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Effective dynamic moment of inertia of 118Xe and 130Ba
Volume 158B, number 6
PHYSICS LETTERS
5 September 1985
E F F E C T I V E D Y N A M I C M O M E N T O F I N E R T I A O F n S x e A N D 13°Ba H. E L - S A M M A N , V. B A R C I , A. G I Z O N , J. G I Z O N , g. K O S S A K O W S K I , Th. L I N D B L A D t Institut des Sciences Nucl~aires (IN2P3, USMG), 53 Avenue des Martyrs, 38026 Grenoble Cedex, France
and T. B E N G T S S O N Department of Mathematical Physics, Lund Institute of Technology, S-22100 Lund, Sweden
Received 29 April 1985
The effective moment of inertia Je(f2~of 118Xeand 13°Bahas been measured using sum-spectrometer techniques. The data are discussed using arguments based on particle alignments and results of cranking calculations, ~band r~2) and Jeff rt2) are compared in order to estimate the contribution of particle alignments to the total increase in angular momentum.
The possible methods which can be used to obtain information on the nuclear structure at high angular momentum, include the studies of "discrete" as well as "unresolved" 7-ray spectra. The first method deals with "fine" details and brings out very precise data on the nuclear level structure, etc. The second method is related to the continuum o f 7-rays and concerns essentially the gross properties of the nuclei and makes it possible to determine e.g. the moments of inertia. The present paper discusses those quantities. Two dynamic moments of inertia, 3"(2) = ~ d l / d w , describing the rate o f change of spin with the frequency can be defined: (i) a collective moment of inertia ~,.J(2).,d = ?t(dI/deO)band related to the bands generated by the collective nuclear motion (measured by "),-ray energy correlation techniques), (ii) an effective moment of inertia J(2)f = ~i(dI/dw)path connected to the decay path along the envelope of these bands. This moment o f inertia takes into account both the collective motion and the particle alignment, which shows up drastically at backbends. The present letter reports o n 4 2 ~ o f llSXe and 13°Ba which are transitional nuclei whose softness rei Permanent address: Research Institute of Physics, Stockholm, Sweden. 0370-2693/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
lative to 7-deformation is well established. We already published data [1] on their collective moment o f inertia J(~2an)d which, in addition, give us the possibility to estimate the alignment effects associated with band crossings [2]. Indeed, if Ai is the increase in angular momentum due to particle alignment only and A I the total increase, then A i / ~ j = 1 --J~2m)d/J(e2)f. Up - ' e o -r(2-) to now, only a f ew re suits a re av ai l aol , a ~ f f [3 - 5 ] and comparisons of both J(~2)md and• d~ff ,t2) are much more scarce. The information in the present work was obtained from 7-ray spectra o f a single detector in coincidence with a sum-spectrometer. This single detector was a 20.3 cm long NaI(T1) crystal having a hexagonal cross section with a 15.2 cm outer diameter. It was placed in the horizontal plane at 38 cm from the target and at 125 ° to the beam direction. It was shielded with lead and its entrance window strongly collimated. The energy resolution was 7.9% at the 662 keV 7-ray of 137Cs. The sum-spectrometer was made o f 12 hexagonal cross section NaI(T1) detectors arranged in a cylindrical geometry around the beam axis. A total 7-efficiency of 0.70 for the 60Co lines was achieved for this arrangement. The "r-ray multiplicity is calculated from the experimental fold-distribution o f the detectors fired. The final nuclei produced in the 459
Volume 158B, number 6
PHYSICS LETTERS
reactions were identified by means of a Ge(Li) diode in coincidence with the sum-spectrometer. About 50 X lo6 events were recorded for each target. The effective moments of inertia were extracted from the continuum y-ray spectra by using the method developed by the Berkeley group [3] which includes a feeding correction. Our data are not shown beyond the 20% feeding point. The experiments were performed at the Grenoble cyclotron by bombarding 4 mg/cm2 targets with 12C ions. The first case studied is the l12Sn + 12C reaction at 112 MeV. The y-ray multiplicity determined from the coincidence between the sum-spectrometer and the Ge(Li) detector shows that I18Xe has the highest angular momentum and the analysis of the singles spectra shows that it is produced in the largest amount. In the second case, we analyse 130Ba from the 122Sn t 12C reaction at 80 MeV. By the same method, one finds that the yrast cascade of this nucleus is the most strongly populated with the largest angular momentum. The determination of@; is valid if