements of importance for the present work (i.e. those involved in Eq. (3)), the quadrupole moment Q(2 +) is approximately reproduced by the calculation [22] but the B(E2,2 + --~0+) value comes out by ..~ 40% too small. In conclusion, uniquely high eccentricities were found for the ground-states of 72-76Ge and 74-78Se among those of nearly seventy nuclei accessible for this study in the region 46< A <192. This feature
4
W. Andrejtscheff, P. Petkov/Physics Letters B 329 (1994) 1-4
Table 1 Comparison of B(E2) ratios to predictions of different geometrical models [7]. The experimental matrix elements used (cols. 7-11 ) are the same employed to derive the data in Figs. 1, 2. B(E2) ratio
Sym. rotor
Vibrator
Asym. rotor = 20 °
22---.2 21---,0:I
<< 1
2.0
0.38
22"21 22"-,01
1.43
cx~
5.33
p o i n t s at t h e m o s t p r o n o u n c e d ( e f f e c t i v e ) t r i a x i a l i t y e m e r g i n g f r o m t h e s e e x p e r i m e n t a l data.
This work has been supported by the Bulgarian National Research Foundation under contracts PH14 and PH31.
References [I] J.L. Wood et al., Phys. Rep. 215 (1992) 101. [2] J.H. Hamilton, in: Treatise on heavy-ion science, vol. 8, ed. D.A. Bromley (Plenum, New York, 1989) p. 3. [ 3 ] M. Vergnes, in: Proc. EPS Conf. Structure of Medium-Heavy Nuclei (Rhodes, 1979), eds. G.S. Anagnostatos et al., IOP Conf. ser. No. 49 (Bristol, 1980) p. 25. [4] M. Girod and B. Grammaticos, Phys. Rev. C 27 (1983) 2317. [51 R. Bengtsson, in: Proc. Intern. Workshop Nuclear Structure of the Zirconium Region (Bad Honnef, 1988 ), eds. J. Eberth et al. (Springer, Berlin, 1988) p. 17.
Z
y = 30 ° 1.43
[6] [7] [8] [9] [10] [11] [ 12] [13] [14] [15] [16] [17] [18] [19]
[20] [21 ] [22]
Neutron number N 40
42
44
46
48
32 34
2.8 1.6
1.4 1.0
1.4 1.0
1.1 0.8
0.3
32 34
526 69
60 34
44 31
37 14
3.7
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