institute of advanced studies
TRANSCRIPT
THE AUSTRALIAN
NATIONAL* UNIVERSITY
RESEARCH SCHOOL OF PHYSICAL SCIENCES
ANU-P/948 April 1986
ELECTRIC QUADRUPOLE MOMENTS OF THE FIRST EXCITED STATES OF 1 9 i +Pt, 2 9 6Pt AND X 9 8Pt
G.J. GYAPONG, R.H. SPEAR, M.T. ESAT and M.P. FEWELL Department of Nuclear Physics, Australian National University,
Canberra, ACT 2600, Australia and
A.M. BAXTER and S.M. BURNETT Department of Physics and Theoretical Physics, Faculty of Science
Australian National University.
INSTITUTE OF ADVANCED STUDIES
ANU-P/948 A p r i l 1986
ELECTRIC QUADRUPOLE MOMENTS OF THE FIRST EXCITED STATES OF 1 9 4 P t , 1 9 6 P t AND 1 9 8 P t
G. J. GYAPONG, R M. SPEAR, M.T. ESAT and M.P. FEWELL
Department of Nuclear Physics, Australian National University, Canberra, ACT 2600, Australia.
and
A.M. BAXTER and S.M. BURNETT
Department of Physics and Theoretical Physics, Faculty of Science, Australian National University, Canberra, ACT 2600, Australia.
Abstract: Coulomb excitation of iHi96,i98Pt by «He, 1 2 C and 1 6 0 projectiles has been used to determine the static electric quadrupole moments Q(2 t
+) of the first excited states of 1 9 4« 1 9 6« 1 9 8 Pt, together with values of BfpliOf^f). It is clearly established that Q(2 t+) is positive for each nucleus, having values of 0.48(14)eb and 0.66(12)eb for 1 9 4 Pt and 1 9 6 Pt, respectively, and 0.42(12)eb or 0.54(12)eb for 1 9 8 Pt, depending on whether the interference term P 4 (2 2
+ ) is positive or negative. Results obtained for B ( E 2 ; 0 J + - > 2 J + ) are 1.661(1 l)e 2b 2 , 1.382(6)e2b2 and 1.090(7)e2b2 for iHi96,i98P t > respectively. The results are compared with the predictions of various nuclear models.
NUCLEAR REACTIONS 1 9 4 ' 1 9 6 - 1 9 8 Pt(a ,a ' ) , E =» 14.0 - 15.6 MeV, 9 = 174.8°; 1 9 6 P t ( a , a ' ) , E = 16.8 - 18.6 MeV, 9 = 90.0°; i94,i96,i98p t (i2C fi2C ) > E = 4 1 0 . 45.0 MeV, 8 = 174.8°; I94, i96 ( i98p t ( i6 0 i6 0 . ) t E = 5 5 0 . 6 3 0 M e V ) e = i74.g=; measured Coulomb excitation probabilities of first 2 +states. iM,i96,i98pt deduced B(E2;01
+->21
+) and Q(2j+). Enriched targets.
Accepted for publication in Nuclear Physics A
2
1. Introduction
The so-called "transition region" near A=190 has, for more than a decade, been a
remarkably active area for nuclear spectroscopic investigations. Theoretical and
experimental studies of nuclei in this region have produced major advances in our
understanding of nuclear strucrore.
In 1968, Kumar and Baranger calculated1) that the nuclear shape changes from
prolate to oblate in the A = 186-192 region. Their microscopic calculations involved
the application of the pairing-plus-quadrupole model to Bohr's collective Hamiltonian.
In particular, they predicted that the static electric quadrupole moment Q(2 1
+) of the
first excited state of even-even nuclei should change from negative (prolate charge
distribution) to positive (oblate) in proceeding from the osmium isotopes to the
platinum isotopes, and that it should remain positive for the mercury isotopes. They
calculated2) that me prolate-oblate shape transition should occur at A = 192.
Within a few years these predictions were experimentally verified by Saladin and
his collaborators at Pittsburgh. By studying Coulomb excitation with various
projectiles they found that Q(2j +) is clearly negative for 184.186,188,19O,1920S ^ . e f s
3- 4) and positive for i ' 4.i 96. 1 98p t ( r ef s 5,6) Subsequently, Cline and collaborators at
Rochester reported7) substantial confirmation of the Pittsburgh results; however, a
detailed presentation of the Rochester data has not been published. In addition,
Coulomb excitation measurements at Canberra8'1 0) and at Koln11) have shown that
Q ( V ) is positive for »W«>.MV04Hg>
The discovery of the A = 192 prolate-to-oblate transition has triggered a very
large amount of experimental work on the spectroscopy of the Os, Pt and Hg isotopes.
In parallel with this, an equally large amount of related theoretical work has been
published. For example, the development of the interacting boson model (IBM)
[ref. 1 2 )] was greatly stimulated by the suggestion13) that the properties of the
low-lying levels of 1 9 6 Pt are in remarkably good agreement with those predicted by
the 0(6) limit of the IBM, and in this context the transition region has been
3
interpreted14) in terms of a progression from the 0(6) ["/-unstable15)] limit for the
heavier Pt isotopes toward the SU(3) limit (axially symmetric rotor) for the lighter Os
isotopes. Other approaches to understanding the tr?nsition nuclei include
interpretations in terms of boson expansion theory 1 6 , 1 7), the rigid asymmetric-rotor
model 1 8 , 1 9), rotation-vibration models, both symmetric20) and asymmetric21), and
other more complex geometric models [e.g. refs 2 2" 2 4)].
