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Anomalous rotational energies and transition probabilities at high spin in the ground-state band in 159Ho
Nuclear Physics A252 (1975) 315-332;
@J North-Holland Publishing Co., Amsterdam
Not to be reproduced by photopriut or microfilm without written permission from the publisher
ANOMALOUS
ROTATIONAL
AND TRANSITION
ENERGIES
PROBABILITIBS
AT HIGH SPIN IN THE GROUND-STATE
BAND IN ‘s9Ho
I. FORSBLOMt , S. A. HJORTH and A. SPALEK tt Research Institute for Physics, Stockholm 50, Sweden Received 27 January 1975 (Revised 26 June 1975) Abetract:
Levels with spin values up to T in the ground-state band of 15gHo are populated when a lsgTb target is bombarded with 49 MeV a-particles. The observed pattern of rotational energies is discussed within a model where a quasiparticle is coupled to a rotor with a variable moment of inertia. The analysis suggests that the core exhibits back-bending similar to the behaviour in 160Er and lssDy. However, the angular velocity giving back-bending is larger than in the adjacent even nucleides unless the introduced ad hoc attenuation of the Coriolis force disappears at high spin, or more likely, unless the matrix elements of 1’ are also reduced. This uncertainty associated with the motion of the odd particle precludes the analysis of small effects on the back-bending mechanism due to the blocking of an h+ orbital. The relative y-ray intensities support the notion that g, decreases with increasing values of spin. Furthermore, in the region of back-bending an additional drop of the E2/Ml branching ratio suggests that the B(E2) values are here reduced by about 50 %.
E
NUCLEAR REACTION rsgTb(a, 4ny), E = 43-51 MeV; measured Er, $(E,, O), w-coin. lsgHo deduced levels, Z, rc, moment of inertia, rotational angular velocity, y-branching, Coriolis talc.
1. Introduction
According to a theoretical model proposed by Stephens and Simon ‘) the strong anomalies exhibited by the nuclear moment of inertia at high rotational angular velocities in doubly even nucleides 2*“) are due to a sudden decoupling of a pair of high-spin nucleons from the pair correlated state. The decoupled nucleons align their angular momenta in the direction of the collective rotation and one may in fact achieve a situation where the nuclear rotational angular velocity temporarily decreases slightly with increasing total spin of the nucleus. Because of the characteristic shape of the curve depicting the nuclear moment of inertia as a function of the square of the angular velocity the latter phenomenon is often referred to as backbending. It is an important feature of the Stephens and Simon model that backbending can only occur when intrinsic orbitals of high j and low K occur in the t Permanent Finland. tt Permanent
address: Accelerator
Laboratory,
address: Nuclear Physics Institute,
Department
of Physics, University
Praha, Czechoslovakia. 315
of Helsinki,
316
I. FORSBLOM et al.
nei~bourhood of the Fermi surface. In the classical region of nuclear deformations, i.e. for 150 2 A. 5 190, these conditions are met for the Nilsson orbitals emanating from the i+ neutron shell around A = 160 and for the orbitals emanating from the h, proton shell for A !.z 180. In addition, for nuclei around A = 190 there is evidence that back-bending is caused by i+ neutrons or h+ protons in connection with an oblate shape of the nucleus. In comparison with the more traditional approach describing back-bending as the result of a pairing phase transition from the superfluid ground-state band into a band where the pairing correlations have completely disappeared 2-4), the “decoupling model” may give similar predictions of rotational energies and nuclear ~a~ition probabilities. One possibility of deciding between the two models is then to compare situations where high-j orbitals of low K are available for low-lying pair excitations with situations where such orbitals are not available. A very direct test of the rotation-alignment (RA) model is thus to compare light deformed rare-earth nucleides of even and odd mass. Here, the i, neutron orbitals with low K-values are near the Fermi level and the doubly even nucleides exhibit back-bending at critical angular momenta of about 16 units. In the odd-N nucleides the strongly decoupled positive-parity band, based on the i, neutron intrinsic state, is strongly populated in (particle, m) reactions and has been followed as high as v units of spin ‘, 6). Because of the fact that one i, neutron now occupies and