the nucleus has a collective behaviour, i.e., the continuum y-spectra we are dealing with are constituted of collective E2 transitions. For this reason we performed a complementary experiment to measure the a22 coefficient of the angular distribution of the continuum spectrum. It consisted in recording to response of 4 NaI crystals in coincidence with a sum-spectrometer made of 14 hexagonal detectors. The 4 additional crystals were placed in the horizontal plane at 90’ and backwards (135’, 163’ and 235”) relative to the beam direction and at 50 cm from the target. The spectra were sorted in coincidence with the same slices of the total y-ray spectrum as previously chosen for the 4:; analysis. In the case of the l12Sn t 12C reaction, the angular distribution of the unfolded spectra shows that the a22 coefficient is positive for Ao M 0.190.65 MeV. In the Aw = 0.30-0.45 MeV range, a22 reaches 0.30 which is typical of strongly aligned highspin states deexcited by stretched E2 transitions. Similar results are obtained for ljoBa in which we also find a quadrupole component starting at 0.27 MeV and extending up to a 0.70 MeV rotational frequency. The amplitude of a22 is similar to that found in l18Xe i.e., 0.30 + 0.05. Consequently, the continuum spectra of l18Xe and 130Ba exhibit clearly a component made of stretched E2 transitions which 460
5 September 1985
6C.l (MeV)
function of Aw for llsXe. Fig. 1. J(‘) moments of inertia The experimental (calculated) is shown by a heavy solid (d shed) line. The open circles correspond to experimental @kd v&es. extends approximately up to 0.65 and 0.70 MeV, respectively, establishing in this way the validity of the .J$i measurement. On the J$ curves, one observes bumps which are associated with intense y-lines between the lowest levels of the yrast-cascade and with an accumulation of y-rays due to particle alignment. For ll*Xe (fig. l), the bump at Ao = 0.39 MeV corresponds to the first backbending. Its energy matches perfectly with the one of a bridge found in the y-7 correlation matrix [ 11. Its origin is not yet established but the calculations made in a cranking model [6] indicate that crossings of h11i2 p roton and neutron orbitals should appear at nearby frequencies, the latter being probably a little lower. The bumps at 0.52 and 0.62 MeV fit also with bridges in the correlation matrix but there is no evidence for the nature of the particles which align their angular momentum. One should point out that the width of these bumps is energy resolution dependent and, in the case of a NaI detector, the .~_ poor resolution washes out a little the alignment effect-on the J$ curve. The peaks which show up at 0.27 and 0.34 MeV (fig. 2) correspond to the 4+-2+ and 6+-4+ y-
100
-Z a0 _:
60
"
LO
-k -i 20
‘?3a
,,I' ',/, ':YT--.-. D ~a '.~._~..._ 0 ..___,,--.._' _I PO CP
otale! 0
,‘., :.’ ,,
,,,I,
0.2 0.4 0.6 60 (b&V)
j
0.8
1.0
Fig. 2. Similar to fii. 1 but for 13’Ba.
Volume 158B, number 6
PHYSICS LETTERS
rays in the 130Ba yrast cascade. The bump at 0.40 MeV contains the two first band crossings of 130Ba which originate from h l l / 2 neutrons and h11/2 protons [7]. In the J(b~d data [1], a strong bridge appears very clearly at 0.54 MeV and it was assigned to an i13/2 neutron alignment. This effect is very likely contained in the broad bump around h e -~ 0.53 MeV on the 4 2) curve. The amplitude and variation ofj(.2)~ resemble those of 130Ce [5]. As a comparison, we have calculated,/(2) for both nuclei as described in ref. [8], where the individual E2 transitions within a cascade are "smeared" around the discrete transition energy. A smearing width of A ( h e ) = 0.05 MeV has been used. The contributions from all such smeared transitions are then summed to form j(2) eff" For the high-spin cascades, we have used the four (lr, a) bands calculated as in ref. [9]. This means that the (n, a) bands are calculated without pairing but with energy minimization with respect to e, 7, e4 deformations for each spin state. For ll8Xe, our previous studies [1] of42)a~d indicate that the low observed values are due to moderately collective rotation with e "~ 0.25,7 = 15 ° - 3 0 °. In the yrast cascades that are used to calculate J(e2~in fig. 1, it is only below h e = 0.4-0.5 MeV that such states contribute. At higher spins, all cascades attain a particle-hole structure (7 = 60°), which distribute the spin contribution in a large frequency interval and thereby reduce 4 2). If the triaxiality continued to 2) w ould be larger. higher spins in most cascades,gff In the low-spin region, the pairing reduces the observed values. Thus, it is only in the region/~e = 0 . 3 0.5 MeV that the observed and calculated values are similar. The cascades of 130Ba have been calculated for a predominantly prolate shape. This means that the cascade will go through the strongly deformed minimum at e "~ 0.35 as was interpreted by our 42~) d data on 128,130Ba [ 1]. In this case, the experimental amplitude of 42~ is nearly reproduced by the calculations. The big peak at h e = 0.5 MeV is created by deformation changes, mainly 7 which goes from ~ 0 ° for high frequencies to ~ - 3 0 ° at low frequencies. The data at high frequency are again much higher than the calculated values. In summary, the calculated values of 42)f have a similar magnitude as the observed ones for an intermediate spin range only. For low spins, the calculated