In all of these considerations, the value of Q(2j+) for the even-even nuclei
involved plays a crucial role. For example, Q(2j+) = 0 in the strict 0(6) limit of the
IBM. Therefore, the reported positive value of Q(2j+) for 1 9 6 Pt [refs.5"7)] raises
complications25) for the suggestion that this nucleus is an 0(6) paradigm. However, a
recent redetermination of Q(2j+) for 1 9 4 Pt by the Pittsburgh group26) has produced a
value consistent with zero (0.13 ± 0.17 eb), in contrast to their earlier value of 0.64 ±
0.16 eb. They attribute the discrepancy to difficulties associated with impurity
subtractions owing to the lower isotopic enrichments of the targets used in the earlier
work. Their published target compositions6) suggest that those problems could haw,
been at least as great for 1 9 6 Pt and 1 9 8 Pt, which raises doubts about their results for
those nuclei, and, incidentally, about the nature of the prolate-oblate transition.
Furthermore, there have been some difficulties26) in reconciling values of Q(2j+)
obtained for some of the Cs and Pt isotopes using muonic X-ray techniques27*29) with
those obtained from Coulomb-excitation data. It is therefore highly desirable to make
new and independent measurements of Q(2j+) for the Pt isotopes. The present paper
presents the results of such measurements for 1 9 4 Pt, 1 9 6 Pt and 1 9 8 Pt. A brief report
of the results for 1 9 6 Pt has already been published30).
2. Experimental Procedure
The use of Coulomb excitation to determine nuclear properties such as Q(2j+)
and the reduced transition probability B(E2;0 1
+-»2 1
+) has been comprehensively
described in several review articles3 1'3 3). The basic experimental procedures used in
4
the present work have been described in previous publications from this laboratory
[e.g. refs. 3 4 " 3 7 ) ) .
Although two independent measurements of the excitation probability of the 2 j +
state usually suffice to determine Q(2j+) and B(E2;0 1
+->2 1
+), we have, in order to
obtain substantial redundancy and increased confidence in the results, made
measurements under four different experimental arrangements for 1 9 o P t
(backscattering of 4He, 1 2 C and 1 6 0 and scattering of 4He at 90°) and three different
arrangements for 1 5 4 Pt and 1 9 8 Pt (backscattering of 4He, 1 2 C and 1 6 0) .
Charged-particle beams were obtained from the ANU MUD pelletron accelerator,
the beam energy having been previously calibrated38) to better than 0.1%. Targets,
which consisted of isotopically enriched Pt metal evaporated onto thin carbon foils,
had thicknesses in the range 2-15 (ig cm"2. Isotopic compositions of the enriched Pt
used are listed in table 1. Backscattered particles were detected with an annular silicon
surface-barrier detector, the mean laboratory scattering angle being 174.8 ± 0.2°.
Particles scattered at 90° were analysed using an Enge split-pole magnetic spectrometer
with a position-sensitive multi-electrode proprotional counter at its focal plane3 9).
Since the Coulomb-excitation probability varies rapidly with angle at 90°, the
scattering angle must be measured accurately. This was done using the kinematic
technique described by Kuehner et al.4 0) and resulted in a value of 90.0 ± 0.1°.
3. Analysis and Results
3.1 SPECTRUM ANALYSIS
The experimentally determined Coulomb-excitation probability P M O of the first V~
state is defined as
Thus for each spectrum obtained, the number of counts in each of the "elastic" (0j+)
and "inelastic" (2,+) peaks must be determined. Representative spectra are shown in
5
figs. 1 and 2. As is normal in this type of experiment, the 2 j + peak sits on a tail
extending down in energy from the much larger elastic peak. Peak areas were
extracted using well-established procedures34,35*37), with the modification that x 2 was
replaced by the log-likelihood function x 2i p of ref.40) as a measure of goodness of
fit. The elastic-scattering peak was fitted with a lineshape consisting of a skewed
gaussian plus one or more exponential functions tc represent the low-energy tail of the
peak. This lineshape was used to estimate the magnitude of the elastic-peak tail
underneath the inelastic peak. When analysing spectra from the Enge spectrometer,
allowance was made for the variation of peak shape along the focal plane. Small
contributions to the spectra from Pt isotopes other than the one of primary interest
("isotopic impurities") were accounted for using the supplier's assay (table 1) and
B(E2) values from the literature 2 5 , 4 2 , 4 3); these contributions are shown by the broken
curves in figs. 1 and 2.
Values obtained for P are listed in table 2. The bombarding energies (E) given
are the values obtained after applying small corrections (ranging from 1 to 15 keV) for
the effects of finite target thickness. Target thicknesses were determined from
Rutherford-scattering measurements. The experimental uncertainties assigned to
values of P arise from statistical uncertainties, and from uncertainties involved in
estimating the background beneath the 2 j + peak and in correcting for isotopic
impurities.
3.2 INVESTIGATION OF TARGET CONTAMINANTS
Elastic scattering from target contaminants other than Pt isotopes could distort the
results by contributing to the spectrum in the region of the Pt inelastic peaks. For each
experimental configuration, the masses of possible interfering nuclei were calculated,
and spectra obtained at low bombarding energies were examined to set upper limits on
the contributions of these contaminant nuclei to the measured excitation probabilities.
For example, in the case of i 2 C on 1 9 6Pt, the offending contaminants would range in
mass r.umber from 185 to 191, and spectra obtained with 18-MeV 1 6 0 ions were used
to set an upper limit of 0.5% on their contribution to P (at the 95% confidence
level;. Other limits ranged from 0.1% for 4Heon 1 9 4 Pt at 174.8° andfor 4Heon 1 9 6 Pt at 90°, to 0. /% for 1 6 0 on 1 9 6 Pt at 174.o°. The small surplus of counts in the
region between the 2 j + and 1^ peaks in fig. 1(a) is attributed to elastic scattering
from isotopes of Ba; their effect on the determination of P is negligible.