blocks a “decoupled” i, orbital a further decoupling and alignment of i, neutrons from the core is strongly hindered. Thus, in the model of Stephens and Simon back-bending will not occur in these decoupled bands of positive parity and, indeed, experimentally they are quite regular ‘-*). Of course the interpretation of the data for the odd-N nucleides depends to some extent on our treatment of the motion of the odd particle. It thus appears urgent to verify this treatment by studying rotational bands in odd-mass nucleides which are expected to exhibit back-bending. Since ispHo is situated in a region where the doubly even nucleides have a strong tendency of showing a back-bending behaviour and since high-spin members of the $- [523] ground-state rotational band should be strongly populated in (particle, xn) reactions it was decided to employ the ’ 5‘Tbfa, 4n) ’ 5‘Ho reaction to probe the properties of the bond-state band at high-spin values in ’ 59Ho. During the conrse of the present experiment similar studies of the properties of rotational bands in lgl* lg3Hg iref. ‘)I, ‘*‘, ls9* 191*193Pt [ref. ‘“)I, 181*1830~, islRe [ref. ““)I, 16’Lu [ref. ‘“)I, 165Yb [ref. “)J and 157*159*161H~ {ref. ‘“)I have come to our knowledge. With the exception of the i+ decoupled band observed in 187-193pt, 1830~ and 165Yb and the &-[541] band in 16’Lu and ‘*‘Re backbending is observed in all cases. These observations certainly support the idea that the i, neutrons and the h, protons play a critical role in the mechanism for nuclear back-bending. It is, however, somewhat surprising that the h, protons seem to play this role for A = 167.
’ -Ho
317
The present investigation parallels the work on ’ “-’ 61Ho in ref. 13) to a considerable extent. However, in that investigation only levels having I++ equal to an even integer were reported and the theoretical model used to extract the moment of inertia was very crude. With the improved treatment of the particle-rotation coupling introduced in the present work it should be possible to verify the previous analysis and, furthermore, to look for evidence that the h+ protons in analogy with the h, protons are involved in the mechanism for back-bending. For these reasons it was felt to be both urgent and justified to continue and publish the results of the present investigation. 2. Experimental procedure and results A metallic terbium target was bombarded with u-particles accelerated to various energies between 43 and 51 MeV employing the 225 cm Stockholm cyclotron. At the same time the emitted y-rays were detected at an angle of 125” with respect to the incident beam using Ge(Li) detectors and a pulse sorting system. These y-ray spectra suggested that an a-particle energy of about 49 MeV was the most favourable for production of ’ 5‘Ho and beams of this energy were accordingly employed for the rest of the investigations. A two dimensional yy coincidence experiment was performed with two 40 cm3 Ge(Li) detectors viewing the target from angles of about +90” with respect to the beam. The coincidence events were stored on magnetic tape by the TRASK computer and the 4000x 4000 channel matrix was analyzed by introducing gates on one of the axes. A description of the electronic equipment employed in the coincidence experiment is given in ref. r4). In order to obtain information about the multipolarities of the observed transitions angular distributions of the emitted y-rays were measured by recording spectra at 56”, 90”, 120”, 135” and 149” with respect to the beam. The normalization of the spectra was based on the intensity of the K.# X-rays, the distribution of which was assumed to be isotropic. This normalization was checked by recording the number of cl-particles which were elastically scattered in the target through an angle of +40” by using two monitor detectors. Furthermore, the angle between the target normal and the detected y-rays was kept constant in order to minimize normalization errors due to absorption of photons in the target. The observed angular distributions were fitted with the expression w(e) = A, +A,P,(cose)
+A,P,(cose).
(1)
Conclusions about the initial and final spin and the multipolarity of the y-ray can be drawn from the ratios AZ/A, and A,/A, [ref. ‘“)I. The results of the present experiments are summarized in tables 1 and 2 and in figs. 1 and 2. Fig. 1 gives one example of the singles y-ray spectra observed and fig. 2 shows two coincidence spectra. The energies, the relative y-ray intensities and
318
I. FORSBLOM
et al.