5 September 1985
values are too large, whereas for high spins, they are too small. These deviations can be seen to be related as we expect that the effective kinematic moment of inertia, J(e~~, should approach ]rigid at high spins. As ](2) = d(wj(1))/de, we find t,O
](1) = 1 e
f ]¢:)(e')de' +raCe0)/e
(1)
co0
where I ( e 0 ) is a constant of integration. By applying eq. (1), we find that both the calculated and experimental J(el)fvalues do approach drilglCidat high frequencies. But the calculated values of j(2) at low frequencies are rather large so that J(e~ is larger than ]rigid at h e = 0.4 MeV. This has to be compensated, thus J(e2~has to decrease radically below ]ri id at higher frequencies. This means that most o~the aligned angular momenta is brought into the system at low frequency and just a little is left to align at higher spins. The experimental data, on the other hand, are lower than ]rigid below h e -~ 0.3 MeV and this allows them to increase all over the observed region. It thus seems that the experimental data reflect a shift to higher spins of the alignments that are calculated to occur at low spins. Such a shift is a natural consequence of pairing, but a larger single-particle energy gap would also give this effect. In fact, from the 42a~)d studies [ I ] we do expect a larger singleparticle gap for the deduced [ 1] deformations of both llSXe and 130Ba, since the high-spin states belonging to these deformations never come out yrast with the present parameter set [9]. It is also interesting to note that the observed J ( ~ d values, which should be affected by pairing, can be reproduced rather closely by these unpaired calculations. The J(e2~values, on the other hand, indicate that pairing could be at least partly responsible for the large values at high spins. As mentioned in the introduction, we can determine the contribution of particle alignments Ai to the total change in spin A/since we measured both 42a~)d and fie2f~.The variations of the Ai/AI ratio are very similar in 118Xe and 130Ba. This ratio reaches 0.40 around h e ~ 0.40 MeV at the first band crossing. After a clecrease down to 0.30 near h e ~. 0.45 MeV, Ai/AI increases again to 0.50 for i~e ~ 0.52 MeV where alignments take place. Then, Ai/AI~ 0.50 for frequencies up to h e ~ 0.70 MeV. This 461
Volume 158B, number 6
PHYSICS LETTERS
5 September 1985
40
Z-
118Xe
2v
...... "" / //" ~,
'30 :D ,
F
"
I
t
f
130Ba
! " ~ =0.2~
~.30
£=0
Z tu
/y/
/~r/a--
///,~
' £ =0.35 '
~=0
<
~1o
~io
Z
0 Z
o
o
I
02
I 0.4
I
06
I
08
1.0
means that the alignments contribute for about 50% to the total increase in angular momentum. Therefore, particle alignment can be considered as one of the main phenomena which influences the behaviour of ll8Xe and 130Ba at high rotational frequency. Some additional comments can be made concerning the analysis of experimental and theoretical data. From the kinematic moment of inertia J (1) = hi/60, it is possible to define an effective angular momentum ]hreff(60 ) = W41}(60) where 4 ~ i s obtained from eq. (1). Ieff can thus be deduced by integrating the 4~(60) curve. The representation of an experimental effective angular momentum ~ P as a function of 6o can be considered as the analog of plots made for discrete "plines. One sees in figs. 3 and 4 that the ~ P curves fit remarkably well with the five first known levels of ll8Xe and 130Ba when taking, in eq. (1), a constant ~t/(600) equal to 3h and 2h, respectively. These constants compensate in fact the low-frequency part of the 42~ curve eliminated by electronic threshold set on the detector response. The agreement mentioned above comfort us in the degree of confidence we have about the properties of llSXe and 130Ba deduced at high frequency. ~ f P is 5 - 1 0 h units smaller than the theoretical effective angular momentum l~pf and barely reaches the spherical rigid-rotor value at the highest frequencies. Up to now, it is difficult to explain the difference between the experimental and theoretical curves but the absence of pairing could 462
./
0/
~
0
0.2
!