3.3 DETERMINATION OF SAFE BOMBARDING ENERGIES
It is essential for the valid application of Coulomb-excitation theory that the data
analysed should be obtained at bombarding energies sufficiently low for
Coulomb-nuclear interference to be negligible44). In the present work the maximum
safe bombarding energy was determined for each experimental configuration by
plotting the ratio P e x p/Pc o ui ^ a f u n c t i o n of s, the distance of closest approach of the
nuclear surfaces, defined by the expression
s(6 ) = c m .
0.72 Z iZ 2 An 1 + —
A 2
J
[ l+cosec(*9 )] -l.25(.kx
xl* + A 2
1 / 3 ) f m , c m .
where Zj,Aj and Z j ^ are the atomic numbers and masses of projectile and target,
respectively, 9 C m is the scattering angle in the centre-of-mass system, E is the
laboratory bombarding energy in MeV, and the nuclear radius is taken to be 1.25 A 1 / 3
fm. The quantity P C o u l is the excitation probability calculated, assuming a pure
Cculomb interaction, with the de Boer-Winther multiple Coulomb-excit?tion code45).
The resulting plots are shown in figs. 3-5. In calculating P C o u , it was assumed
that Q(2j+) and B(E2;0j f -»2 1
+ ) have the values obtained in the present work. For
each experimental configuration, the deviation of Pexp/Pcoui ^ r o m a constant value
(unity for the particular nuclear parameters used in calculating P C o u l ) indicates the
onset of Coulomb-nuclear interference as the bombarding energy is increased. The
maximum bombarding energies deemed to be safe are indicated by arrows in figs.3-5.
Data obtained at energies greater than the maximum safe energy were not used in the
determination of Q(2j+) and B(E2;01
+-»21
+).
7
3.4 MATRIX ELEMENTS USED IN ANALYSIS
Energy levels included in the multiple-Coulomb-excitation analysis are identified
in table 3. Other levels were found, using available experimental information, to make
negligible contribution (<0.1%) to the Coulomb excitation of the 2 j + state. Values
adopted from the literature for the relevant matrix elements are listed in table 4.
It is usually found 3 1 , 3 2) in the analysis of reorientation-effect experiments for the
determination of Q(2j+) that interference effects from higher states are dominated by
the contribution of the 2 2
+ state, which depends on the product P 3 (2 2
+ ) =
(Of I M(E2) I 2 2
+> {If1 M(E2) I 2 2
+> ( 0 ^ I M(E2) 12^). Since the signs of the
matrix elements are usually not known, an ambiguity arises in the value of the
extracted quadrupole moment In this context it is convenient to consider the product
P 4 (2 2
+ ) = <2!+1 M(E2) 121+) P 3 (2 2
+ ) , the sign of which is independent of the phase
convention used in the definition of the reduced matrix elements. For Q(2j+) > 0, a
positive (negative) value of P 4 (2 2
+ ) corresponds to constructive (destructive)
interference. Most nuclear models predict tiiat P 4(2 2
+) is negative for nuclei in the
transition region (see, for examples, refs. 5 2 " 5 4 ) . However, there is strong
experimental ev idence 4 8 , 5 5 , 5 6 ) that P 4 (2 2
+ ) is positive for 1 9 4 P t The 2 2
+
interference contribution is negligible for 1 9 6 Pt because of the very small experimental
upper limit on (Of1 M(E2) 122+) (table 4), and so the usual ambiguity does not arise.
For the nuclei studied in the present work, interference effects involving the 4 t
+
state were found to be unusually significant The relevant matrix element product is
P 3 (4 t
+ ) -,. (O^ I M(E4) I 4j +) <2j+ I M(E2) 14,+> ( 0 ^ I M(E2) I If), the sign of
which is independent of phase conventions because alternative definitions of
(0j +1 M(E4) 14j*) differ only be the factor i 4. It has been shown experimentally57)
that ?$(4f) is negative for iHi96,i98p t Thi s r e s u i t j s adopted for the present
analysis, although the effects of assuming P3(4j+)>0 are also given. Interference
effects due to states other than 2 2
+ and 4 j + are relatively minor, and uncertainties
arising from associated sign ambiguities are incorporated into the uncertainties quoted
8
for 0(2^) and B(E2K)1*-»21
+).
3.5 DETERMINATION OF Q(2j+) AND B(E2;0 1
+^2 1
+)
For each nucleus, values of the excitation probability were calculated using the de
Boer-Winther program, and values of Q(2j+) and B(E2;0 1
+->2 1
+) were variea to
obtain the best fit to all the data obtained at safe bombarding energies. Small
corrections were applied for the effects of electron screening 5 8), vacuum
polarisation59), nuclear polarisation60), and use of the semiclassical approximation,
i.e. the quantal correction61). In addition, a correction for the effects of the
giant-dipole resonance (GDR) was applied assuming that k = 1.0 ± 0.5, where the
parameter k represents the effect of virtual excitation of states in the GDR relative to
that calculated from the hydrodynamic model [for a detailed discussion, see ref. 3 7 ) ] .
The net c^ect of these corrections and of corrections for the effects of target thickness
was to increase Q(2j+) by amounts ranging from 0.01 eb for 1 9 8 Pt to 0.03 eb for 1 9 6Pt, and to increase B(E2;01
+-»21
+) by from 0.001 e 2 b 2 for 1 9 6 Pt to 0.003 e 2 b 2 for 1 9 4 P t Relativistic corrections were not applied; however, a recent calculation62), as
yet unverified by experiment, suggests that effects on Q(2j +) values would be
negligible, although B(E2;0 1
+->2 1
+) values could be reduced by as much as 0.03
e 2b 2.
