The cranking model applied to Yb bands and band crossings

Nuclear Physics A347(1980)141-169. @North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission

from the publisher.

THE CRANKING MODELAPPLIEDTO Yb BANDSAND BAND CROSSINGS L. L. Riedinger Department of Physics The University of Tennessee Knoxville, Tennessee 37916

In-beam Y-ray spectroscopy experiments have been performed on 160,161,162Yb in Copenhagen and on 16% in Oak Ridge. Fifteen rotational sequences (bands) were observed, most of them beyond I =20. Thirteen band crossings (upbends or backbends) were found, and these are classified according to the frequency and the gain in aligned an lar momentumin the crossing. Six negative-parity side bands in l6 %”+162Yb and the ils/, band in lslYb experience a similar crossing around w = 0.36 MeV/h, a value intermediate to the frequencies of the first and second yrast backbends in 16%b. Cranking model calculations are used to suggest quasiparticle assignments for the bands and to explain the nature of the three basic types of band crossings observed in our measurements.

One of the major discoveries in nuclear physics in the past decade was the observation of the backbending phenomenon in many deformed nuclei, especially those in the rare-earth region. That is, a discontinuity in the normal rotational pattern is often observed for states of angular momentum, I, around lo-14 h. As a result of a few key blocking experiments on bands in adjacent odd-A .nuclei [ 1,2] and a great deal of theoretical work [e.g., Stephens and Simon 133; Mang, Ring and colleagues; Bohr, Mottelson, and coworkers], we now understand this backbend in the A % 160 region to result from the selective rotation alignment of a pair of i13,.!2 neutrons. In other words, a crossing between the ground-state rotational band and a 2 quasiparticle (q.p.) band of larger effective moment of inertia takes place around I = 12 h, changing suddenly the nature of the yrast line. But, band crossings should not be symptomatic of only i13/2 neutrons and only the ground-state band. Indeed, there should exist near the yrast line many 2 q.p. bands, most of which would experience band crossings due to the alignment of i13i2 neutrons or other high-j quasiparticles near the fermi surface, e.g., h9/2 neutrons, h9/2 or hll 2 protons. Likewise the yrast line should experience more band crossings at hig her spins and rotational frequencies as a result of other Backbends or upbends in side bands in deformed nuclei have rarely particles. been observed previously. Several of these few cases have involved vibrational This paper bands, e.g., the y band in 164Er [4,5] and the B band in ’ 56D [6]. describes recent in-beam spectroscopy experiments on 160,161* r 62Yb, where we observe many bands and band crossings, and the theoretical analysis ofthese effects using cranking-model calculations emphasized in the last few years b the A preliminary report of this work on 16’* r 61Yb has Copenhagen school [7-lo]. been recently published [ll] . Using this example of a new set of rather complete data and concurrent calculations, I hope to demonstrate the power of this theoretical approach in classifying and understanding band crossings in rotational nuclei.

142

L.L. RIEDINGER

The work described here is the result of several large collaborations, and the people involved are listed in Table 1. Most of the experimental and theoretical work was performed at the Niels Bohr Institute; my participation in this effort helped to provide a perfectly delightful sabbatical year. More recently, we have performed follow-up experiments at Oak Ridge in order to observe higher spin states in the 160Yb bands. The large amount of effort by all of my colleagues is gratefully acknowledged. TABLE 1.

Participants in work described in this paper.

Niels Bohr Institute Work

Oak Ridge Work

Ole Andersen Stefan Frauendorf--Rossendorf,GDR Jerry Garrett Jens-JBrgen Gaardhaje Gudrun Hagemann Bent Herskind Yuri Makovetzsky--Kiev,USSR Lee Riedinger--Univ. of Tennessee Magne Guttormsen Univ. of Oslo, Per Tjam 1 Norway Jim Waddington--Univ. of Bergen, Norway and McMaster Univ., Canada

David Haenni Sven Hjorth--Stockholm Noah Johnson I. Y. Lee Paul Luk--Univ. of Tennessee Lee Riedinger--Univ.of Tennessee Russell Robinson

Central to the understanding of rotating nuclei is the ability to fingerprint or classify a particular band crossing and thus hopefully learn the physical phenomenon which produces that change in the normal rotational mode of a nucleus. The classical backbending plot is illustrated in the top part of Fig. 1, a plot of effective moment of inertia versus the square of the rotational frequency. The crossing between the ground and S (super bands) is signified by a large increase in 9. Perhaps a better way to demonstrate this crossing is in a plot of spin of the rotational level, I, versus Rw. The S and ground bands do not really differ in collective moment of inertia, but rather in angular momentum for a particular rotational frequency. This difference is defined to be the aligned angular momentum, i, of the S band relative to the ground band. The crossing of two bands results in a certain measurable increase in alignment, Ai, and the crossing occurs at a certain rotational frequency, wc. These two quantities provide the observable fingerprint of a band crossing and are calculable in the cranked Nilsson approach described briefly in this paper and elsewhere [7-lo]. The rotation alignment of different particles is generally predicted to result in different Ai and wc values, which provides a way to deduce the origin of an observed band crossing.

EXPERIMENTS - 160,162Yb - NBI Most of the spectroscopy experiments relating to this paper were performed at the Niels Bohr Institute Tandem Accelerator using (160,3n) reactions, as described in Table 2. Quantity and quality of the data are both very important in trying to observe weakly populated side bands in heavy-ion induced reactions. The quality is achieved by using discrete NaI counters as a y-ray multiplicity filter or a large NaI annulus as a total y-energy (sum) filter. One can thus discriminate against events which have low angular momentum or low total energy input, for example those resulting from unwanted (160,4n) reactions. The total-energy filtering technique is demonstrated in Fig. 2, which displays the total s ectrum for a certain experiment in addition to those parts connected with the (lg0,3n) and (160,4n) reactions channels. The higher sum energy for the 3n channel allows one to use a simple digital or analog threshold to discriminate against the 4n channel.

TEE CRANKINGMODEL APPLIEDTO Yb BANDS AND BAND CROSSINGS

143

of the ground-Sband crossing; i is Fig. 1. Frequencyclassification the alignedangularmomentumof the S band.

TABLE 2. Experiments performed. Accelerator: Reactions:

NBI Tandem

ORIC

147~(160,3n)160~ 148Sm(160,3n)161Yb 14gSm(160,3n)162Yb 80-84MeV

144~(zoNe,4n)160~ 108 MeV

Measurements: Coincidence(4 Ge's) AngularDistribution Polarization ConversionElectrons Filter Techniques:

(1) 20 cm x 25 cm NaI Sum Crystal (2) MultipIicity Counters

Coincidence(5 Ge's)

25 cm x 25 cm NaI

The y-y coincidencedata in Copenhagenwere accumulatedwith an array of 4 Ge(Li) and 3 NaI counters. Three detectorswere requiredto fire to allow Ge-Ge events word for these 7 counterswas also being writtenon tape. The multiplicity recordedso that playbackin differentfolds could be performed. A large number of tapes were filledin each experiment, resulting,for example,in h 240 million

144

L.L. RIEDINGER

TOTAL SUM SPECTRUM

-0

240

CHANNEL

560

NUMBER

Fig. 2. Spectra from a NaI total-energy detector, demonstrating portions due to the (160,3n and 4n) reaction products.