I
i
0.4
0.6
0.8
h co
1~C0 ( M e V )
Fig. 3. Angular momentum I in tlaXe as a function of hto. The solid (dotted) line is dec~qced from the integration of the experimental (theoretical)Jet,f; the crosses are for angular momentum from discrete v-lines in the experimental level scheme. The thin solid lines correspond to rigid rotors.
..... ~:;"
10
( MeV )
Fig. 4. Same as for fig. 3,but here the nucleus is lS°Ba.
perhaps account for at least one part of it. In conclusion, we have measured the effective dynamic moment of inertia of two transitional nuclei and discussed the results in the frame of a cranking model. The alignment of particles and shape-changes ~,~2)f at hi gh frequency. This could explain the rise OZaef is in agreement with the amplitude of the Ai/A1 ratio deduced from our 42) d and 42f~ values which shows that alignments participate for about 50% to the total increase in angular momentum. We thank Mr. G. Margotton for technical assistance and the cyclotron crew for providing the beams. The experiments have been possible only thanks to a loan of detectors. We thank Dr. C. Detraz, Director of GANIL, who gave us the possibility to use the NaI detectors of his laboratory. This work was supported in part by Centre National de la Recherche Scientifique and by the Swedish National Science Research Council under contract U-FR 3219-114.
References [1] H. EI-Samman et al., Nucl. Phys. A427 (1984) 397. [2] A. Bohr and B. Mottelson, Phys. Scripta 24 (1981) 71. [3] M.A. Deleplanque et al., Phys. Rev. Lett. 50 (1983) 409. [4] A.O. Macchiavelli et al., Nucl. Phys., to be published. [5 ] D. Jerrestam et al., Phys. Ser., to be published. [6] R. Bengtsson and S. Frauendorf, Nucl. Phys. A327
(1979) 139. [7] Sun Xiangfu et al., Phys. Rev. C28 (1983) 1167. [8] T. Bengtsson, Phys. Lett. 126B (1983) 411. [9] T. Bengtsson and I. Ragnarsson, Nucl. Phys. A436 ((1985) 14.
PHYSICS LETTERS
5 September 1985
E F F E C T I V E D Y N A M I C M O M E N T O F I N E R T I A O F n S x e A N D 13°Ba H. E L - S A M M A N , V. B A R C I , A. G I Z O N , J. G I Z O N , g. K O S S A K O W S K I , Th. L I N D B L A D t Institut des Sciences Nucl~aires (IN2P3, USMG), 53 Avenue des Martyrs, 38026 Grenoble Cedex, France
and T. B E N G T S S O N Department of Mathematical Physics, Lund Institute of Technology, S-22100 Lund, Sweden
Received 29 April 1985
The effective moment of inertia Je(f2~of 118Xeand 13°Bahas been measured using sum-spectrometer techniques. The data are discussed using arguments based on particle alignments and results of cranking calculations, ~band r~2) and Jeff rt2) are compared in order to estimate the contribution of particle alignments to the total increase in angular momentum.