The results obtained for Q(2j+) and B(E2;0 1
+-»2 1
+), assuming P4(22+)>0 (as
found experimentally for 1 9 4Pt) and P3(4j+)<0 (as found experimentally for all three
nuclei) are shown in table 5. Changes which would result from the alternative choices
for the signs of P ^ * ) and P 3(4j +) are given in table 6. It should be noted that in
our earlier report3 0) on this work for 1 9 6 Pt, the option P 3(4j +)>0 was preferred
because we were not awa/e at that time of experimental evidence to the contrary57).
Also, the value contained in that paper for P 3(4 1
+)<0 differs slightly from the value
given in table 5 because we are now using an improved value for <0j+1 M(E4) 14j+),
obtained by analysing more data than previously. The uncertainties quoted in table 5
include, in addition to statistical uncertainties and uncertainties in spectrum analysis,
9
the effects of uncertainties in the beam energy, the scattering angles, the higher-state
matrix elements, and the GDR correction.
In order to visualize the influence of each set of data on the determination of
Q(2j+) and B(E2;01
+-*21
+), an approximate expression for the excitation probability
P of the form
P = f B ^ O ^ V M l +pQ(2,+)]
is useful. The quantities f and p (the sensitivity parameter) are functions of
experimental parameters (energy, angle, etc.) and are calculated from the de
^oer-Winther program. Fig. 6 shows plots of P e x p/f as a function of p. The fits to
the data are represented by straight lines with intercepts on the vertical axis equal to
B(E2;01
+-^21
+) and slopes of B(E2;01
+-»21
+).Q(21
+).
4. Comparison with previous work
4.1 VALUES OF B(E2;01
+->21
+)
Values of B(E2;01
+-»21
+) compiled from previous work by Ramaniah et al.63)
are plotted in fig.7, together with those of the present work, which are the most
precise yet published. It appears that, prior to the present work, the best determined
value of B(E2;01
+->21
+) was that for 1 9 4 Pt The weighted mean of the most precise
previous values for that nucleus, i.e. the Coulomb-excitation results of refs. 6 , 4 3 , 6 4), is
1.640 ± 0.008 eV, which is in reasonable agreement with the present value of 1.659
±0.011 e V .
4.2 VALUES OF Q(2X
+)
The present results for Q(2j+) are compared with those of previous work in table
7. The value of Grodzins et al.65) was obtained using the reorientation precession
technique; however, the large experimental uncertainy reduces its usefulness. The
results of Glenn et al.6) were obtained at Pittsburgh using Coulomb excitation by
42-MeV 1 6 0 ions (and also 6-MeV protons for 194Pt) and detecting scattered particles
10
with an Enge split-pole spectrograph. Their analysis ignored possible E4 matrix
elements of higher states, and in the case of 1 9 8 Pt contributions from higher states
were not considered at all. The subsequent work of Chen et al.2 6), also at Pittsburgh,
was undertaken because of concern that the relatively low enrichment of the targets
used might have produced errors in the analysis of the earlier experiments. Their value
of Q(21
+) for 1 9 4Pt was obtained by scattering 53-MeV 1 6 0 ions from more highly
enriched 1 9 4Pt targets, measuring Coulomb excitation probabilities with an Enge
spectrograph, and analysing the results with the assumption that B(E2;0j+—>2j+) =
1.620 e^b2, as determined from 4He scattering data by Baktash et al.4 3). As discussed
in Sect 1 above, the concern about the effects of relatively low target enrichment also
raises doubts about the earlier results of Glenn et al. for 1 9 6Pt and 1 9 8 Pt
The present value of Q(2X
+) for 1 9 4 Pt disagrees with that of Chen et al.
Although the 1 6 0 particle spectra of those authors are of excellent quality, that is not
the only prerequisite for obtaining reliable results from Coulomb excitation
experiments. For example, the precise determination of scattering angles becomes
very important for 0 90°. Furthermore, the result obtained by Chen et al. is very
sensitive to their choice of B(E2;01+-»21
+).
The results obtained at Rochester by Cline and his collaborators7) using Coulomb
excitation by 5 8Ni beams and detecting de-excitation y-rays have not been included in
table 7. This is partly because they have apparently never been published, and partly
because Wu and Cline66) have recently performed a more sophisticated analysis of the 1 9 4Pt data, which changed their result for 1 9 4Pt from 0.63 ± 0.06 eb to 0.45 ±0.08
eb. Presu-nably this means that the Rochester data for 1 9 6Pt and 1 9 8Pt should also
be re-analysed. It seems prudent to suspend judgement on the Rochester data until
their analysis is complete and the results published.
In summary, die present work has substantially clarified the situation concerning
the values of Q(2j+) for 1 9 4Pt, 1 9 6Pt and 1 9 8 Pt In particular, they are not consistent
with zero. The positive values obtained show that the prolate-oblate transition occurs
in the region of mass 192. Furthermore, the value obtained for 1 9 4Pt is only about one
11
standard deviation larger than the value 0.25 ± 0.17 eb stated by Chen et al.2 6) to be
the result obtained from muonic X-ray work by Hoehn et al.2 9) (the latter report docs
not give any numbers). Thus, as far as can be ascertained from available publications,
there seems to be no significant disagreement between Coulomb excitation and muonic
X-ray results for platinum.
5. Discussion of results
5.1 SYSTEMATICS OF 0.(2^) VALUES IN THE A = 192 TRANSITION
REGION
The variation of Q(2j+) as a function of mass in die transition region near A =
192 is shown in fig. 8. The values plotted are obtained from experiments involving
the reorientation effect in Coulomb excitation, except for seme from muonic X-ray
work (MXR). The tungsten data are from unpublished Rochester work6 7), but they
are the only tungsten data known to us. The Coulomb-excitation data for osmium are
from the same Rochester work, except for a recent result from Pittsburgh for 1 9 2 0 s
[ref. 2 6 ) ] . Also shown are the MXR results for osmium reported by Hoehn °t al. 2 8).