Ge-Ge pairs for the 16% measurement. This experiment was especially successful in populating high-spin yrast states, as demonstrated in Fig. 3. This figure contains parts of 8 Y-ray coincidence spectra resulting from gates on individual transitions. The first yrast backbend occurs at I = 12, and a second backbend is observed at I = 28, as was first reported by Beck et al. [12]. The usual problem --. in this type of data is trying to assign the next transition, here the 30+ + 28+ line. The 865.5-keV peak was suggested for this transition in Ref. 12, but the spectra in Fig. 3 indicate that this line feeds the 16+ level. Later in this paper, we will discuss higher yrast states from the Oak Ridge measurements. The level scheme of 160Yb is shown in Fig. 4. Many experiments were performed in order to deduce the spins and parities of the observed levels. Both singles and multiplicity-gated angular distribution data were acquired, but y-ray polarization and conversion-electronmeasurements were necessary to deduce the parities of the side bands in 160Yb. The measured K-shell conversion coefficients for the 744.3and 780.1-keV lines indicated El transitions requiring negative parities for bands 2 and 3. The angular distribution of the 436.1-keV line indicates a mostly quadrupole 8 + 7 transition, thus requiring negative parity also for band 1. The level scheme resulting from our measurements on 162Yb is shown in Fig. 5. The spin and parity assignments were made from angular distribution measurements on Pb-backed targets. The two negative-parity bands seem to be similar to some of the 3 or 4 negative-parity bands observed in 160Yb. In contrast, the analog of the odd-1 positive-parity band in 162Yb was unobserved in 160Yb. Another difference between these two cases is that the yrast line is observed up to I = 28 in 160Yb, but only 22 in 162Yb. Such a difference in population pattern must be related to structure effects.

EXTRACTION OF PERTINENT VALUES FROM DATA - 160,162Yb Our measurements have thus led to the assignment of 8 non-yrast rotational cascades in 160,162Yb, 6 up to I > 19. We thus have a unique opportunity to analyze

THE CRANKINGMODEL APPLIEDTO Yb BANDS AND BAND CROSSINGS

subtracted)from Fi8' 3. Coincidenceyrast gates (background (I 0,3n)16*Ybexperimentin Copenhagen.

145

the

these bands for evidenceof alignmenteffects. As discussedabove,an important quantityis the alignedangularmomentum,i, of each of these excitedbands. The extractionof i from the data is more completelyillustratedin Fig. 6. In this figure,Ix is the angularmomentumalong the rotationaxis and is given by Ix m [(I * l/Z)*-K21fi2.

L.L. RIEDINGER

146

,7-

Fig. 4.

j

4432.0

Level scheme of 160Yb from (160,3n) measurements [ll].

Not knowing the bandheads for the non-yrast cascades in 160,162Yb, we assyme R = 0 for all. The rotational frequency is extracted from the transition energies, dE W=aI,=

E(I + 1) - E(I - 1) = + bEy. I,(1 + 1) - I,(1 - 1)

THE CRANKING MODEL APPLIED TO Yb BANDS AND BAND CROSSINGS

(22')

19'

Fig. 5.

56625

22-

147

58134

62610

Level scheme of 16%

from (160,3n) measurements.

The angular momentum of the excited band relative to the ground configuration at a certain w is i=IxI xg' From Fig. 6, one can see immediately that i - 10 for the S band (the 14 -t 12 and 4 + 2 transitions have nearly the same energy!) and i E 8 for side bands 1 and 2. Such highly aligned side bands have rarely been observed previously. In extracting i values for the excited bands, we find it necessary to use a VMItype description of the ground band rather than the ground band itself [lo],

Ixg = a.0 + &~W2)lo, where do and &I are parameters for this Harris formula [13]. This is necessary for two reasons. First, the ground band is not definitely knownaboveho=0.32 MeV,

148

L.L. RIEDINGER

I

’

160Yb YRAST

I

)

I

g

I 26’

/

8AND

BAN0 I BAN0 2

f Y

Fig. 6. Angular momentum along the rotation axis versus rotational frequency for some bands in 160Yb.

and even the IO+ level position may be affected by the i~inent band crossing. Secondly, in this region of rather small deformations, the ground and excited bands likely have different deformations and collective moments of inertia. We are-concerned mostly about the nature of the excited bands, so we chose a reference (ground) configuration, Ixg, that more closely approximates the collective $.of the excited bands. We use $1 = 90 MeVT3 and $0 = 16, 18, and 20 MeV-1 for '60,161~162Yb,respectively. The resulting alignments for the bands in 160Yb are graphed in Fig. 7. Side bands 1, 2, and 4 have rather constant i values down to the lowest observed frequencies. In contrast, band 3 begins with rather low i but rapidly gains alignment. It is possible that this band begins as a K = 0 octupole structure but then changes into a 2 q.p. band as an i13j2 neutron aligns along the rotation axis. Similar changes or crossings are observed at I = ll- in the N = 88 nuclei [14]. Of interest, of course, is the quasi-particle composition of these side bands, a point which will be addressed in the calculations. Rut a very striking feature of Fig. 7 is the fact that side bands 1, 2, 3, and 4 all experience a sudden increase in i (i.e., an upbend) around w = .36 MeV/h. This common crossing frequency is clearly intermediate to the frequency at which the first (% 0.27 MeV/h) and second (% 0.43 MeV/h) yrast backbends occur. Even band 5 may be a manifestation of the same crossing. That is, if the 763.9-keV 12+ + lO+ transition is interpreted as a continuation of the ground band beyond the first backbend, then it appears that the ground band immediately experiences another crossing around ho = 0.35 MeV. It is then a challenge to pinpoint the alignment process which apparently affects bands 1-5, but not the S band.

THE CRANKING MODEL APPLIED TO Yb BANDS AND BAND CROSSINGS

149

Fi 7. Aligned angular momentum versus frequency for bands observed in l6%* Yb, using a0 = 16 MeV-1 and $1 = 90 MeVm3 in the parameterization of the ground configuration.

CROSSING FREQUENCIES Before discussing the calculations, it is perhaps good to illustrate more clearly the importance of describing an observed band crossing as a crossing in rotational frequency rather than angular momentum. The level energies are plotted versus the level spins for 160Yb in Fig. 8. The crossing between the ground and S bands

I

I

I

ML/n BLlc.o.335

,a

*t

i =7.2h

”

7

4

3

2

I

0

0

-2

5

IO

15

20

25

LEVEL SP!N (h) Fig. 8. Level energy versus spin for yrast band and band 5 in 160Yb. Note crossings at two distinct frequencies.

