VOLUME 79, NUMBER 4 PHYSICAL REVIEW LETTERS 28JULY 1997
Tilted Rotation and Backbending in an Odd-Proton Nucleus
C. J. Pearson,
1
P. M. Walker,
1
C. S. Purry,
1
G.D. Dracoulis,
2
S. Bayer,
2
A. P. Byrne,
2
T. Kibédi,
2
F. G. Kondev,
2
T. Shizuma,
3,4
R. A. Bark,
3
G. Sletten,
3
and S. Frauendorf
5
1
Department of Physics, University of Surrey, Guildford, GU2 5XH, England
2
Department of Nuclear Physics, Research School of Physical Sciences and Engineering, Australian National University,
Canberra ACT 0200, Australia
3
Niels Bohr Institute, Risø, DK-4000 Roskilde, Denmark
4
Department of Physics, Kyushu University, Fukuoka 812, Japan
5
Institute for Nuclear and Hadronic Physics, Research Centre Rossendorf, 01314 Dresden, Germany
(
Received 13 February 1997)
Tilted-axis rotation, arising from Fermi-aligned configurations, has been observed for the first time
to cause backbending in an odd-proton nucleus. In
181
Re, two t-bands are found to be energetically
favored relative to the usual rotation-aligned s-bands, presenting an alternative form of cold nuclear
rotation. Interactions between the bands are weak, and unambiguous comparisons with tilted-axis-
cranking calculations can be made. [S0031-9007(97)03669-7]
PACS numbers: 21.10.Re, 21.10.Jx, 23.20.Gq, 27.70.+q
Rotational excitations in atomic nuclei have long been
a source of information about the underlying nucleonic
structure. A dramatic increase in apparent moment of
inertia, known as backbending, was first discovered [1] in
160
Dy at an angular momentum of I ø 16 ¯h, and is now
well established as a general feature of nuclear rotation.
In well deformed, axially symmetric nuclei, backbending
has been understood [2] to be due to the alignment of
the angular momentum of a pair of high-j nucleons along
the rotation axis, so that the mean angular-momentum
component along the symmetry axis (perpendicular to the
rotation axis) is kKlø0. The aligned structure (s-band)
becomes favored in energy at high spin and crosses the
nonaligned structure (g band).
In the A ø 180 mass region, where backbending is
due to a pair of i
13y2
neutrons, it has been proposed
[3,4] that a new structure arises due to the Fermi-aligned
[4,5] coupling scheme which is active in the middle of
the neutron shell (N ø 104). In this scheme the nucleon
angular-momentum precesses about an axis lying between
the rotation and symmetry axes. The projections of the
angular momentum onto the symmetry axis, K, and the
rotation axis, i, are both localized at a nonzero value.
This differs from the strong-coupling scheme in which the
nucleon angular-momentum precesses about the nuclear
symmetry axis resulting in good K, with kilø0. States
formed from a pair of Fermi-aligned nucleons have K ø
jK
1
6 K
2
j and i ø i
1
1 i
2
, resulting in the usual s states,
having K ø 0, and also t states having large K and a
total angular momentum tilted between the rotation and
symmetry axes. The s- and t-bands compete for “yrast”
status (the lowest energy for a given angular momentum)
corresponding to cold nuclear rotation. The nature and
extent of this competition is not yet understood, and
the possibility of there being large-amplitude high-K
components in the yrast bands of nuclei with N ø 104
is contrary to the usual interpretation of backbending as
intrinsically a low-K phenomenon.
Evidence for the role of t-band structures in backbend-
ing has been observed in
179,180
W [3,6] and
181,182
Os [7]
(with proton numbers Z 74, 76, respectively). How-
ever, only in
179
W are the band interactions sufficiently
weak to enable a clear signature to be obtained of the high-
K band crossing, and in this unique case a chance near de-
generacy at the band crossing could be responsible for the
special features. Furthermore, rotational models [4] have
failed to reproduce the band crossings in
179
W. Conse-
quently, the generality of t-bands being a significant struc-
tural feature in the A ø 180 region has been in doubt.
As yet no evidence for t-bands has been found in odd-
Z nuclei. Apart from the need for additional information
about t-bands, the odd proton can be expected to enhance
M1 transitions, and allow M1yE2 branching ratios to
be measured for rotational states with the potential of
providing new structural information. Odd-Zt-bands
would also have different pairing and residual spin-spin
interactions compared to the odd-N and even-even cases,
testing the concepts that have so far been proposed [3,4].