The value plotted for 1 9 2 Pt is also from unpublished Rochester work7). The results
for 1 9 4 Pt, 1 9 6 Pt and 1 9 8 Pt are from the present work, assuming P 4(2 2
+)>0 and
P 3(4 1
+)<0. In addition, the MXR result reported for 1 9 4 Pt in rcf.26) is shown. The
data for mercury are ANU results8"10), as also are those for lead [ref.68)]. The values
plotted for 1 9 8 Hg, 2 0 0 Hg and 2 0 2 Hg assume that P4(22
+)<C, as predicted by most
collective models, since there is no experimental evidence to the contrary. Adoption of
the alternative values would not significantly affect the general trends shown in fig.8.
The MXR result obained by Hahn et al.6 9) for 1 9 8Hg is also included.
Perusal of fig.8 prompts the following comments: (i) The prolate-oblate
transition near A = 192 is strikingly evident, as also is the rapid decrease in Q(2j+)
a* shell closure is approached near A = 208. (ii) There is a clear need for a new
measurement for 1 9 2 Pt, particularly in view of the discussion of the unpublished
12
Rochester work in Section 4.2 above. Similarly, there is a need for confirmation of
the reported results for the isotopes of tungsten, (iii) There is a tendency, noted by
Baktash et al. 7 0), for the results of muonic X-ray experiments on nuclei in this region
to produce values of Q(2j+) more negative than those obtained from reorientation
effect experiments. However, the differences between the results from the two
techniques are barely significant, and do not in themselves provide substantial support
far the suggestion70) that the analyses of existing MXR experiments may have taken
inadequate account of possible axial asymmetries in nuclear charge distributions.
5.2 COMPARISON WITH MODEL PREDICTIONS
There now exists a very large amount of experimental information on the
spectroscopic properties of nuclei in the A = 192 transition region. A thorough
comparison of this information with the predictions of the many nuclear models which
have been applied in the region is beyond the scope of the present paper. Instead,
discussion will be largely concentrated on the significance of the results obtained in
this work. This is not as restrictive as might at first appear. For example, the value of
Q(2j+) provides a sensitive test of nuclear models in that it depends on the wave
function of a single state (and the lowest excited state at that), and consequently is less
subject to ^nbiguity of interpretation than are many other spectroscopic quantities
which involve the wave functions of more than one state. If a model of collective
behaviour is unable to account for the value of Q(2j+) it can hardly be said to have
correctly predicted the state vector of the first excited state.
We shall try to isolate, at least partially, those predictions which are intrinsic to a
particular model from those which depend on the particular parameters chosen for the
model. A simple example is the ratio Rj = Q(2j*)/VB(E2;0, +->2X+), which the
axially-symmetric rigid-rotational model predicts to equal 0.906, regardless of the
values chosen for the two parameters of that model (the intrinsic quadrupole moment
and the moment of inertia). This may be compared to the values for I94,i96,i98pt
obtained in the present work, which lie in the range 0.3 - 0.6.
13
Most models do not make absolute predictions of Rj, the 0(6) and SU(3) limits
of the IBM being interesting exceptions, but absolute predictions can be obtained from
some if Rl is considered in conjunction with another nuclear property. There is some
freedom in choosing this other quantity, but, adopting the attitude implied above that
collective models should, in the first instance, be judged by their ability to predict the
properties of the lowest excited states of collective nuclei, we choose to use the ratio
Rj = [B(E2;0 1
+->2 2
+)/B(E2;0 1
+-»2 1
+)] 1 / 2. Figs. 9(a) - 9(c) are plots of Rj against Rj
for the three nuclei I94.i96,i98p^ showing the experimental values (obtained from
tables 4 and 5) and various theoretical predictions.
As mentioned above, the axially-symmetric rigid-rotational model (RF) predicts
Rj = 0.906; we take it to predict R2 = 0 since there is no 22+ state in this model. All
three dynamical symmetries of the IBM predict Rj = 0. For the 0(6) limit, Rj is also
zero71). The SU(3) limit predicts72)
RL = y (4N+3) / 2 T T / ( 1 0 N 2 + 1 5 N ) ,
where N is the total number of bosons. Since the U(5) limit makes no prediction of Rj
[ref.73)], it is not plotted in fig.9.
The curves in fig.9 show the predictions of those models in which Rj versus Rj
may be parameterized by a single quantity. For the asymmetric rotor model (ARM),
this quantity is the asymmetry angle y, which runs from 30° at the origin to 60° at the
axially symmetric value 7 4). The curve marked CQ shows the predictions of a
simplification of the IBM known as the consistent-Q formalism75). The parameter in
this case is the ratio x of the coefficients of the two terms of the quadrupole operator;
X varies from zero at the origin or 0(6) limit to V35/2 at the SU(3) limit, and
intermediate values of % correspond to mixing of the two limits. It is emphasized that
the consistent-Q formalism represents just one path through the IBM parameter space,
albeit one which has been held to be broadly appropriate in this mass region75). The
rotation-vibration model20) (RVM) considers harmonic vibrations in both the (3 and y
14
directions about an axially symmetric equilibrium. The ratios Rj and R2 are given by
two parameters; the ratios EJe and E /e of the quanta cf P and y vibrations,
respectively, to the characteristic rotational energy e. The curves in fig.9 show the
variation with E le, \ \ich runs from infinity at the maximum value of Rj toward zero
as Rj decreases. F01 hose marked RVM-1, EJe = 90, corresponding to almost
complete absence of P vibrations. To obtain the curves marked RVM-2, values of
EJe were chosen which placed the P bandhead at known excited 0 + states, the 1479.3
keV, 1402.7 keV and 1481.5 keV states of 194,196,198^ respectively, being chosen as
the most likely candidates. Parts of the RVM curves are shown dashed to reflect
uncertainties of interpolation in the tables of ref.20).