L.L. RIEDINGER

150

(backbend) occurs near I = 12. It may be difficult to calculate the angular momentum at which this crossing occurs in view of the interaction between levels of the same spin in the two bands. However, the frequency at which the crossing occurs is specified by observable levels which differ in spin by the alignment gain in the crossing. In Fig. 8, the rotational frequency is essentially the instantaneous slope of the parabola for each band. The line of common tangents between the parabolae for the ground and S bands in the lower part of Fig. 8 then defines the crossing frequency wc = .27 MeV/h. Of course, a physical crossing between the bands cannot occur there, since the bands differ in angular momentum by 10.1 h at this common frequency (this difference in I is the alignment, i, of the S band). Instead, the nucleus gains I (and w) until the spins of the two bands are comparable, at which point the band crossing occurs. The upper part of Fig. 8 illustrates the higher frequency crossing possibly experienced by the ground band. As discussed in the previous section, the 763.9-keV line could be assigned as the 12+ + lo+ transition within the ground band, and so the 3139.1-keV level is plotted in Fig. 8 as the 12+ member of the ground parabola. The higher levels in band 5 (see Fig. 4) then indicate a crossing with another band, as is apparent in Fig. 8. The common tangent between these 2 curves yields wc = .34 MeV/h. The frequency of a particular band crossing can be more easily deduced from the plots in Fig. 9. In the top part of the figure, the average level energy is plotted versus frequency, where E(1) = ; (E(I+l) + E(I-1)) for the I+1 + I-l transition. The two crossings experienced by the ground band are quite obvious, but the precise crossing frequencies are somewhat difficult to extract. The g band in Fig. 9 is not a real fit to the ground-state band but functions as the reference configuration. It is drawn according to the Harris expansion and corresponds to a collective moment of inertia more similar to that of the 2 q.p. bands. The value of the crossing frequency can be better defined if one plots the level energies in the rotating frame. As discussed in Refs. 7 and 10, the energy of rotation is WI,, and so the energy in the rotating frame can be written as E' = E - WI, for the observed levels and as

for the reference configuration (g band). The S band has a larger Ix than the g band (that is the alignment of the S band); the larger wIx for the S band results in a crossing of the S and g bands at Rw = 0.28 MeV. We will discuss later the calculations [lO,ll], where it is convenient to predict the q.p. energies relative to the ground configuration. We can also quote the experimental energies in a similar way, et

=

E’

-

E

’ g

where et is the energy in the rotating frame relative to the-g band. In the e' vs. w plot in Fig. 9, the alignment of a band is the negative of the slope, i = -de'/dw.

CALCULATIONS The abundance of observed side bands in 160s162Yb emphasizes the need for detailed calculations of their quasiparticle components and the causes of their crossings.

THE CRANKING MODEL APPLIED TO Yb BANDS AND BAND CROSSINGS

151

Fig. 9. Ground, S, and #5 bands in 160Yb plotted in terms of average level energy (E), absolute energy in rotating frame (El), and relative energy in rotating frame (e') versus frequency.

A successful recent approach has been the cranked-Nilsson calculations of the quasiparticle energies in the rotating frame of the nucleus relative to the ground configuration. Bohr and Mottelson [7] described this approach in analyzing the bands found in 164Er,Bengtssonet --*al [8] discussed the oscillating nature of the sharpness of the Yb backbending nuclei, and Bengtsson and Frauendorf [lo] have systematicallydemonstrated the success of the calculations for odd-A Er and Yb nuclei. The single-quasiparticleHamiltonian in the rotating frame is written as [7-lo] H' = H - I&,, where the Hamiltonian in the laboratory frame, H, contains the Nilsson single particle terms, a term for BCS pairing between the quasiparticles, and a term to conserve the expectation value of the particle number. This H' is generally called the Routhian. One normally performs a theoretical calculation with this Hamiltonian in an HFB approach, employing a self-consistentdetermination of the parameters (deformationsand strength of the pairing field). Such a self-consistent cranking calculation can be very difficult, and so several variations to the usual approach are adopted here. A self-consistentcalculation is not performed; instead, the parameters are fixed at certain realistic values. Also, absolute q.p. energies are not calculated, but only q.p. energies in a rotating frame relative to a reference configuration. Comparisons to experiment are made by

152

L.L. RIEDINGER

transforming measured level energies and spins to the rotating frame, instead of converting the calculated q.p. energies to the laboratory frame. In the calculations described here, the deformation parameters, cp and ~b, correspond to the equilibrium values calculated by the Strutinsky method [13]. These deformations, the pairing gap, A, and the chemical potential are all assumed to be independent of w in the calculation. The data analyzed by Bengtsson and Frauendorf [lo], in addition to those presented here, support the assumption of substantial pair correlations up to the rotational frequencies observed in this work. Furthermore, the calculations of Neergzrd --* et al [16] and Anderson -et al. [17] (without pairing) indicate a fairly constant shape up to I % 30. The calculated quasi-neutron energies in the rotating frame e', are plotted as a function of hw in Fig. 10 for parameters representative of 16Oyb . The 4 different

hw (r&v) Fig. 10. Cranking model calculation [ll] of e', the neutron q.p. energy in the rotating frame, vs. hw. The parameters used in this calculation are ~~ = 0.2, E4 = -0.02, A = 1.06 MeV, A = 6.38 Awe, and N = 90. The curves labeled A, B, C, and D represent the il,/, states, those labeled E, F, G, and H are N = 5 levels. The solid lines correspond to CUT = + 1/2+, dotted lines - l/2', dashed lines + l/2-, and dashed-dotted lines - l/2-.

types of curves in Fig. 10 refer to a = + l/2, IT= f, as defined in the figure caption, where the signature, a, is defined such that the level spin is a plus an even number [9,10]. The more important levels are labeled by letters for ease of discussion. At w = 0, A and B correspond to 0:= + l/2 and - l/2, respectively, for the 3/2[651] Nilsson level, while E and F are the 2 signatures of the 3/2[532] level. The slope of a q.p. level in this figure is related to the alignment, i = -de'/dw. The largest slopes are obviously those correspondingto the i13/2 orbits. In classifying the configurations in Fig. 10, we use the occupation-numberrepresentation of Ref. 10, which is especially suitable for crossings between levels of the negative and positive energy regions. As discussed in Ref. 10 in detail, this representation deviates somewhat from the familiar representation for fermions.