TABLE 1 Summary of the y-ray data obtained in the lsgTb(a,
zvfor Z&=49MeV (65.0) 66.1 89.0 97.5 (107.2) 119.8
8.6 20.7 9.8 1oOb) 12.1 34.0
121.2
296.6
127.5 129.6 130.6 141.8 149.0 150.4 (162.0) 165.0 166.1 167.4 175.6 191.3 195.3 199.4 201.8 203.5 204.8 206.6 215.7 219.0 (225.0) 233.3 (241.6) (249.5) 251.3
9.7 8.6 13.4 10.6 17.3 114.1 6.5 8.6 39.0 91.1 12.2 8.4 14.2 10.4 13.8 90.9 54.9 18.0 18.6 23.2 7.1 29.5 8.6 11.0 22.1
253.0
166.5
255.0 257.2 258.7 267.5 269.3 271.9 283.1 284.5 289.4 292.5 295.2 297.0 302.5
17.8 20.7 8.7 6.9 16.6 52.4 5.8 7.4 5.6 6.4 34.2 42.6 8.2
Z,(E, = 20 MeV)
Z&E,= 49 MeV)
4n)i5gHo
reaction ‘) Assignments
Angular distribution coeaticients
AlAo
11 4
iAc/Ao
Zr x;
0.00~0.08 2.8 1.0’) 3.6
0.03 kO.03 0.19f0.10
1.4
0.06kO.03
Q %- 4 -H-
t+H-
8 b-;
radioactivity
0.42f0.10 (3.2) 2.5 0.45
3.2
1 1.6
6.7
(0.3)
1.7
0.9
0.45 3.4
2.4
0.16f0.06 I -0.08&0.11 0.03 &to.01 0.37&0.10 0.12f0.10 0.06f0.04 0.07zkO.12 0.28&0.15 -0.42&0.15 0.19~0.15 -0.11 rF_0.09 0.14po.11 -0.28f0.12 0.4o,f,o.o7 0.17hO.25 -0.02f0.15 0.04~0.04 0.24&0.12 -0.30*0.30
0.01 kO.01 0.02*0.09 0.26&0.10 0.29rfrO.16 -0.26&0.40 0.09f0.10 0.29 f0.02 0.02rjIo.20 0.07f0.20 0.28f0.30 -0.11 rto.40 0.07~0.04 0.08f0.04
-0.06&0.02
y
ic- J#
(3? 4’
H-;
ik St+)*
radioactivity q f- zj g-
part radioactivity 9 )- y )-0.04f0.03
-0.01 kO.05 -0.01 f0.05
y 4
%- aie %H- 6 f-
319
’ 5gHo TABLE 1 (continued) Zr for
Z,.(E, = 20 MeV)
Z% = 49 MeV
z,,(E, = 49 MeV)
Angular distribution coefficients AlAo
304.0 308.0 315.9
25.1 9.7 11.6
0.32
0.2950.03
317.8
72.7
0.26
0.24hO.03
323.2 349.3 353.0 (369.0) 370.8 (383.9) 385.6 389.0 408.2 (419.0) (425.0) 427.8 (441.3) 451.0 (455.8) 457.9 461.3 468.4 471.0 484.4 486.5 490.9 496.4 505.4 518.1 528.6 (534.4) 536.7 541.1 547.4 550.3 (554.5) 559.0 575.8 578.0 586.2 587.9 592.0
11.7 22.0 23.0 16.7 73.5 8.8 20.9 25.5 95.1 5.1 5.6 29.4 18.9 12.3 10.6 78.7 16.7 16.9 20.0 19.2 71.9 13.0 31.3 11.0 11.2 53.6 12.6 9.1 5.6 20.1 54.0 14.6 17.1 6.8 24.6 12.1 13.2 22.4
AlAo
Assignments Zt &
Zr Kr
-0.04*0.04
0.45i-0.25
0.49 1.4 0.27 0.46
-0.09~0.08 0.29f0.05 0.08f0.04 -0.30*0.50 0.42&0.10 0.41*0.15 0.30&0.07 0.37f0.05 0.31 f0.04 0.40f0.25 0.26&-0.20 0.30f0.05 0.15~0.08 -0.26&0.15 0.29f0.02 0.17&0.06 0.43 ho.08 0.35kO.06 0.41 kO.15 0.29hO.04 0.42&0.06 0.23 kO.05 -0.27f0.27 0.04&0.24 0.38f0.04 0.34*0.20 0.27f0.20 0.46hO.30 0.25&0.10 0.32kO.04 0.23f0.14 -0.10*0.12 -0.21 f0.14 0.42+0.07 0.96kO.23 -0.40~0.30 0.13*0.13
-0.06~0.04 -0.07*0.07 0.01*0.05
-0.09*0.07 -0.11 *to.06
-0.07&0.08 -0.13*0.11
-0.11*0.05 -0.10*0.07
-0.05*0.05
-0.14*0.14 -O.OS=kO.O5
‘) Transitions assigned to other final nuclei and transitions with relative intensity less than five are excluded from this table. Parentheses around energy value indicates that assignment to 159H~ is judged to be uncertain. s) Used for normalization.