Also shown in fig.9 are predictions of Rj and Rj from various more complex
models. We have only shown results of those calculations for which either the
relevant matrix elements have been published or sufficient information is available to
enable the required matrix elements to be calculated. For example, refs. 1 4 , 7 6) do not
quote values of Q(2 1
+), but these have been calculated using PHINT7 7) and the
published parameters. The values shown for refs. 7 8 , 7 9) do not appear in these papers,
but are quoted in ref. 7 0). The only published calculations for 1 9 8 Pt are those
performed with the boson-expansion theory 1 6 , 1 7) (BET). We have extended the
IBM-1 calculations1 4 , 7 6) to this nucleus using the systematics recommended in the
respective papers. We have similarly extended the IBM-2 calculations of Bijker et
al. 8 1) using the computer program NPBOS82).
For the quantities considered in fig.9, the models seem to do best for 1 9 4Pt. All
have increasing difficulty as neutrons are added. Although each of the more complex
models contains degrees of freedom not present in the simple models, these extra
degrees of freedom do not always improve the fit to the data. For example, adding a
variable moment of inertia to the ARM [ref.79)] has no perceptible effect on the
relationship between R{ and R^ whereas adding a «iatic hexadecupole
deformation57 ,80) produces a change in the direction required by the data. A similar
change can be obtained by abandoning the requirement that the vibrations be
15
harmonic24), although die generalized collective model (GCM) of ref.24) is so complex
that it is difficult to pick any one of the additional degrees of freedom as being an
important one.
Baktash et al.7 0) concluded from an extensive discussion of the electromagnetic
properties of heavy transitional nuclei that the general experimental trends are best
reproduced by the microscopic pairing-plus-quadrupole model (PPQ) of Kumar and
Baranger1 , 2) and the boson-expansion theory (BET) of Weeks and Tamura 1 6 , 1 7).
While the PPQ gives satisfactory values of Rj and R2 for 1 9 4Pt, it performs poorly for 1 9 6PL The BET generally gets 1^ about right, but not R^
Generally the IBM-2 [ref.81)] predicts larger values of Rj than does the IBM-1
[refs. 1 4- 7 6)], but, for 1 9 4 -* 9 6 Pt, still not as large as those observed. Thus, the
expectation expressed by Casten83) and others that the application of the EBM-2 would
rectify the failure of the IBM-1 to account for the Q(2j+) values of the Pt isotopes is
at best only partially fulfilled. It is clear that the consistent-Q formalism is not
appropriate for I96,i98pt a t l e a s t
Our result for Q(2j+) for 1 9 6 Pt is of particular interest because this nucleus has
been proposed as an excellent empirical manifestation13) of the 0(6) limit of the IBM.
The non-zero value obtained for Q(2j+) is certainly inconsistent with the requirement
of the unperturbed 0(6) limit It is conceivable that, as suggested by Bolotin et al. 2 5),
a small admixture of SU(3) to 0(6) might reproduce the observed value of Q(2j+)
simultaneously with other properties of the low-lying states of 1 9 6 Pt. How* :er, we
have been unable to reproduce the numerical results of Bolotin et al, with the program
PHINT77) and their published parameters. Furthermore, all of our attempts to fit Rj
and R2 with some coronation of the 0(6) and SU(3) limits have failed; when we
choose parameters to fitT .v we obtain values of Rj which are much too large.
6. Conclusion
Static electric quadrupole moments of the 2 , + states of 1 9 4 , 1 9 6 , 1 9 8 P t have been
16
determined using Coulomb excitation by 4He, 1 2 C and 1 6 0 projectiles. All known
corrections were applied in the analysis; in particular, it was found that
B(E4;0j+—>4j+) values are large enough to produce significant interference effects.
The quadrupole moments of all three nuclei were found to be positive, thus locating
the prolate-to-oblate transition at or near A = 192. Comparison with the predictions of
various well-known nuclear models shows that all have difficulty in describing the
systematics of the E2 properties of the first two excited states of these nuclei. Within
this restricted compass, none of these three nuclei, including 1 9 6Pt, appears to be a
good example of the 0(6) symmetry of the IBM.
The authors are grateful to F. Todd Baker for drawing to their attention the
experimental evidence that P3(41
+) < 0 for the even-mass Pt nuclei.
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21
TABLE 1
Percentage compositions of Pt target materials as provided by the supplier (Oak Ridge National Laboratory)
Target ( i.e. major isotope] I
Isotope 194R
1 9 6 Pt( l ) 1 9 6Pt(2) 198ft
192ft 0.04 (1) <0.01 <0.05 0.01 (1) 1 9 4 Pt 95.06 (15) 0.63 (1) 0.78 (2) 0.79 (1)
IMpt 3.78 (10) 1.57 (2) 2.39 (5) 1.18(1) 1 9 6 Pt 0.97 (5) 97.51 (3) 96.54(5) 2.18(2)
IMpt 0.17 (2) 0.29 (1) 0.29 (2) 95.83 (5)
Targets of 1 9 6 Pt were prepared using two different isotopically enriched samples.