The possible methods which can be used to obtain information on the nuclear structure at high angular momentum, include the studies of "discrete" as well as "unresolved" 7-ray spectra. The first method deals with "fine" details and brings out very precise data on the nuclear level structure, etc. The second method is related to the continuum o f 7-rays and concerns essentially the gross properties of the nuclei and makes it possible to determine e.g. the moments of inertia. The present paper discusses those quantities. Two dynamic moments of inertia, 3"(2) = ~ d l / d w , describing the rate o f change of spin with the frequency can be defined: (i) a collective moment of inertia ~,.J(2).,d = ?t(dI/deO)band related to the bands generated by the collective nuclear motion (measured by "),-ray energy correlation techniques), (ii) an effective moment of inertia J(2)f = ~i(dI/dw)path connected to the decay path along the envelope of these bands. This moment o f inertia takes into account both the collective motion and the particle alignment, which shows up drastically at backbends. The present letter reports o n 4 2 ~ o f llSXe and 13°Ba which are transitional nuclei whose softness rei Permanent address: Research Institute of Physics, Stockholm, Sweden. 0370-2693/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
lative to 7-deformation is well established. We already published data [1] on their collective moment o f inertia J(~2an)d which, in addition, give us the possibility to estimate the alignment effects associated with band crossings [2]. Indeed, if Ai is the increase in angular momentum due to particle alignment only and A I the total increase, then A i / ~ j = 1 --J~2m)d/J(e2)f. Up - ' e o -r(2-) to now, only a f ew re suits a re av ai l aol , a ~ f f [3 - 5 ] and comparisons of both J(~2)md and• d~ff ,t2) are much more scarce. The information in the present work was obtained from 7-ray spectra o f a single detector in coincidence with a sum-spectrometer. This single detector was a 20.3 cm long NaI(T1) crystal having a hexagonal cross section with a 15.2 cm outer diameter. It was placed in the horizontal plane at 38 cm from the target and at 125 ° to the beam direction. It was shielded with lead and its entrance window strongly collimated. The energy resolution was 7.9% at the 662 keV 7-ray of 137Cs. The sum-spectrometer was made o f 12 hexagonal cross section NaI(T1) detectors arranged in a cylindrical geometry around the beam axis. A total 7-efficiency of 0.70 for the 60Co lines was achieved for this arrangement. The "r-ray multiplicity is calculated from the experimental fold-distribution o f the detectors fired. The final nuclei produced in the 459
Volume 158B, number 6
PHYSICS LETTERS
reactions were identified by means of a Ge(Li) diode in coincidence with the sum-spectrometer. About 50 X lo6 events were recorded for each target. The effective moments of inertia were extracted from the continuum y-ray spectra by using the method developed by the Berkeley group [3] which includes a feeding correction. Our data are not shown beyond the 20% feeding point. The experiments were performed at the Grenoble cyclotron by bombarding 4 mg/cm2 targets with 12C ions. The first case studied is the l12Sn + 12C reaction at 112 MeV. The y-ray multiplicity determined from the coincidence between the sum-spectrometer and the Ge(Li) detector shows that I18Xe has the highest angular momentum and the analysis of the singles spectra shows that it is produced in the largest amount. In the second case, we analyse 130Ba from the 122Sn t 12C reaction at 80 MeV. By the same method, one finds that the yrast cascade of this nucleus is the most strongly populated with the largest angular momentum. The determination of@; is valid if the nucleus has a collective behaviour, i.e., the continuum y-spectra we are dealing with are constituted of collective E2 transitions. For this reason we performed a complementary experiment to measure the a22 coefficient of the angular distribution of the continuum spectrum. It consisted in recording to response of 4 NaI crystals in coincidence with a sum-spectrometer made of 14 hexagonal detectors. The 4 additional crystals were placed in the horizontal plane at 90’ and backwards (135’, 163’ and 235”) relative to the beam direction and at 50 cm from the target. The spectra were sorted in coincidence with the same slices of the total y-ray spectrum as previously chosen for the 4:; analysis. In the case of the l12Sn t 12C reaction, the angular distribution of the unfolded spectra shows that the a22 coefficient is positive for Ao M 0.190.65 MeV. In the Aw = 0.30-0.45 MeV range, a22 reaches 0.30 which is typical of strongly aligned highspin states deexcited by stretched E2 transitions. Similar results are obtained for ljoBa in which we also find a quadrupole component starting at 0.27 MeV and extending up to a 0.70 MeV rotational frequency. The amplitude of a22 is similar to that found in l18Xe i.e., 0.30 + 0.05. Consequently, the continuum spectra of l18Xe and 130Ba exhibit clearly a component made of stretched E2 transitions which 460
5 September 1985
6C.l (MeV)
function of Aw for llsXe. Fig. 1. J(‘) moments of inertia The experimental (calculated) is shown by a heavy solid (d shed) line. The open circles correspond to experimental @kd v&es. extends approximately up to 0.65 and 0.70 MeV, respectively, establishing in this way the validity of the .J$i measurement. On the J$ curves, one observes bumps which are associated with intense y-lines between the lowest levels of the yrast-cascade and with an accumulation of y-rays due to particle alignment. For ll*Xe (fig. l), the bump at Ao = 0.39 MeV corresponds to the first backbending. Its energy matches perfectly with the one of a bridge found in the y-7 correlation matrix [ 11. Its origin is not yet established but the calculations made in a cranking model [6] indicate that crossings of h11i2 p roton and neutron orbitals should appear at nearby frequencies, the latter being probably a little lower. The bumps at 0.52 and 0.62 MeV fit also with bridges in the correlation matrix but there is no evidence for the nature of the particles which align their angular momentum. One should point out that the width of these bumps is energy resolution dependent and, in the case of a NaI detector, the .~_ poor resolution washes out a little the alignment effect-on the J$ curve. The peaks which show up at 0.27 and 0.34 MeV (fig. 2) correspond to the 4+-2+ and 6+-4+ y-
100
-Z a0 _:
60
"
LO
-k -i 20
‘?3a
,,I' ',/, ':YT--.-. D ~a '.~._~..._ 0 ..___,,--.._' _I PO CP
otale! 0
,‘., :.’ ,,
,,,I,
0.2 0.4 0.6 60 (b&V)
j
0.8
1.0
Fig. 2. Similar to fii. 1 but for 13’Ba.