An experiment to investigate the Z 75 nucleus
181
Re,
an isotone of
180
W and
182
Os, with N 106, was car-
ried out at the Australian National University 14UD tan-
dem accelerator. An
11
B beam of energy 77 MeV was
incident on a self-supporting 5 mgycm
2 176
Yb target. The
beam energy was chosen to maximize production of high
angular-momentum states in the
176
Ybs
11
B,6nd
181
Re reac-
tion. Gamma rays were detected using the CAESAR ar-
ray of six Compton-suppressed germanium detectors and
two unsuppressed planar LEPS detectors. The beam was
bunched and chopped to give 1 ns wide pulses separated by
1.7 ms. This allowed measurement of g-ray coincidences
across isomers with half-lives of up to 10 ms in strongly
populated cases. A total of 3 3 10
8
gg coincidences was
0031-9007y97y79(4)y605(4)$10.00 © 1997 The American Physical Society 605
VOLUME 79, NUMBER 4 PHYSICAL REVIEW LETTERS 28JULY 1997
recorded—including energy and time relative to the beam
pulse for each g ray. In addition, singles-g-ray energy and
time measurements were performed in which a beam with
1 ms pulses separated by 20 ms was used, allowing the
observation of longer half-lives. The beam-pulsing and
time information allowed highly sensitive measurements
of g rays above and below isomers, which proved vital
in deducing the level structure.
181
Re has a complex de-
cay scheme with two strongly populated long-lived iso-
mers with half-lives of 1.4 and 11 ms, three isomers having
half-lives of ,100 ns, and one short-lived isomer with a
half-life of ,20 ns.
A complementary data set was obtained at the Niels
Bohr Institute Tandem Accelerator Laboratory, for
prompt (in-beam) events. The
150
Nds
36
S, p4nd
181
Re re-
action was employed at a beam energy of 166 MeV, with
stacked targets of approximately 1 Mgycm
2
total thick-
ness. The Nordball array, equipped with 20 germanium
detectors, and a “Si-ball” charged-particle array, consist-
ing of 21 DE detectors of 170 mm thickness, was used to
collect approximately 3 3 10
7
gg-particle coincidences.
The results are consistent with the CAESAR data. A
detailed comparison of the two data sets will be made in
a later full report on this work.
Fourteen sequences of rotational states were identified
based on one, three, and five quasiparticle structures, of
which three were known [8] previously. The partial decay
scheme relevant to this Letter, shown in Fig. 1, focuses on
two pairs of interacting one- and three-quasiparticle bands
associated with the 5y2
1
f402g and 9y2
2
f514g Nilsson
orbitals which are close to the proton Fermi surface. The
5y2
1
f402g band was known up to spin
25
2
¯h [8] but is now
observed up to spin
45
2
¯h. A new band based on an I
p
21y2
1
state at 1858 keV is observed up to spin
53
2
¯h.It
has a well defined bandhead which decays via a 164 keV
transition to a previously identified [8] I
p
s17y2
1
d
state at 1694 keV. The band based on the 1858 keV state
crosses the K
p
5y2
1
band at I 25y2, where out-of-
band transitions to the K
p
5y2
1
band compete with
in-band transitions.
A second pair of interacting one- and three-quasiparticle
bands is observed. The band based on the 9y2
2
f514g
Nilsson orbital is now identified up to spin
43
2
¯h compared
to the spin
23
2
¯h known from previous work [8]. The
new band based on the I
p
25y2
2
state at 2225 keV is
observed up to spin
55
2
¯h. It has a well defined bandhead
which decays via a 345 keV transition to a known [8]
11 ms isomer at 1881 keV. The band based on the
FIG. 1. Partial decay scheme for
181
Re, showing the rotational bands based on the 9y2
2
f514g and 5y2
1
f402g one-quasiparticle
states, the associated t-bands, and their decay paths. Energies are in keV.
606
VOLUME 79, NUMBER 4 PHYSICAL REVIEW LETTERS 28JULY 1997
2225 keV state crosses the K
p
9y2
2
band at I
27
2
¯h, where out-of-band transitions to the K
p
9y2
2
band compete with in-band transitions. Strong DI 1
transitions are seen through each band.