I. FORSBLOM
320
ef al.
TABLE 2 Qualitative results of the coincidence measurement Coincident prays
Energy of gate y-ray (keV)
97.5 121.5 1so.4 167.4 203..5+204.8 215.7 219.0 233.3 253.0+255.0 271.9 295.2 304.0 317.8 349.3 370.8 389.0 408.2 457.9 448.4 486.5 490.9 528.6 534.4f536.7 541.1 550.3 578.0 586.2+ 587.9 592.0
122 98 98 550 98 122 208 150 541 122 458 116 272 98 110 208 98 253 122 540 207 122 122 216 122 253 150 253 216 122 592 150 150 550 150 529
(keV)
150 150 122
167 167 167
204 204 204
253 233 219
272 253 233
318 295 253
371 318 295
408 371 371
458 408 408
487 458 458
529 487 487
550 529 496
550 529
122 150 253 167 550 150 550 122 295 167 122 211 122 257 150 JSO 216 150 150 253 IS0 2.57 167 257 253 150
150 167 304 204
204 204 389 233
219 233 468 240
233 253 541 253
253 272
272 408 29.5 318
458 371
487 458
529 487
550 529
550
295
297
304
318
371
408
487
529
167
204
208
219
253
272
276
280
295
318
371
408
130 304 204 150 216 204 428 204 578 253 167 167 304 167 349 204 349 304 167
142 318 233 167 233 219 491 219
150 349 239 204 253 253 537 233
167 371 242 216 257 267
204 389 253 284 389 269
208 408 295 329 468 295
216 428 371 358 541 408
219 487 408
233 491 484
249 529 487
253
257
529
550
458
487
529
550
592
253
272
295
353
408
441
458
487
529
304 233 204 389 204 386 253 388 389 204
468 541 253 272 233 255 541 604 25.5 272 428 496 255 272
295 272
318 318
406 371
471 529
487 550
J18 J78
529
550
295
318
371
408
550
578
318
371
458
578
167 167 559 167 544
204 204 578 204 550
233 233 592 219 559
468 233
253
272
318
323
371
408
458
487
586
253 253
269 269
318 295
458 323
487 371
529 408
588 458
487
529
544
233 586
253
267
272
295
318
371
408
458
487
the angular distribution coefficients for all the lines one may assign to 159Ho are tabulated in table 1. Included in this table is some information derived from an experiment in which l5 ‘Ho is produced in a (p, 2n) reaction ’ “). The occurrence of a high value of the ratio I,@$, = 20 MeV)/f,(E, = 49 MeV) indicates that the transition is associated with the decay of low-spin levels in r5gHo. The qu~itative summary of all the coincidence relations observed is given in table 2.
lsgHo __, X103
321
_r-?--__r--_.----_?~~-~~-,-i-r-
15’Tb + 49 MeV
I-2035
6012i.5.R
60 1
1 1.99J i( I
R'i
CHANNEL
1
/ /
*2OL.6 253.0*R
I
a-particles 0=125”
I
I
NUMBER
Fig. 1. Spectrum of y-rays recorded during the bombardment of lsgTb with 49 MeV a-particles. The notations (a, a’), R and B indicate that the peak is assigned to the target nucleus, to the radioactive decay of rS9Ho, and to the background, respectively. 3. The level scheme of 159Ho
The rotational been established
levels belonging to the ground-state band in ’ 5 ‘Ho have previously q by Diamond t ‘) and Bout& “). With the aid of the up to spin
present experimental information it is a straightforward procedure to extend our knowledge of this band up to the member with spin y as shown in fig. 3. For the levels where I++ is an even integer these assignments agree with the recent findings
I. FORSBLOM
322
et af.