Volume 158B, number 6
PHYSICS LETTERS
rays in the 130Ba yrast cascade. The bump at 0.40 MeV contains the two first band crossings of 130Ba which originate from h l l / 2 neutrons and h11/2 protons [7]. In the J(b~d data [1], a strong bridge appears very clearly at 0.54 MeV and it was assigned to an i13/2 neutron alignment. This effect is very likely contained in the broad bump around h e -~ 0.53 MeV on the 4 2) curve. The amplitude and variation ofj(.2)~ resemble those of 130Ce [5]. As a comparison, we have calculated,/(2) for both nuclei as described in ref. [8], where the individual E2 transitions within a cascade are "smeared" around the discrete transition energy. A smearing width of A ( h e ) = 0.05 MeV has been used. The contributions from all such smeared transitions are then summed to form j(2) eff" For the high-spin cascades, we have used the four (lr, a) bands calculated as in ref. [9]. This means that the (n, a) bands are calculated without pairing but with energy minimization with respect to e, 7, e4 deformations for each spin state. For ll8Xe, our previous studies [1] of42)a~d indicate that the low observed values are due to moderately collective rotation with e "~ 0.25,7 = 15 ° - 3 0 °. In the yrast cascades that are used to calculate J(e2~in fig. 1, it is only below h e = 0.4-0.5 MeV that such states contribute. At higher spins, all cascades attain a particle-hole structure (7 = 60°), which distribute the spin contribution in a large frequency interval and thereby reduce 4 2). If the triaxiality continued to 2) w ould be larger. higher spins in most cascades,gff In the low-spin region, the pairing reduces the observed values. Thus, it is only in the region/~e = 0 . 3 0.5 MeV that the observed and calculated values are similar. The cascades of 130Ba have been calculated for a predominantly prolate shape. This means that the cascade will go through the strongly deformed minimum at e "~ 0.35 as was interpreted by our 42~) d data on 128,130Ba [ 1]. In this case, the experimental amplitude of 42~ is nearly reproduced by the calculations. The big peak at h e = 0.5 MeV is created by deformation changes, mainly 7 which goes from ~ 0 ° for high frequencies to ~ - 3 0 ° at low frequencies. The data at high frequency are again much higher than the calculated values. In summary, the calculated values of 42)f have a similar magnitude as the observed ones for an intermediate spin range only. For low spins, the calculated
5 September 1985
values are too large, whereas for high spins, they are too small. These deviations can be seen to be related as we expect that the effective kinematic moment of inertia, J(e~~, should approach ]rigid at high spins. As ](2) = d(wj(1))/de, we find t,O
](1) = 1 e
f ]¢:)(e')de' +raCe0)/e
(1)
co0
where I ( e 0 ) is a constant of integration. By applying eq. (1), we find that both the calculated and experimental J(el)fvalues do approach drilglCidat high frequencies. But the calculated values of j(2) at low frequencies are rather large so that J(e~ is larger than ]rigid at h e = 0.4 MeV. This has to be compensated, thus J(e2~has to decrease radically below ]ri id at higher frequencies. This means that most o~the aligned angular momenta is brought into the system at low frequency and just a little is left to align at higher spins. The experimental data, on the other hand, are lower than ]rigid below h e -~ 0.3 MeV and this allows them to increase all over the observed region. It thus seems that the experimental data reflect a shift to higher spins of the alignments that are calculated to occur at low spins. Such a shift is a natural consequence of pairing, but a larger single-particle energy gap would also give this effect. In fact, from the 42a~)d studies [ I ] we do expect a larger singleparticle gap for the deduced [ 1] deformations of both llSXe and 130Ba, since the high-spin states belonging to these deformations never come out yrast with the present parameter set [9]. It is also interesting to note that the observed J ( ~ d values, which should be affected by pairing, can be reproduced rather closely by these unpaired calculations. The J(e2~values, on the other hand, indicate that pairing could be at least partly responsible for the large values at high spins. As mentioned in the introduction, we can determine the contribution of particle alignments Ai to the total change in spin A/since we measured both 42a~)d and fie2f~.The variations of the Ai/AI ratio are very similar in 118Xe and 130Ba. This ratio reaches 0.40 around h e ~ 0.40 MeV at the first band crossing. After a clecrease down to 0.30 near h e ~. 0.45 MeV, Ai/AI increases again to 0.50 for i~e ~ 0.52 MeV where alignments take place. Then, Ai/AI~ 0.50 for frequencies up to h e ~ 0.70 MeV. This 461
Volume 158B, number 6
PHYSICS LETTERS
5 September 1985
40
Z-
118Xe
2v
...... "" / //" ~,
'30 :D ,
F
"
I
t
f
130Ba
! " ~ =0.2~
~.30
£=0
Z tu
/y/
/~r/a--
///,~
' £ =0.35 '
~=0
<
~1o
~io
Z
0 Z
o
o
I
02
I 0.4
I
06
I
08
1.0
means that the alignments contribute for about 50% to the total increase in angular momentum. Therefore, particle alignment can be considered as one of the main phenomena which influences the behaviour of ll8Xe and 130Ba at high rotational frequency. Some additional comments can be made concerning the analysis of experimental and theoretical data. From the kinematic moment of inertia J (1) = hi/60, it is possible to define an effective angular momentum ]hreff(60 ) = W41}(60) where 4 ~ i s obtained from eq. (1). Ieff can thus be deduced by integrating the 4~(60) curve. The representation of an experimental effective angular momentum ~ P as a function of 6o can be considered as the analog of plots made for discrete "plines. One sees in figs. 3 and 4 that the ~ P curves fit remarkably well with the five first known levels of ll8Xe and 130Ba when taking, in eq. (1), a constant ~t/(600) equal to 3h and 2h, respectively. These constants compensate in fact the low-frequency part of the 42~ curve eliminated by electronic threshold set on the detector response. The agreement mentioned above comfort us in the degree of confidence we have about the properties of llSXe and 130Ba deduced at high frequency. ~ f P is 5 - 1 0 h units smaller than the theoretical effective angular momentum l~pf and barely reaches the spherical rigid-rotor value at the highest frequencies. Up to now, it is difficult to explain the difference between the experimental and theoretical curves but the absence of pairing could 462
./
0/
~
0
0.2
!
I
i
0.4
0.6
0.8
h co
1~C0 ( M e V )
Fig. 3. Angular momentum I in tlaXe as a function of hto. The solid (dotted) line is dec~qced from the integration of the experimental (theoretical)Jet,f; the crosses are for angular momentum from discrete v-lines in the experimental level scheme. The thin solid lines correspond to rigid rotors.
..... ~:;"
10
( MeV )
Fig. 4. Same as for fig. 3,but here the nucleus is lS°Ba.
perhaps account for at least one part of it. In conclusion, we have measured the effective dynamic moment of inertia of two transitional nuclei and discussed the results in the frame of a cranking model. The alignment of particles and shape-changes ~,~2)f at hi gh frequency. This could explain the rise OZaef is in agreement with the amplitude of the Ai/A1 ratio deduced from our 42) d and 42f~ values which shows that alignments participate for about 50% to the total increase in angular momentum. We thank Mr. G. Margotton for technical assistance and the cyclotron crew for providing the beams. The experiments have been possible only thanks to a loan of detectors. We thank Dr. C. Detraz, Director of GANIL, who gave us the possibility to use the NaI detectors of his laboratory. This work was supported in part by Centre National de la Recherche Scientifique and by the Swedish National Science Research Council under contract U-FR 3219-114.
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