Given the previous one-quasiparticle band assignments
[8], the present spin and parity assignments of the
higher-spin levels are determined by the regularity of
the rotational sequences, the systematically competing
DI 1 and DI 2 transitions, and a consistent set
of g-g angular correlation (DCO) ratios. In particular,
the 514 keV (25y2
1
! 21y2
1
) and 544 keV (27y2
2
!
23y2
2
) interband transitions have stretched quadrupole
character. The assignments are also consistent with the
band-crossing interpretation (see below) whereby close-
lying states of equal spin and parity mix sufficiently to
enable interband transitions to compete with collective
intraband transitions.
The two pairs of bands behave in a similar way,
as illustrated in Fig. 2 (top). In each case the one-
quasiparticle band is crossed by what appears to be a
band of the same parity and with an additional eight units
of K (assuming that the K value is equal to the spin
of the bandhead). Since the Fermi surface at N 106
FIG. 2. Energies and BsM1dyBsE2d ratios corresponding to
the bands shown in Fig. 1. The upper panels show experimen-
tal energies, with an arbitrary rigid-rotor reference subtracted
in order to highlight the band-crossing features. The middle
panels show the corresponding energies from TAC calculations.
The lower panels show the experimental BsM1dyBsE2d ratios
(symbols) for the energetically favored states, together with val-
ues from TAC calculations (full and dotted lines).
is between the 7y2
1
f633g and 9y2
1
f624g Nilsson orbits,
each of i
13y2
character, it is reasonable to attribute the
increase in K to the excitation of this neutron pair coupled
to K
p
8
1
. At each band crossing a moment-of-inertia
increase is indicated by the change in slope [Fig. 2
(top)]. This corresponds to an alignment increase of
,6¯h, due to the Fermi alignment of the high-j neutrons.
The crossing bands are therefore interpreted as t-bands,
composed of two i
13y2
neutrons and a spectator proton,
analogous to the t-bands found in the neighboring odd-
N nuclei,
179
W [3], and
181
Os [7], where the spectator
is a 7y2
2
f514g neutron. A significant difference is that
in
181
Re the band crossings involve almost negligible
perturbations of the level energies, with interaction matrix
elements less than 8 keV. At first sight it appears
surprising that, with such a small interaction, each high-
K band has strong g-ray branches to its partner one-
quasiparticle band at the crossing. This is, however,
consistent with the K values discussed above, with band
interactions of only 4 keV (c.f. [3] for detailed discussion
of the methods used to determine the interactions) and
arises because the high-K intraband transition strengths
are strongly retarded by their small angular-momentum-
coupling (Clebsch-Gordan) coefficients.
A consistent picture emerges of high-Kt-bands that
become yrast relative to their one-quasiparticle g-band
partners. In each case the structural change at the band
crossing is the excitation of two i
13y2
neutrons coupled
to K ø 8, where the approximation sign is used to
acknowledge that the Fermi alignment leads to some
spreading of the K distribution. This is precisely the
situation that has been modeled by tilted-axis-cranking
(TAC) calculations [4] for odd-N
179
W. The TAC model
is in contrast to the usual principal-axis-cranking (PAC)
model [9] in which the high-spin states are generated
by cranking about an axis perpendicular to the nuclear
symmetry axis. The PAC model could not be expected to
give a consistent description of high-K rotational bands,
since these have a significant fraction of the angular
momentum aligned with the symmetry axis. However, the
TAC model permits the total angular-momentum vector
to be at an angle intermediate between the symmetry axis
and the rotation axis. This additional degree of freedom
is required if a cranking model is to be able to describe
high-K structures. Nevertheless, in the application to the
odd-N case,
179
W [4], strong interactions between the
calculated bands resulted in ambiguities in the comparison
with experiment, and the calculated s-band appears to be
lower in energy than the t-band at high spin, contrary to
experiment.
TAC calculations have now been performed for odd-
Z
181
Re, using standard deformation and pair-gap param-
eters (´
2
0.225, ´
4
0.046, D
p
0.87 MeV, D
n
0.67 MeV). The calculated and experimental
181
Re level
energies are compared in Fig. 2. Although no fitting has
been attempted and the gradients are not well reproduced,
607
VOLUME 79, NUMBER 4 PHYSICAL REVIEW LETTERS 28JULY 1997
other qualitative features are in good accord with the data.
An unambiguous comparison between observed and cal-
culated t-band crossings can be made for the first time.