200
100 v) I-
z 3
0 ”
00
LL 0 CT
w
r” 200 2
300
-T-v-m-500 CHANNEL
7009
NUMBER
Fig, 2. Examples of the coincidence spectra obtained.
of Grosse et a2. ‘“) who in addition tentatively give the energy of the y level as 3707.3 keV. In addition to the ground-state band the experimental data (tables 1 and 2) suggest the existence of two additional cascades of E2 transitions namely the 253.0, 349.3, 427.8,490.9 and 536.7 keV cascade and the 215.7, 304.0, 389.0, 468.4 and 541.1 keV cascade. Due to the low energy of the low-spin transitions and the isomerism of the 4, f’ [41 l] level good coincidences with lower lying transitions are lacking and it has not been possible to establish without ambiguity the nature of these two cascades. Most probably they are associated, as suggested by Funke et al. 19) with the 3’ [41i J and $- [541] rotational bands. 4. Discussion 4.1. THE ROTATIONAL
ENERGIES
At low-spin values the energy of the E2 cross-over transitions in the ground-site band increases with increasing angular momentum roughly in agreement with the I(I+ 1) rule. At higher spin values this trend is broken and the data suggest that the transition energy may here even decrease a little as the spin of the nucleus becomes higher, The absence in the singles spectrum of prompt peaks with energies
l=Ho
323
‘-Ho
4865
233.3
21/2-. 19/2-
I 2530 4Y7Q
.L I 204.8 / ---+-370.6 203.5
4m2
t712m/213/2-
317.8 *01
7/2-
l6z4
1 2h.Q 1
do.4 12J.5
1
sg.5
1198.1 945.0 7403 5368 36ql 219.0 97.5 0
7/2- Et5231 Fig. 3. Level scheme of lsgHo showing the ground-state rotational band.
larger than 592 keV and of sufficient intensity (> 2 %) is a beautiful verification of this statement. Thus the y-ray spectrum resembles very closely the spectra that are found in (‘particle, xny) reactions leading to the neighbouring doubly even nucleides. The anomalous properties of ground-state bands in doubly even nuclei are perhaps best exposed when the moment of inertia is plotted as a function of the square of the rotational angular velocity 2P3). For such bands the formulae 24 -= R2
M-2
(2)
E(I)-E(I-2)’
tr202 = 4(P--1+1)
(E(I)-E(I-2))2 (4f-2j2
’
(3)
offer an approximate but reasonably accurate possibility of deriving the quantities desired. For the odd nuclei these formulae are not immediately applicable since the odd particle makes a contribution to the total angular momentum of the nucleus which must be subtracted out before the desired relationship can be derived. In a first
324
I. FORSBLOM
et al.
rough approximation one can find the moment of inertia and the angular velocity for the odd nucleus by assuming the validity of the rotation-aligned coupling scheme 20*21)* In this scheme the angular momenta of the particle and the rotation are completely aligned for the levels where I-j is an even integer and one has the relation R = I-j, (4) with R being the angular momentum of the core. For the other levels (where I-j is odd) the angular momentum of the core is R = I-j+l.
(5)
By replacing I with R in eqs. (2) and (3) and by assuming j = q- (the s- [523 J orbital originates from the h9 shell-model state) the analysis becomes identical to the one employed for the doubly even nuclei. The results are presented in fig. 4 (curves A) together with the curves for the surrounding doubly even nucleides ’ s8Dy and 16’Er. Because the assumption of complete decoupling is a poor approximation for the low-spin states in 159Ho, the curves for 1-j even and I-j odd (= 1+$ even or odd) differ and show too low moments of inertia for low angular velocities. However, in the region where the curves for ’ “Dy and r6’Er rise sharply, the curves for I5 ‘Ho follow the ones for the even nuclei amazingly well. It is possible to improve on the previous analysis by treating the full dynamics of quasiparticle motion in a deformed field which is assumed to have a variable moment of inertia. Such calculations are described in ref. 7). Very briefly the present calculations imply an approximate diagonalization of the Hamiltonian H = Hqp “i-H,, .