THE CRANKING MODEL APPLIEDTO yb

bms m

BANDCROSSINGS

153

Here we restrict the number of possible configurations by the rule that, if a A) is occupied, then the conjugate level (-A), obtained by reflection level (e.g., about e’ = 0 and by changing CLto -a, must be free. Therefore, half of the levels are always occupied, and a q.p. excitation corresponds to occupying a level at ei and freeing its conjugate partner, with a resulting change of energy of ei. At the lowest frequencies, the ground band corresponds to the configuration where all e’ > 0 levels are empty and all e’ < 0 levels are filled. A 2-q.p. i13i2 band AB, formed by p1acingoneq.p. in A and one in 6, has a = + l/2 - l/2 = 0 (even spins) and lowers in energy, as the nucleus rotates and gains alignment due to the Coriolis interaction. At hw = 0.23 MeV, the AB configuration crosses the ground band and becomes the vacuum state. It contains two i1s/2 particles which have become unpaired, with a resulting angular momentumof approximately 10 ?I along the x-axis. The observed bands in 160a162Yb are 2 q,p. bands, and must be compared with predicted energies resulting from the combination of two curves in Fig. 10. Before one can confidently consider 2 q.p. bands (and their crossings), it is important levels in odd-N nuclei in this region. to consider measured single-q.p. 161Yb EXPERIMENT - NBI In order to probe the origin of the three distinct crossing frequencies in lfioYb, we performed in-beam y-ray measurements on 161Yb through the 1’t8Sm(160,3n) reaction. The experimental arrangement was similar to that described for 160Yb. The results of the Y-y coincidence and y-ray angular distribution experiments are shown in the level scheme in Fig. 11. The yrast band is assigned as the favored part of the i,s/, band, based upon the systematics of similar bands in 163*165Yb [18,2]. Members of the yrast band up to 37/2’ had been reported previously by Hershberger -et -* al [191. The Iground state of 161Yb is assumed to have Ia = 3/2- according to the spins of the 1 3,165Yb isotopes [18] and the expected trend of neutron states in this region. A long string of stretched E2 transitions is observed and assigned as the ground state band. The structure of the low-spin part of this band and its connection with the excited i1sj2 band was deduced by a coincidence experiment involving an x-ray counter. The 232.3-keV transition in the ils/z band is in delayed coincidence with the 44, 67, a half-lifeof20-3Onsec. We assume the pres100, and 110 keV lines, indicating ence of an unobserved low-energy E2 (13/2+ + 9/2+) transition to explain the observed lifetime and to complete the path between the 3/2- ground state and the lowest observed (13/2+) state in the yrast band. The highest members of the observed bands have been assigned from a very recent multi-detector experiment of Gaardhaje -et al. [20], one which resulted in better statistics than measurements which we performed earlier. COMPARISON To CALCULATIONS - 161Yb The first question to consider is: how do the observed bands in 161Yb compare To facilitate this comparison, we have with the one-q.p. calculations of Fig. lo? plotted the yrast band in addition to bands 1, 2, and 3 (ls1Yb) in the Fig. 12 graph of e’ vs. hw. The yrast band and band 1 both have positive parity and look unmistakably like the favored and unfavored members of the ils/z band, levels labeled A and B in Fig. 10. The slopes of the curves (alignments) and the energies, e’, for these two observed bands (Fig. 12) agree rather well with the predicted values (Fig. 10). As listed in Table 3, the observed alignments (i) for the yrast band and band 1 are 5.7 and 4.4 h, respectively (at Rw 2: 0.25 MeV) , The calculations thus do whereas the cranking calculations yield 5.8 and 4.1. Success is also well at explaining the bottom parts of the observed iIs/ bands. achieved in explaining band 2, the ground band at very low frequencies but not

154

L.L. RIEDINGER

Fig. 11.

Level scheme of 161Yb from (160,3n) measurements [ZO].

yrast for I > 13/2. The lowest predicted negative parity band has a = + l/2 and i = 3.0 PI. Band 2 has a = + l/2 and i = 2.5 h, and so it assigned as configuration E, the mixture of the 3/2 [521] and 3/2[532] Nilsson states. The other signature (a = - l/2) of this configuration, labeled F in Fig. 10, is unobserved experimentally at low spin, but may correspond to band 3 at higher spins.

THE CRANKING MODEL APPLIED TO Yb BANDS AND BAND CROSSINGS

155

16’Yb

ha (MeV) Fig. 12. Experimental energies in the rotating frame vs. AU for bands in 161Yb. The dashed lines mark the crossing frequencies.

The experimental alignments, i, are plotted versus frequency in Fig. 13 for 161Yb. As can be easily seen in this alignment plot or in the level spacings in Fig. 11, a strong backbend is observed in the ground band (band 2). A similar effect was observed in the negative-parityband in 163Yb [18] and in 165Yb [2,18]. The frequency of this obvious crossing in 161Yb is seen to be 0.24 MeV/h in Fig. 12, similar to the frequency (0.27) of the first yrast backbend in 160Yb. The alignment gain in the crossing is measured as 8.6 h in 161Yb and 10.6 h in 160Yb. The yrast band (A configuration)has no backbend at that frequency, but instead experiences a strong upbend at hw = 0.35 MeV, resulting in a Ai of approximately 6.5 h. The phenomenon of the i1si2 band not backbending like the even-A ground bands has been seen in many cases in the light Sm-W region (for example, in 165Yb [2], l'j3Yb [18], and several Er isotopes [21]). This has previously been explained [21] as the result of a blocking effect; that is, the alignment of 2 i13j2 neutrons, which is responsible for the first yrast backbend in even-A nuclei, is blocked by the presence of the valence neutron in an i13/, Nilsson orbital in the adjacent odd-A nucleus. The blocking has been previously observed, but in lsl~b we observe for the first time a delayed, but complete, upbend of the i13i2 band. Looking back to Fig. 7, the experimental i vs. w plot for 160Yb, one realizes that the upbend experienced by the yrast band in 161Yb occurs at the same frequency as the crossings seen in the 5 side bands in 160Yb. The nature of such a general crossing is not immediately obvious in the previous blocking picture, and so we

156

L.L. RIEDINGER

TABLE 3.

Summary of quasiparticle assignments and band crossing information. Crossing Band

Confa

a~'

i(h) Calc EXP

Conf

wc(MeV/h) Calc EXP

Ai(W Calc EXP

5.7 4.5 2.5 ?

5.8 4.1 3.0 2.0

ABC BAD EXB FB -

.36 .33 .24 ?

.36 .37 .23 .23

6.5 ? 8.6 'L9

6.6 7.2 9.9 9.9

10.6 8 7.5 7.0 6.5

9.9 8.8 7.8 7.1 6.1

AB x&J AEBC AFE BEm BFE 5:Bc -

.27 .42 .38 .37 .33 .35 .34

.23 .44 .36 .36 .37 .37 .36

10.6 5.5c 6.6C ~6~ ? ? 7.6

9.9 6.0 6.6 6.6 7.2 7.2 6.6

8.4 7.9 8.5

AB EBC AFBC -

.31 .33 .30

.23 .33 .33

10 ? ?

9.9

6.5 7.5 6.8

161yb

A E E F

+ + -

l/2+ 1/2+ 1/21/2-

16Oyb

GSB S:AB 2:AE l:AF 4:BE 3:BF GSB

0+ o+ loOlo+

162yb

GSB AE AF AC

o+ lo1+

7.0 7.0

aConfigurations for each nucleus listed in order of increasing energy. A and B: 3/2[651]; E and F: 3/2[532], 3/2[521]. b The signature, CI,is defined such that I = a + an even number 1101. For a multi-q.p. state, a is the sum of l-q.p. values; c1= 0 and 1 correspond to even and odd spins. 'Deduced from Oak Ridge measurement.

Fig. 13. Aligned angular momentum vs. hw for 161Yb, using do = 18 MeV-' and $1 = 90 MeVm3 in the parameterization of the ground configuration.

THE CRANKING MODEL APPLIED TO Yb BANDS AND BAND CROSSINGS

157

must consider an explanation involving the cranking calculations displayed in Fig. 10. A simplified version of the calculated e' vs. w plot of Fig. 10 is shown in Fig. 14 for only the positive-parity quasi-neutron states. In an even-even nucleus, the 2 q.p. il,/, Nilsson states, (3/2[651])2,is shifted up above the vacuum by the pairing gap. However, as the nucleus rotates, the core iI,/, particles begin aligning their angular momenta along the rotation axis until the pairing correlations for two il,/, neutrons go to 0.

UC=.23

36.37

hW(Mev) Fig. 14. Cranking model calculation identical to that presented in Fig. 10, except that here only the il,/, states are included in order to illustrate the crossings between levels.