22
TABLE 2
Measured excitation probabilities P for the 2X+ states
o f 194,196,198^
Target Projectile 0 l a b E(MeV) P e x p x l ( ) 2
1 9 4 Pt 4He 174.8°
1 9 4 Pt 1 2 C 174.8°
1 9 4 Pt 1 6 0 174.8e
1 9 6 Pt 4He 90.0°
1 9 6 Pt 4He 174.8°
14.199 1.887 (16) 14.399 1.944 (17) 14.599 2.022 (17) 14.799 2.185 (19) 14.999 2.230 (19) 15.199 2.363 (21) 15.599 2.532 (22)
40.999 11.95 (ID 41.999 12.97 (10) 42.999 13.97 (14) 43.999 14.94 (13) 44.999 15.86 (10) 45.999 17.07 (11) 47.999 19.41 (19) 49.999 21.59 (22) 40.999 11.54 (15) 41.999 12.77 (12) 42.999 13.99 (14) 43.999 14.99 (15) 44.999 16.07 (14)
54.998 19.03 (28) 55.998 20.27 (28) 56.998 21.58 (29) 57.998 22.39 (44) 58.998 23.87 (41) 59.998 25.00 (37) 60.998 25.22 (41) 62.998 27.69 (44)
16.799 0.905 (8) 16.999 0.927 (8) 17.399 0.995 (9) 17.799 1.056 (10) 17.998 M53 (21) 18.199 1.154 (10) 18.599 1.202 (12)
14.199 1.156 (13) 14.399 1.582 (13) 14.599 1.669 (14) 14.799 1.781 (15) 14.999 1.840 (16) 15.199 1.894 (16) 15.399 2.000 (17) 15.599 2.099 (18)
23
TABLE 2 (continued) 1 9 6 Pt 1 2 C 174.8°
196pt 16Q 1 7 4 > 8 O
1 9 8 Pt 4 He 174.8°
198pt 12c 1 7 4 _gc
1 9 8 Pt 1 6 0 174.8°
40.985 9.75 (12) 41.986 10.57 (12) 42.987 11.42 (13) 43.988 12.36 (13) 44.989 13.22 (14) 45.990 14.32 (14) 46.991 15.37 (14) 47.991 16.22 (14) 48.992 17.25 (15) 49.992 18.15 (15) 51.993 19.86 (16) 53.995 20.66 (17) 55.996 19.72 (21) 40.997 9.70 (10) 41.997 10.58 (10) 42.997 11.49 (11) 43.998 12.36 (10)
54.993 15.75 (25) 55.993 16.69 (28) 56.994 17.93 (25) 57.994 18.55 (28) 58.994 20.27 (22) 59.994 20.60 (28) 60.994 22.18 (29)
13.999 1.059 (11) 14.199 1.112 (12) 14.399 1.167 (16) 14.599 1.209 (15) 14.799 1.289 (13) 14.999 1.326 (11) 15.199 1.415 (13) 15.399 1.488 (13) 15.599 1.547 (17) 40.998 7.10 (7) 41.998 7.86 (7) 42.998 8.48 (8) 43.998 9.19 (8) 44.998 9.95 (9) 45.998 10.63 (11) 47.998 12.33 (13) 49.998 13.72 (i:) 40.998 7.22 (9) 41.998 7.88 (7) 42.998 8.47 (11) 56.997 13.55 (18) 57.998 14.41 (17) 58.998 15.07 (18) 59.998 15.87 (19) 60.998 16.67 (21) 61.998 18.00 (22) 62.998 18.33 (22)
24
TABLE 3
Excitation energies (Ex) and spin-parity values (J^) of energy
levels included in multiple-Coulomb-excitation analysis
Nucleus Reference Ex(keV) Jn»
194ft 46^
196pt 47^
198^ 48,49)
0 o, +
328.5 V 622.0 V 811.3 V
1229.6 V
0 0 i +
355.7 V 688.7 V 876.9 V
1135.3 o 2
+
1293.3 V
0 o, +
407.4 V 774.2 V 914.4 o 2
+
985 4 i +
1287 V
25
TABLE 4
Magnitudes of matrix elements included in multiple-Coulomb-excitation analysis
Matrix Element 194pt 196pt 198pt
(O^IM^IV) 0.090 (2) < 0.002 0.039 (7)
(O^lMO^jilV) 0.23 (8) 0.18 (7) 0.09 (9)
(01HlM(E4]l|42
+> 0.13 (3) <0.14 <0.2
<21HlM(E2)j|22
+> 1.455 (25) 1.30 (4) 1.02 (5)
(21
+ilM(E2)l|41
+> 2.17 (2) 1.94 (2) 1.56 (7)
(21HlM(E2)||02
+> 0.15 (3) 0.44 (6)
(21^M(E2)||42
+> 0.30 (7) 0.164 (27)
(ZjlMfrljlOf) 0.38 (10)
<22HlM(E2)||42
+> 2.5 (7) 1.26 (13)
<4^|M(E2)||42
+> 1.32 (33)
M(E2) matrix elements are given in units eb and M(E4) matrix elements in units eb2.
The values are taken from refs.2 5'2 6'4 3-4 8-5 0-5 1).
26
TABLE 5
Results obtained for Q(21
+) and B(E2;01
+-»21
+)
assuming P ^ + X ) and P3(4j+)<0
Q(21 +) B(E2;01
+-»21
+)
(eb) (eV)
1 9 4 Pt 0.48 (14) 1.661 (11) 1 9 6Pt 0.66 (12) 1.382 (6) 1 9 8Pt 0.42 (12) 1.090 (7)
27
TABLE 6
Changes which would be produced in the present results for Q(2j+) and
B(E2;01
+-»21
+) if the alternative signs were adopted for P 4(2 2
+) and P3(4j+)
ifP 4 (2 2
+ )<0 i f P 3 ( V ) > 0
AQ(2,+) AB(E2) AQ(2,+) AB(E2) nucleus (eb) (eV) (eb) (e2b2)
1 9 4Pt +0.36 -0.001 +0.10 +0.001 1 9 6Pt 0 0 +0.09 -0.001 1 9 8Pt +0.12 0 +0.03 0
28
TABLE 7
Published results for Q(21
+) (in eb) for 1 9 4Pt, 1 9 6Pt and 1 9 8Pt, assuming P4(22+)>0 and P3(4j+)<0 where appropriate.
