against I in figure 5.7(a) (compare with figure 4.5) The increasing

W, in which the negative-parity yrast band has a relatively high

excitation energy, the ^ 2.3/2^ t>and maY also be expected to be

relatively high, so that backbending is not seen in this nucleus.

It is reasonable to take this a stage further, at least for the

isotones Hf and W. Here the neutron structures should be

similar, although there are considerable apparent differences in the

behaviour of their positive-parity yrast bands at high spin. In these

cases an estimate of the unperturbed ^2.3/2^2 band is m a d e by taking

the 14+ -*12+ , 16+ ->-14 + , 18+ -*16 + , etc. transitions as being equal in

energy to the observed 13 11 , 15 ^ 13 , 17 15 , etc. transitions

in the negative-parity yrast band, and in this way allowing for the

expected higher moment of inertia: the positive-parity band in the

two-quasiparticle-plus-rotor calculation (see figure 5.13) has a

4- . 2

the excitation energy of the 12 state in the postulated (i^3/2^

. 174

is taken as 75 keV higher than the observed 11 state in W, and 100

keV higher than the 11 state in Hf. These are judicious choices

since the final results depend sensitively on this energy, but it is

noted that the single-quasiparticle i-^3 / 2 state is about 100 keV

— 1 7 4 1 7 5

higher than the 5/2 [512] ground state in both Hf and W.

With these postulated bands, together with extrapolations of the

gsb's in Hf and W, a two-band-mixing calculation has been

carried out for each nucleus. The moment of inertia of the gsb in

each case has been extrapolated from the spin 12+ gsb state since the mixing matrix elements are small, and this state (and lower states) are negligibly perturbed by the band crossing. Mixing matrix elements


in the band-crossing region have been found to be ~ 38 keV in Er

[Le 76], ~ 24 keV in 154Gd [Kh 73a] and ~ 9 keV in 156Dy [An 74]. The moments of inertia of the unperturbed bands in 174W are illustrated in figure 5.14, together with the moment of inertia of the mixed yrast band, after mixing the two bands with matrix elements of 40 keV and 20

keV. The agreement between this simple "theory" and the experimental

backbending is good for the weaker mixing. 172

The results for Hf with the same 20 keV matrix elements are

also shown in figure 5.14. The significant rise in the experimental

moment of inertia at high spin is reproduced by the calculation, along

with the change in slope at the 22+ 20+ transition. It is emphasised

that the calculation was originally performed before the experimental observation of the 22+_>20+ transition, and that the predicted

22+ 20+ energy (694 keV) was within a few keV of the subsequent

experimental value (699 keV). Indeed, the multiplicity filter

(described in section 2.5.3) was used specifically to extend the known

1 7 9 + -f

yrast band in Hf, in order to test this prediction of the 22 -> 20


It is further noted that the highest levels (20+ and 22+) in 17?

Hf are in this way found to be predominantly of the configuration

of the high moment-of-inertia crossing band. This association is in

agreement with the results of the two-quasiparticle-plus-rotor

experiment 40 keV matrix elements 20 keV •• postulated 2 q p b a n d extrapolated gsb ( t w ) 2 (M e V 2 ) 1 7 2 1 7 4

Figure 5.14: Moment-of-inertia diagram for Hf and W

illustrating the results of the band-crossing model described in the text. A postulated upper band (O) is mixed with an

extrapolated gsb using 20 keV (x) and 40 keV (+) matrix elements. The experimental values (•) are included for comparison.

states are two-quasiparticle states (see figure 5.13), while the lower states are members of the g s b .

The success of this simple approach for the two isotones, 174W and 172Hf, is encouraging. For these nuclei, the intrinsic neutron states should be similar, but application of the same prescription to

172W does not give as good agreement with experiment, and may be a result of the different intrinsic structure of the neutron states. The important role of CAP effects in 172W has already been discussed. These effects are again apparent in the rising moment of inertia of

the negative-parity yrast states, eompared to the more constant values at high spin in the other nuclei (sec figure 5.9). However, the

change of slope in the moment-of-inertia curve of W, at the

20+ ~> 18+ transition, does suggest that a crossing band has been reached in these positive-parity yrast states.

It is again emphasised that the details of backbending are

sensitive to the underlying intrinsic particle configurations, and in particular to the excitation energy and moment of inertia of the crossing band, as well as the moment of inertia of the gsb. The observation of extensive quasiparticle bands in both even-even and odd-mass nuclei gives the hope that the parameters of more detailed calculations may ultimately be fixed, with a consequent improved

understanding of both these parameters and the rotating nucleus itself.

Before concluding, it should be mentioned that band crossing is not confined to positive-parity yrast bands in even-even nuclei (which may even have double crossings, as in 158Er [Le 77]) and single-quasi-

particle bands in odd-mass nuclei, as considered above; band-crossing

anomalies are also found in 3-vibrational [Kh 73a, An 74] and y -

vibrational [Ki 78, Jo 78] bands. The present results indicate

similar changes in octupole-vibrational bands, but not a clear

transition in any individual nucleus. The systematics through the.

tungsten isotopes indicate that the low-spin structure results from the 2 octupole vibration, which is well established in the heavier

tungsten isotopes. The nucleus W presents an interesting case

because its moment of inertia is initially low, but is rising

significantly with spin and may approach the decoupled limit above the

presently known states (12 or possibly 13 [Do 76]). It is

particularly interesting because this octupole band is known from its 2 band head upwards, and both even-spin and odd-spin states are

populated. The experimental observation of higher-spin states in this well-deformed nucleus would appear to be valuable.


The nuclear states observed in this study have been selectively

favours the states with high angular momentum, and it has been possible to see the different ways in which the nucleus is able to accommodate this angular momentum. At high spins the favoured

configurations are ones in which the particles align themselves with the rotation, thus contributing a large fraction of their intrinsic

angular momentum to the total. The deformation-coupled configurations,

although significantly populated in some of the even-even nuclei studied, are unable to accommodate high angular momenta in such an efficient manner.

Two of the deformation-coupled bands in Hf (6 and 8 ) have

been found to fit well with the systematically observed isomers in the heavier hafnium isotopes. Other deformation-coupled bands are

observed (in the even-even nuclei) but their weak population makes it difficult to obtain spectroscopic information.

1 72

Rotation-aligned bands are observed in each of 172'174'176^ and Hf, and they may be interpreted within the particles-plus-rotor

l 7 3.

model. When taken with the results for the odd-neutron nuclei *'“W

17 5

and W, a dominant role of the i^ ^ neutrons in the decoupling is

found. The anomalous behaviour in the yrast bands of 174W and 17CW is

explained on this basis, while the yrast band of 1 7 2Hf behaves

c o n s i s t e n t l y w i t h its e x c i t e d - b a n d struc tu re, a l t h o u g h it d oes not b a c k b e n d like its i s o t o n e 174W. The n e g a t i v e - p a r i t y y r a s t b a n d s in

1 7 2 , i 7 4 , i 76W ancj 172Hf h a v e a d e c o u p l e d s t r u c t u r e c o n s i s t e n t w i t h the

rotation-alignment model, and this is seen to change to a more regular structure based on the 2 octupole vibration in the heavier tungsten isotopes.

In document High-spin states in well-deformed nuclei : rotational bands and quasiparticle structure in ¹⁷²,¹⁷³,¹⁷⁴,¹⁷⁵,¹⁷⁶W and ¹⁷²HF, observed following (¹⁶O,xn) reactions (Page 174-179)