At hw = 0.23 MeV, levels -A and -B (filled in the ground band) meet A and B (empty) and there is a crossing, as a q.p. is transferred from -A to B and from -B to A. As discussed above, this crossing occurs as a result of the alignment of the most favorable air of i13i2 neutrons, AB, with i - 10 h. However, the situation is changed in 1% lYb when a q.p. is present in A (the yrast ila/z band). There can be no band crossing at 0.23 MeV, since A and -B are filled and B and -A are empty. This crossing is blocked, and the rotation continues. However, at hw, = 0.36 MeV, the filled level -B crosses with the empty level C (an a = + l/2 crossing), and the empty level B crosses with a filled -C (a = - l/2), resulting in an alignment of the BC pair from the core with Ai = 6.6 h. We write this as the A + ABC crossing. On the other hand, the AB crossing is not blocked by the presence xthe valence neutron in the E level, and so the negative-parity band in 161Yb backbends at around 0.23 MeV, i.e., E + EAB. This feature of two distinct crossing frequencies associated with pair breaking and alignment of 1131~ neutrons is actually a general phenomenon which is operational for other values of the fermi surface (OS nuclei, for example) and for other particles. The S band (AB) is the lowest energy, most aligned a = 0 state. However, these calculations demonstrate that one can make higher energy a = 0 states using less aligned single q.p. excitations, e.g., BC and AD. Thus, in an odd-A nucleusrthe A Q 160 region, every single-q.p. band will experience a band crossing due to the alignment of ila 2 neutrons, although the frequency at which this crossing occurs depends upon wh. ich pair is aligned, i.e., AB, BC, orAD.

158

L.L. RIEDINGER

In the calculations performed here, the crossing frequency and the alignment gain for BC and AD are nearly the same and thus indistinguishable. The BC crossing frequency is predicted to be slightly lower than for AD. Experimentally, we have little information about this in 161Yb, although there is some indication that the unfavored i,,/, band (B) begins to upbend (B + BAD) at If real, xis would slightly lower frequency than the favored band (A -fABC) indicate an ordering of crossing frequencies opposite"?-othat predicted. As argued for band E above, the unfavored (o = - l/Z) 3/2[521] band (F) should experience the BC crossing. Although the lower part of band F is unobserved, band 3 (see Fig. 11) could be the part of that structure occurring after the backbend (FAB). As seen in Fig. 13, the alignment of band 3 is slightly lower than that ofband EAB (at least for hw a .25 MeV). Also, the energy of band 3 in the rotating frame mg. 12) is slightly higher than that of band 2 (EAB). Both of these facts point to the FAB assignment to band 3 in lslYb. The puzling aspect of this situation is w&band 3 decays to the yrast structure near the F + FAB crossing rather than to the members of band F below the backbend. It may be pozible that the rather large energy (922 keV) available for the 2?/2- + 25/2+ decay allows this transition to dominate a lower energy 27/2- + 23/2- line at the band crossing.

COMPARISON TO CALCULATIONS - 160Yb Now that the lower observed bands and their crossings have been found to compare well with the cranking calculations, it is appropriate to apply the calculations of Fig. 10 to 2 q.p. bands in 16*Yb. The assignments are summarized in Table 3. The lowest predicted 2-q.p. state at moderate frequencies is AB (i = 9.9 PI),which compares well with the observed S band. The next higher-lying 2-q.p. bands should be 4 negative-parity bands: AB (a = l), AF and BE (nearly the same energy and alignment, both a = O), and BF (a = l), that is, four AI = 2 cascades resulting from the coupling of the 3/2[651] and the 31215321, 3/2[521] levels at w = 0. The energies of the observed 2-q.p. bands in the rotating frame are plotted as a function of Rw in Fig, 15 in order to facilitate comparison with the l-q.p. theoretical diagram of Fig. 10. Note the order in energies of the 3 negative-parity side bands at hw * 0.3 MeV: band 2 (a = l), band.1 (a = 0), band 3 (e = 1). The ordering and the i values for these bands (see Table 3) are very suggestive of the predicted bands AE, AP, and BF, respectively. The parity of band4isunknowu, but it has roughly the correct i to be configuration BE. The calculations sug-' gest that AF and BE would be nearly degenerate, but inclusion of a residual interaction could explain the position of BE (band 4) above BF (band 3). Another low-lying level in the calculated spectrum in Fig. 10 is C (as = + l/2+), which is the third of the four il3/2 levels emanating from the 3/2[651] and 1/2[660] orbits. One would expect bands AC (01%= l+, i = 8.2 h) and BC (O+, 6.6 A), only slightly above the observed negative-parity bands. Band 5 is a strong candidate for BC, in view of its alignment (6.6 TX),position (see Fig. 15), and crossing with the ground band (discussed below). On'thk other hand, there is no obvious candidate for AC. Possibly the 4716- and 5370-keV levels (see Fig. 4) could be the 17+ and 19+ members of band AC. In that case, the alignment for the 654-keV transition would be 8 h and e' would be 0.114 MeV, slightly below the position of band 5 at that w. Both values would be consistent with the AC configuration, but it is puzzling why the band is not more strongly populated. Another candidate for AC is band 4, if the odd-spin and positive-parities are chosen. The i of this band would also then be 8 h, and the e' values (see Fig. 15) would decrease by 161 keV for the first transition and 324 keV for the last, thereby occupying a position slightly below band 5.

TBE CRANKING MODEL APPLIEDTO Yb BANDSANDBANDCROSSINGS

159

lsoYb

Fig. 15. Experimental in 160,162yb.

energies

in the rotating

frame vs. hw for bands

One must also consider proton levels in explaining the nature of the observed highly-aligned side bands in 160Yb. In Fig. 16, we have reproduced a calculated proton diagram from Bengtsson and Frauendorf [lo] for a set for input parameters similar to those of 160Yb. The lowest levels with high i result from the 7/2[523] There is small signaturehII/, Nilsson state (i 9 3 for both signatures). and thus one might expect at hi her excitation splitting in this configuration, energy 2-q.p. a = 0 and 1 positive-parity bands with i - 6 in l6 f Yb. We evidently do not observe such bands, and so it seems that the neutrons are responsible for the yrast and near-yrast phenomena below hw z 0.4 MeV. The most striking aspect of the multiple side bands observed in 160Yb is that they all experience an upbend at a rotational frequency larger than that of the first yrast backbend (see Fig. 7). Bands 1, 2, 3, and 4 experience band crossings in the range of hw = 0.33 - 0.38 MeV, which is very similar to the crossing in the yrast il3 2 band in 161Yb (Fig. 13). As discussed above, side bands l-4 in 160Yb are proba b ly composed of one q.p. in A or B and one in the negative-parity level E or F. The AB crossing at 0.23 MeV is blocked since either A or B is filled, but Thus, the upbend the BC or AD crossing takes place around 0.36 MeV (see Fig. 14). in band 2 could be described as AE + AEBC, that in band 3 as BF -f BFAD. The calculations illustrated in Figs. 10 or 14Tndicate that the AD crossingshould occur slightly higher in frequency (.37 MeV) than BC (.36) and result in a greater From the Copenhagen data, we have no alignment gain (7.2 compared to 6.6 h). measure of the Ai for the crossing in bands l-4, but the WC for bands 3 and 4 (AD crossings) are slightly lower than the. values for bands 1 and 2 (BC crossings), predicted in the calculations. In the previous section, a similar possible

160

o mPRMONS. A

l

BENGTSSON AND FAAUENDORF

5.86 ftti,

0.15

-0.10

-0.15

-0.20 0

Fig.