Authors 1 9 4Pt 1 9 6Pt 1 9 8Pt
Glenn et al.6)
Grodzins et al.6 5)
Chenetal.26)
Present work
0.64 (16)
0.77 (50)
0.13 (17)
0.48 (14)
0.51 (18) 1.22 (50)
0.66 (12) 0.42 (12)
29
FIGURE CAPTIONS
Fig. 1. Representative spectra obtained with an annular counter. In each case the full curve shows a fit to the data obtained as described in the text, and the broken curve shows the calculated contribution due to scattering from Pt isotopes other than the primary isotope in the target Peaks corresponding to excited states in the target nucleus are indicated by J ^ values. Pulse pile-up effects are evident on the high-energy side of the 0 j + peak in (a).
Fig. 2. Typical spectrum obtained using an Enge split-pole magnetic spectrometer. The full and broken curves are as for fig.l.
Fig. 3. Safe-energy plots for 1 9 4 Pt.
Fig. 4. Safe-energy plots for 1 9 6 Pt.
Fig. 5. Safe-energy plots for 1 9 8 Pt.
Fig. 6. Plots of P e x p/f against the sensitivity parameter p. In each case it is assumed that P4(22+) > 0 and P 3(4j +) < 0. For simplicity of presentation, each data point shows the average of results obtained at all safe energies for the target-projectile-angle combination concerned.
Fig. 7. Summary of experimental values of B(E2;0 1
+ -»2 1
+ ) taken from the compilation of ref.63). Coulomb-excitation data are shown by circles, and other data by crosses. The results indicated by G, R, B and P are due to Glenn et al.6), Ronningen et ai. 6 4), Baktash et al. 4 3) and the present work, respectively.
30
Fig. 8. Quadrupole moments of the first excited states, Q(2 1
+), of even-mass nucleiin the transition region near A = 192. The data are taken from experiments involving the reorientation effect in Coulomb excitation, except for those from muonic X-ray work (MXR). The value shown for 1 9 8 Pt for the present wonc assumes P 4 (2 2
+ ) > 0; the alternative assumption, that ¥A(2£) < 0, would increase Q(2j+) by 0.12 eb and would not change the overall picture significantly. As noted in the text, refs. 7 , 6 7 > are to unpublished work.
F ;g. 9. Comparison between experiment and theoretical predictions for the relationship between the quantities Rj and Rj, where Rj = Q(2 1
+)/VB(E2;0 1
+->2 1
+) and Rj = [B(E2;0 1
+ -»2 2
+ )/B(E2;0 1
+ -»2 1
+ )] 1 / 2 . References are indicated by the numbers in square brackets. The key to the less than obvious model acronyms is as follows: RR-axially symmetric rigid rotator; PPQ - pairing-plus-quadrupole; GCM - generalised collective model; ARM - asymmetric rotor model; ARM + VMI - asymmetric rotor model with variable moment of inertia; ARM + p"4 - asymmetric rotor model with static hexadecupole deformation; DCM - dynamic collective model; RVM - rotation vibration model; CQ - consistent-Q formalism. The experimental value shown for I 9 8 Pt assumes that P4(22+) > 0; the alternative assumption,that P 4(2 2
+) < 0,would increase Rj from 0.40 to 0.52 and would not affect the conclusions.
31
• P H I 6 0
59.0 MeV
174.8°
10'
• » •
M » 4 M M * I
CHANNEL
P"; V I
32
S1ND00
r » <\ . x
33
JT—|—i—i—i—i—| i i i i | E ( M e V )
1.0
0.9 7
I. A
t l 9 4 P t + 4 H e
174.8°
1.0
O O
Q- 0.9
T r 45
T 1 1 1 r 50 E(MeV)
t i i i
I 9 4 p t + I 2 c
\ 174.8°
^ . CL X 4>
55 60 _ , . , w l
71—i—i—i—i—|—i—i—r E(MeV)
1.0
0.9
H 1 1 1 1 1 t
I I I 9 4 p t + I 6 0
174.8°
i l 6 5 4
s (fm)
Fi V 2. u
34
0 0
T
0.90
LOO I- h i ' i . , ,
0 9 0 > " , • « ° . E(MeV)
1.00 o
cf 0.95
x o
0-1.00-
0 .90
0 .80
17 18 19 E ( M e V ) -t—i—|—i—i—i—i—[ i i i i | i c- \ ' » ' c » /
9 6 P t +
4 H e
90.0°
| 4 i • • . 'p . • . .'f E(MeV) \
i , 9 6 P t +
4 H e
t 174.8*
- { j I t f , 1 , 9 6 P t +
, 6 o \ \
t 45
174.8°
50 55 1 1 1 r T 1 — | 1 — i — i — l — | — r E (MeV)
* I * I | M
t i i
9 6 P t • , 2 C
174.8°
6 5 s (fm)
t ' V ^
35
1.0 -I l *
°**
1.0 e o
* 0.9 • <? 14
1.0
0.9
T 1—i 4 ? i—i . • 5 | ° E (MeV)
55
i i
t
60
i I
, 9 8 p t +
l 2 c 174.8°
T 1 — i — i — | — i — i — r E(MeV)
I! " < \
I 9 8 p t + I 6 Q
174.8°
t T—i I i i i i I E (MeV)
L 1 1 ' . ' , . . , , 9 8 Pt + 4He
174.8°
t
5 s (fm)
i F.^.r
36
6 6 F
4 8 h
.44h -O
\ 1.40
QT
1.36
C (1748°)
J I I L J L 1
010
F't t
57
CHRONOLOGICAL ORDER OF PUBLICATION
FiY 7
38
1 ' r HfrH
* < ^ H
- * -
> - * - " * * -
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K j i
J L
1—r T~r
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X
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o: x
^ O O O Q - C L Q - X X Q -
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I
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cc 0.2
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/
0.1 -
0
(c) . . 198
/ \ /
RVM-I[20]
EXPT
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