The t-bands are calculated to become energetically fa-
vored, as observed, and the gradual gradient decrease in
the unfavored g-band extensions can be identified with
s-band crossings. The comparison between calculation
and experiment shows a considerable improvement over
the
179
W situation [4], which appears to be related to
the interactions between the
181
Re bands being weaker
(most likely due to less blocking of the neutron pairing
in the odd-Z nucleus). The calculated t-band wave func-
tions are also of interest. For example, close to the band
crossing, the t-band that crosses the 9y2
2
f514g band has
kKl 12.2, which is close to the strong-coupling limit of
K 12.5 (for two i
13y2
neutrons coupled to K 8, to-
gether with the K 9y2 proton).
The present data contain many DI 1 transitions from
which estimates of the magnetic dipole strength can be
made. It is therefore possible for the first time to make
comparisons with the TAC model calculation of the
M1 transition rates [4,10], relative to the collective E2
transitions, through the t-band crossings. This is illus-
trated in Fig. 2 (bottom), for the energetically favored
sequences, i.e., for the states having the lowest energy
for a given spin, above the 9y2
2
f514g and 5y2
1
f402g
one-quasiparticle states. The TAC calculations (full and
dotted lines) give a reasonable description of the data,
though in the absence of detailed level-energy agree-
ment, the quality of the transition-rate comparisons is dif-
ficult to judge. The poorer agreement for the 5y2
1
f402g
t-band may be due to interactions with another (in this
case unidentified) positive-parity sequence, possibly asso-
ciated with the observed 1694 keV intrinsic state. The
rapid decrease with spin of the BsM1dyBsE2d ratio close
to each bandhead can be understood as arising from the
geometrical factors (angular-momentum coupling coef-
ficients) associated with the K value, or, in the TAC
calculations, from the increasing tilt angle, u, where
BsM1dyBsE2d ~ 1y sin
2
u in the strong-coupling limit
[10]. [Note that the experimental BsM1d values have been
obtained assuming no E2 component in the DI 1 tran-
sitions; these E2 components are estimated to be less than
10%, which is supported by measured g-ray angular dis-
tributions where available.] The lack of detailed quanti-
tative agreement for the t-bands may be, at least in part,
a consequence of the simplified assumption of equal neu-
tron pairing for all bands. Reduced neutron pairing would
be expected in the t-bands. This would reduce the rota-
tional g factors and increase the proton contribution to the
M1 matrix element, but these effects have not yet been
explored in detail.
The involvement of t-bands in the explanation of
anomalous (weakly hindered) K-isomer decays has
already been demonstrated [3] in
179
W. The present
data, together with their qualitative description by the
TAC model, suggest a more widespread influence of
t-bands, which provide a mechanism for the intro-
duction of large-amplitude, high-K components into
A ø 180 yrast bands. Hence, t-bands would have an
important role to play in determining multiquasiparticle
isomer decay rates. It is hoped that the new data for
181
Re will stimulate more detailed theoretical evaluation
of the phenomenon, including the effect of residual
spin-spin interactions which have not been included
in the TAC calculations. Generalized Gallagher-
Moszkowski rules [11] favor parallel intrinsic-spin
couplings for unlike nucleon pairs (n " p " and n # p #)
and antiparallel couplings for like pairs (n " n # and
p " p #). Residual interactions in even-even nuclei would
therefore favor s-bands (n " n #) over t-bands (n " n ")
with regard to bandhead energies (by ,200 keV). How-
ever, there is no such favoring for the
181
Re, 9y2
2
f514g "
and 5y2
1
f402g " s-bands (p " n " n #) compared to their
respective t-bands (p " n " n ") when all pair interactions
are considered. Indeed, the higher spin of the bandhead
for the high-K coupling enables these t-bands to become
yrast. Other couplings, such as to the 7y2
1
f404g # proton,
would form energetically unfavored t-bands (p # n " n ")
which would not become yrast.
In summary, clear signatures have been found for two
t-bands involving si
13y2
d
2
, K ø 8 couplings in
181
Re. In
each case it is the t-band and not the s-band which gives
rise to backbending in the associated yrast sequence. This
is the first observation of t-bands in an odd-Z nucleus, and
the first observation of more than one t-band in a nucleus.
TAC calculations provide a reasonable description of the
experimental data. It is suggested that t-bands are of more
general importance than has hitherto been recognized.
Experiments at the ANU have been carried out through
access under the ANU-EPSRC agreement, and with the
support of EPSRC Grant No. GRyJ95867.
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