16.

rotating

a01

0.02

m3

0.01

a05

0.06

0.M

Cranking model calculation of the proton q.p. energy frame vs. hw [lo], for A = 5.86 hwo and 2 x 69.

in the

conflict was noted for 161Yb, where the crossing in the unfavored i13i2 band In (B + BAD) may occur at a lower fxequency than in the favored band (A -+ ABC). contrast to this disagreement in the order of the BC and AD crossings inThe Yb order is evidently observed in 18*Os, as discussed by region, the ‘korrect” Frauendorf, May, and Pashkevich l22] and by Rainer Lieder at this conference. There, the a = Ii2 i13/2 band [A) upbends at PW = 0.29 MeV, while the a = - 1/Z band (B) upbends at 0.36 MeV. Another crossing is apparent in band 5, where we have tentatively interpreted the 12’ level as the continuation of the ground band. As seen in Fig. 14, the ground band, after iiw, = 0,23 MeV, would have levels -A and -B filled. fiowever, at hwc = 0.36 MeV, filled level -B crosses empty level C, arrd filled level -C would cross with empty B, resulting 2n a BC alignment. The observed crossing frequency (0.34 Mev/A) and alignment gain (7.6 71) are in good agreement with the expected values (0.36, 6.61. In sununary, the cranking-model calculations can clearly explain the five side Due to the presence of bands in 160Yb and the band crossings seen in each case. 2 i13/2 Nilsson orbital.5 near the Fermi surface, one can observe 2 alignment processes involving these neutrons. The most favorable a = 0 pair of i13/p neutrons,

THE CRANKING MODELAPPLIEDTO Yb BANDSANDBANDCROSSINGS

161

AB, is broken at a rather low frequency (0.27 MeV/h), giving a large alignment increase. At a later frequency (0.36 MeV/h) the next most favored pair of i13j2 neutrons, BC, loses its pairing with a resulting Ai that is smaller. Nearly every other 2-q.p. band in the nucleus should experience one of these basic alignment processes at the appropriate frequency, depending on the q.p. composition of the band and the blocking possibilities. COMPARISON TO CALCULATIONS - 16%‘b The success of the calculations in explaining the patterns of the side bands in l6oYb leads one to look for similar effects in 16%. As shown in Fig. 5, 3 side bands are observed in 162Yb, two of negative parity (o = 0 and 1) and one of positive parity (a = 1). The alignments for the measured bands are shown in Fig. 17 and summarized in Table 3. While there are similarities in these bands, with those in 160Yb, one is struck by the differences: The WI = l+ band in 162Yb has no obvious analon among the 160Yb bands. with la). respect to both its spins and parity and its absence-of a {and crossing up to hw = 0.38 MeV, bands in ls2Yb look somewhat similar to two of those in (b) . The negative-parity 16 OYb, although the crossing frequencies in the former are much lower, The ground-S band crossing ,cc&b compared to 162Yb.

is less

sharp and occurs

at a later

frequency

in

hw (Mev) Fig. 17. Aligned angular momentumvs. hw for 162Yb, using&o = 20 MeV-’ and&1 = 90 MeV-3 for the parameterization of the ground configuration. To account for these differences, we performed calculations using the deformation appropriate for 162Yb. The parameters, pairing strength, and chemical potential calculated results are overall rather similar to those shown in Fig. 10 for 160Yb, although one significant difference is a lower position of the C level from the This lowering results from the changing location of the Fermi suriI,/, orbits. face relative to the 1/2[660], 3/2[651], and 5/2[652] levels and the larger signaThe AC configuration (an = l+) should ture splitting between the C and D states.

L.L.

162

RIEDINGER

then be in the vicinity of the negative-parity AE and AF bands. In the experimental rotating-frame plot in Fig. 15, one sees that the observed CITY = 1+ band lies only slightly higher than the 2 negative-parity bands, and so we assign it as the AC band. This assignment also nicely explains the absence of a band crossing in this excitation, since the AB, BC, and AD alignment crossings are all blocked by the presence of a q.p. in A or C. Indeed, of all the side bands one could reasonably hope to observe in 160s162Yb, this is the only one that would not experience a crossing with the ground band or with a 2-q.p. neutron band of higher alignment, at least up to hw Q 0.40 MeV. Concerning point (b) mentioned above, the 2 negative-parity side bands in l’j2Yb are analogous to the lowest 2 such bands seen in 160Yb. As seen in the rotatingframe diagram of Fig. 15, the CZTI= l- band lies slightly below the aa = 0- band up to nw = 0.29 MeV, suggesting the AE and AF assignments, respectively. It is true that at hw 5 0.24 MeV the alignment of the QII = 0- band is slightly greater than that of the l- band, opposite to the pattern for AE and AF in the calculato argue too strongly on the basis tions and in 160Yb. However, it is difficult of i in view of the rather complicated nature of these bands. For example, the odd-spin negative-parity band seems to cross at low frequencies with some band of robably a Kn = 0- octupole band as we discussed also in the case low alignment, The extraction of i for each band is also complicated by the of band 3 in 1 69Yb. rather early crossings observed at higher frequencies. As listed in Table 3, bands AE and AF experience a crossing with 4-q.p. states involving BC at hw, = crossings occur at .33 and .38 and .37 MeV, respectively, in 160Yb. In contrast, These earlier crossings can be understood in the calculations .30 MeV for 162Yb. earlier to be associated with the lowering of level C in 162Yb and the resulting The BC alignment recess is then predicted to occur crossing with the -B level. at hw, = 0.32 MeV, compared to 0.36 MeV for l6 g Yb. While the above points (a) and (b) can be explained in the calculations 166 point (c) The ground-S band crossing occurs at 0.27 MeV in Yb, but at is very difficult. The differences between wcl and wc2 is obvious for A = 160, 0.31 MeV in 162Yb. The decrease in wc2 is logical in view of the calbut unobservable for A = 162. should have essentially the same value in 160~162Yb. We culations, but wcl One can view this problem in a therefore cannot explain the larger wcl for 162Yb. slightly different way through the plot of the band energies in the rotating frame In le”Yb, there is a gap of e’ ‘L 300 keV between the S band and the (Fig. 15). gap is much smaller. lowest side band at hw s 0.33 MeV, but in l62~b the e uivalent Apparently the S band lies higher in energy (e’) in l %2Yb than in 16’Yb, resulting in a crossing with the ground band at a later frequency. There is no apparent reason for a slightly higher-energy AB configuration in the calculations for 162Yb, The large interaction so we have no theoretical explanation of this effect. in a smooth upbend instead strength between the crossing bands in 162Yb, resulting may have an effect on the position of the S band and thus on of a sharp backbend, It is possible that the detailed theoretical analysis the crossing frequency. described here and the extraction of critical parameters fi. Ai. w-1 from the data between crossing kinds is small, are best performed when the interaction strength as in 160Yb.

EXPERIMENTS -

160Yb - ORNL

important for determining The choice of projectile and reaction channel is rather The measurements with the (160, the range of bands seen in a particular nucleus. 3n) reaction in Copenhagen were successful in that the angular momentum input was low enough to avoid excessive funneling of the population intensity along the Kst and thus to allow observation of side bands, but yet high enough to result The measure in population of rather high-spin members (I % 22) of the sidebands. ments on 160Yb were particularly fruitful in view of the significant population of in observation of the second yrast backbend. the yrast line up to I = 28, resulting wedecidedtoperformmeasurements Following the experiments with 160 in Copenhagen,

163

THE CRANKING MODEL APPLIED TO Yb BANDS AND BAND CROSSINGS

with a slightly heavier projectile at Oak Ridge in order to see if one could increase the relative population of the high-spin yrast and non-yrast states and thus observe discrete lines from higher spin states. We discussed in previous sections the observed side band crossings and their theoretical explanations. While the observed crossing frequencies matched the predicted values for i13j2 neutron alignment, we had no measure of the alignment gain in the crossings since we did not observe a completed crossing (upbend) for any sideband. The hi values, along with the wc, are the critical measured parameters from which we can deduce the quasiparticle contribution to a given band crossing. In a similar vein, the second yrast backbend in lsoyb was observed up to I = 28 (see Fig. 7). As will be discussed in the following sections, the observed wc agrees with that expected for hll/z proton alignment (see Fig. 16), but a measure of Ai in the crossing was et al [12] assigned a 30+ -f 28* not possible from the Copenhagen data. Beck --* transition, but our data indicated that this transition actually populated the yrast 16+ level directly. The necessity of observing higher spin states was clear. In-beam y-ray spectroscopy measurements were performed using the 144Nd(20Ne,4n) 16%~ reaction and a 108 MeV 20Ne beam from the Oak Ridge Isochronous Cyclotron. An array of 5 Ge(Li) counters was used to accumulate y-y events in coincidence with the total energy signal from a 25 cm x 25 cm NaI detector. Absorbers and single channels were used to nearly eliminate events occurring with y-ray energies below Q 380 keV in order to maximize the accumulated rate of data at high spins. In Fig. 18 is shown one of the best spectra obtained from sorting the Ge-Ge data with a cut on the upper half of the observed total energy spectrum. This background-subtractedspectrum results from a gate on the 836-keV peak, previously

“‘Nd ( ‘he, 4 n ) “‘Y b ORIC GATE ON 836 keV

sxl

60.00

so.oa

2io.00

3~0.00

YOO.00 ’ 420.00 CHANNEL

510.00 NIJHBER

t’uo.00

710.00

&lo.00

Fig. 18. Background-subtractedcoincidence spectrum resulting from a gate on the 836-keV peak in the (20Ne,4n)160Ybexperiment at Oak Ridge.

164

L.L. RIEDINGER

thought [11,12] to result solely from the 26+ -f 24+ transition. A striking featufe of this spectrum is that the 836-keV peak is in coincidence with itself, leading to our assignment of this line also to the JO* -f 28+ transition, The 878-keV peak is shown from this and from other data to correspond to the 32+ + 30f transition. Possible higher-energy transitions are still under analysis. A remarkable aspect of this 'ONe experiment is the large increase in population of high-spin states compared to the 160 measurement. This is demonstrated in Fig. 19, a plot of measured Y-ray intensity versus I for the yrast band. The extra angular moments imparted to 160Yb by the (20Ne 4n) reaction results i; a 2-5 fold 1.0 increase in the popuA ("0,3n) "'Yb I 8lMeV I NBI lation of states beyond o ?Ne, 4n)'60Yb,107MeV,0RlC the first backbend. .9 l f"Ar,4n 1‘*‘Er, 166MeV, LBL Also included in Fig. 19 is the intensity -8 pattern along the yrast line of 15*Er measured .7 by Lee et al. [23] in an (40AF4T reaction. It is ve& difficult to 6 compare intensities from $j* different experiments, - .5 since a coincidence with t different numbers of =4 detectors is involved. I-r' This difficulty in normalizing the 160Yb .3 and ls8Er data is obvious at low spins. Of 9 course, it is also difficult to compare .I two different final products, although their yrast properties 0 8 12 16 20 4 24 28 32 are very similar. Nevertheless, one has I the feeling that the 4oAr data are only Fig. 19. Plot of y-ray intensities vs. spin of slightly enhanced in the initial level from 3 different experiments, high-spin po ulation the 160 and 2oNe induced reactions to lC"Yb and compared to g ONe, an the 40Ar induced reaction to 158Er [23]. enhancement certainly much less than that obtained by changing from l60 to 2CNe reactions. Much can be learned in the future about the population mechanism of (HI,xn) reactions to yrast and side bands from such comparisons.

HIGH-SPIN STATES - 160Yb Our measurements at Oak Ridge have led to the assignment of additional high-spin transitions not only in the yrast band of 160Yb, but also in side bands 1 and 2, which are the 2 bands at lowest energy, besides the S band, in the rotating frame plot of Fig. 15. Three more transitions in band 1 lead to states with spins up to 28-, while two higher transitions in band 2 possibly indicate levels of spins 25- and 27-. These new transitions are added to the i vs. w plot in Fig. 20. The backbend in band 1 is then completed and an estimate of the alignment gain in the band crossing is 6.6 h. As listed in Table 3, this Ai and the measured UC

THE CRANKING MODELAPPLIEDTO Yb BANDSANDBANDCROSSINGS

l

YNAST

OeANel A BAN02 X

YRAST

aAND

32’

‘=Yb

i,,

I 15 A BIND

I

1

BAND

P

30’f

161Yb I

__”

165

T i Ai.8.5

I

*I

.2

.3

.4

.5

hw IMeW Fig. 20. Aligned angularmomentumvs.hwforselected bandsin 160j161Yb. This is similarto Figs. 7 and13, exceptthatrecentlyobserved higher-s intransitions havebeenincludedfor 160Yb (Oak Ridgemeasurements) and for P61Yb [20]. agree very well with the values predicted for the rotation alignment of the i13/2 pair BC. This agreement leads to the conclusion that the crossing experienced by negative-parity band 1 can indeed be described as an Al? + AEE process. It is interesting that the last transition observed in band 1 has a frequency essentially equal to that of the second yrast upbend at 0.43 MeV/h. If this yrast crossing is caused by proton alignment, then all neutron q.p. bands should experience this type of crossing at 0.43 MeVIA. Band 1 should have a second backbend, and this discontinuity would probably be evident at the next highest transition Indeed, there must exist a kind of a frequency barrier itt 0.43 (unobserved). MeV/h such that any normally rotating (neutron) band must gain alignment at that point. The higher energy yrast states are also entered in the diagram of Fig. 20. The 32+ transition allows an estimate of the Ai value in the second yrast crossing, ‘L 5.5 ?I. It is true that the alignment of the S band is not constant, and thus The fact that the that the extraction of Ai in the crossing appears difficult. S band is not flat in this graph, while side bands 1 and 2 nearly are, is an indication that the collective moments of inertia of these 2 q.p. bands are slightly different, since one extracts 1 by comparing the 2 q.p. band to a Even.though i is not constant reference defined by certaind10 and &I parameters. for the S band, the I = 16 through 24 points define a trend which can rather safely be extrapolated to hw = 0.43 MeV. A second yrast backbend has now been observed in two cases, ls8Er and 160Yb (N = Faessler and Ploszajczak [24] performed calculations to indicate 90 isotones). for this crossing in ls8Er, as that the alignment of h11j2 protons is responsible Bengtsson and Frauendorf [lo] have reached a similar conclualso in 160Yb [25]. sion, and their calculations for ls8Er are shown in Fig. 16. The hi and wc values

166

L.L. RIEDINGER

predicted [lo] for hll/, proton alignment are 6.0 h and 0.44 MeV/h, respectively, as listed in Table 3. An alternative explanation [lo] involves an h11/2h9/2 proton pair which should yield about the same wc, but Ai = 7.0 h. The alignment gain then is an important value to measure. In 15*Er, the crossing is not sharp due to large interaction strength, and so the extraction of a Ai is somewhat difficult. In 160Yb, the sharper crossing allows a safer extraction of Ai c:5.5 h, in sufficient agreement with the predicted values to indicate that hll/, proton alignment is responsible for the second backbend. Another possibility for an Aw Q 0.42 MeV crossing is the alignment of an h9i2 neutron pair, EF in Fig. 10. However, the recent measurement on 161Yb by Gaardhaje et al. [20] in Copenhagen rules out that possibility. Their placementsofhigher --. transitions in band 3 (see Fig. 11) indicate an upbend around hw = 0.43 MeV, as shown in Fig. 20. Since this crossing has the same wc as the yrast crossings in 15aEr and 160Yb, we are inclined to search for a common cause. As discussed in a previous section, band 3 in 161Yb likely has the FAB configuration, and a crossing with an EF configuration is blocked. The 161Yb thE acts as a probe to eliminate h9/2 neutron alignment as a possible cause of the 0.43-MeV crossing and to then strengthen the arguments for an hll/2 proton effect.

TREND OF SECOND BACKBENDERS Two nuclei, 15*Er [23] and 160Yb [11,121,are now known to display a second discontinuity along the yrast line. These cases, along with other nuclei in this region, are shown in Fig. 21. This plot is similar to that shown most recently

96

Fig. 21. Systematics of backbending yrast bands. See the text for data references. In many cases, the spins and parities of levels at the tops of the bands are uncertain, but such uncertainties have been neglected in this figure.

by Lieder and Ryde [26], and so one can refer to their paper for most of the references for these data. More recent data have been entered for 156Dy from Ward -et al. [27], 160Yb [11,12], 162Yb from our Copenhagen work, le'+~bfrom preliminary results of Johnson &al. [28], 160Er from preliminary results of Ogaza et al [301. et al. [29], and 16aHf from preliminary results of Vervier --* --

THE CRANKINGMODEL APPLIEDTO Yb BANDSAND BAND CROSSINGS

167

Most of the nuclei includedin Fig. 21 are not known to high enoughrotational frequencyto test the occurrenceof a secondbackbend. The Dy, ET, and Yb N = 90 isotonesare known high enoughin frequency,and one sees no crassingin ls6Dy, an upbend in 15*Er,and a slightbackbendin 160Yb. In additionto this obvious 2 dependence,there is evidentlyan N dependence,since f64Ybapparentlydues not experiencea secondcrossingup to (h~)~= 0.20 MeV2. While the crossingdue to alignmentof hII/ protonsshouldbe experienced by essentiallyall nuclei in this region,the interactionof the crossingbands and thus the severityof the backbend will be a sensitivefunctionof the chemicalpotentialand the deformation parameters. Such an effectwas first discussedby Eengtsson,Hamamoto,and Mottelson181 with regardsto the severityof the first backbendin the Yb isotopes,by gengtsson and Frauendorf[lo] for i13j2neutron and hlIjz proton crossings,and very FAESSLER. completelyby PLOSZIZAJCZAK , Faessler,Ploszajczak, and S&mid fZ5] for the secondbackbend. The calculationof Faessler-et al. (251 is directlyreproduced in Fig. 22. Here the interaction matrix elementbetweenthe S band and a band containingan aligned pair of h11i2 protons is plottedversus chemicalpotential. Experimentally, 16% has the smallest interaction between the crossingbands, and Faessleret al. --T 1251demonstratem this calculationthat such a trend can be nicelyreproducedby the choiceof g4 = 0.1. An obviousprediction from this calculation is that the interaction in le2Hfcould be small enoughthat a secondbackbend might be observed. As noted in Fig. 21, nothingis known about these lightHf nuclei.

-2.L

-2R

-1.6 k

-1.2

- 0.6

[M&i

CONCL~SI ONS

In this paper, I have tried to describea largebody of data on multiplebands

Fig. 22. Interaction matrix elementversus chemical potentialfor hl1j2 protonalignment(secondyrast backbend),taken from Faessler,Ploszajczak, and Schmid [25].

L.L.

168

RIEDINGER

and band crossings in the light Yb nuclei fied cranking approach in explaining most 13 band crossings (upbends or backbends) have been able to classify these crossings In this classification, these ment gain. separate types and the cranking calculations the crossing:

and demonstrate the success of a modiof the observed features. We observed in different rotational sequences, and according to the frequency and aligncrossings are subdivided into three can readily identify the nature of

alignment of the most favored set of i13j2 neu(1) wc ‘L 0.26 MeV/h, Ai Q 10 h: trons, AB, affecting the ground bands in 160,162Yb and the two signatures of the negative parity band (E and F) in 161Yb. neu(21 wc Q 0.36 MeV/h, Ai Q 6.6 h: alignment of another (r = 0 set of iIs/ trons, BC or AD, affecting the il3/2 band in 161Yb (A and probably B), 4 negativearity side bands in 160Yb (AE, AF, BE, BF), and 2 negative-parity side bands in P 62YB (AE, AF). (3) the

alignment wc 2, 0.43 MeV/h, Ai ‘L 5.5 h: yrast band in 160Yb and a 3-quasiparticle

of

a set of band [Fs]

hll/f6yrotons, in

affecting

As described in this paper, there are still observed features that cannot be For example, the ground-S band easily explained in this theoretical approach. crossing in 162Yb occurs at a slightly higher frequency than expected; the 160a161Yb data indicate that the BC and AD crossings, though close in frequency, may occur opposite in order to that predicted; the interaction strengths of the observed crossings are not reproduced very well in the calculations. While there it is overall rather amazing that this simple cranking are these difficulties, assuming o-independent values of c2, c4, X, and A, works so well in approach, explaining the energies and alignments of the bands, in addition to the characterThe finer details must await further calculations. istics of the band crossings. The success of this frequency classification of band crossings, achieved by an opens up the possibility of studying analysis of discrete-line spectroscopy data, higher-frequency crossings and alignment processes using continuum y-ray techgroup in Copenhagen began the development of the technique niques. Bent. Herskind’s of studying rotational effects at higher spins using y-ray energy-energy correlaand Frank Stephens has described at this conference the rapid adtions [31], As I have described inthispaper, vances of this work in Berkeley and Copenhagen. a certain alignment process should affect many bands in a similar fashion, giving This then lends rise to crossings in many bands at nearly the same frequency. and gives us hope for pushing itself to continuum studies of many parallel bands, our knowledge of the rotation alignment process to higher spins.

ACKNOWLEDGMENTS The author gratefully acknowledge the great amount of help from his collaborators (see Table 1) in these experiments and calculations in Copenhagen and in these A year’s visit to Copenhagen was made possible by measurements in Oak Ridge. the many conversations with Aage partial support from the Niels Bohr Institute; The author’s Bohr and Ben Mottelson were very valuable throughout the year. work at The University of Tennessee and at Oak Ridge is supported by the U.S. Department of Energy and by Oak Ridge National Laboratory.

REFERENCES [l]

E. Grosse, 840.

F. S.

Stephens,

and R. M. Diamond,

Phys.

Rev.

Lett.

-31

(1973)

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