(6)
Here, the quasiparticle Hamiltonian is chosen in the spirit of the procedure developed in ref. ““). This means that the eigenstates of the Nilsson Hamiltonian are found for several values of e2 and .sq using ttcu, = 7.5 MeV, K = 0.0637 and p = 0.6. These single-particle states are then transformed to the quasi~rti~le scheme by means of the Bogo~ubov-Valatin transforn~ation using the BCS approximation with separate blocking, G = 0.1281 MeV and 32 levels included in the gap equation. The rotational Hamiltonian is written
ee*= 2$(I-j)”
+ v,,,,,
(7)
where the moment of inertia of the core, 3, is the constant relating core angular momentum I-j to core angular velocity 0 i.e. h(l-j)
= $a.
(8)
In the following the moment of inertia of the core is allowed to vary with spin or angular velocity as suggested in the VMI model 2*3*23).Hamilton’s equations then
159H~
325
lead to the condition ‘*“) d
do = odl,
/s
(9)
which after some trivial algebra ‘) may be integrated to give
vcore
=
l
-
2 9(&=0)
co* d9.
For low and moderately high rotational angular velocities the success of the classical VMI model 23P24)tells us that the expansion 9 = &)+$uP,
(11)
Vcore = 3h4,
(12)
with gives an amazingly good description of nuclear rotational energies in doubly even nuclei. Only the orbitals emanating from the h, state are strongly connected through H,,,. Therefore, only these six orbitals are needed in the rotational energy matrix. Following an estimate given in ref. * “) a common value of 36 is chosen for j* for the 11, orbitals. It is also assumed that V,,,, is diagonal in the representation chosen and that V,,,, and X are the same for all states of a given spin. In practical calculations the values of a*, 9 and V,,,,, are found through an iterative procedure using eqs. (lo)-(12) and the equation
(13) where Q1, o is the wave function of the lowest level with spin I. When the data for r 59Ho are analysed within the present model it is found that the Coriolis matrix elements connecting the ground-state band have to be reduced while the matrix elements connecting the excited bands must be enhanced if the transition energies in the low-spin part of the band shall be fitted. Such a need for reducing the theoretical Coriolis coupling strength is generally encountered and is here defined as
=
(14
On the other hand, the need for enhancing these couplings is judged to be unphysical and is believed to be an artifact of the calculation. This may be explained as follows. Since the energies of the low-K h, orbitals decrease very strongly with increasing deformation and since these orbitals are hole states in the present calculation one may expect that the deformation is here smaller than in the ground state. With decreasing deformation the importance of the couplings to the high-lying low-K orbitals is rapidly increasing as discussed by Alonso et al. *“). Our actual
326
I. FORSBLOM
et al.
calculations also show that a very good fit to the transition energies below spin J$ can be obtained with all CL~S 1 provided the quadrupole deformation e2 is reduced from the recommended value of z. 0.23 [ref. ““)I to 0.215. In comparison, a variation of sq between 0 and -0.04 has little influence on the calculated energies. Therefore, the values of &2 = 0.215 and E,, = -0.02 [ref. ‘“) J are chosen for the rest of the calculations. By following the aforemention~ assumptions and the approximations implied by eqs. (11) and (12) the experimental transition energies below spin 9 are fitted within an average error of 0.1 keV. In this fit the values of the adjustable parameters are 24,/fr2 = 45.6 MeV-‘, /?,&” = 107.1 MeVW3 and 4 = 0.65 (all cc, with K # 5 are assumed to be unity) in addition to the previously chosen value of s2. Beyond spin y the theoretical transition energies are systematically smaller than the ones experimentally observed and one may assume that these deviations are primarily due to the approximations implied by the use of eqs. (11) and (12). A set of “experimental” values of 2Y/fi2 and h2w2 is here found by maintaining the parameter values s2 = 0.215 and R%= 0.65 while varying the moment of inertia along other straight lines X = ~~(~)~~~(~)~2.
fll’)
The new parameters g&(1) and /?‘(I) are subject to the constraint that the previously calculated moment of inertia of the preceding level f-2 is reproduced.
3
lsoEr ‘Y!?? ,,,’ ,,,_....---,! .f”
Id
I.
*
*
*
1..
on5
*
a
I
I
L
al0 #%?
Fig. 4. Moment of inertia as a function of the square of the rotational angular velocity for ’ sgHo and for lsaDy and 160Er. The curves for rs9Ho labelled A correspond to the derivation assuming the validity of the rotation-aimed coupring scheme while the curves labelled B refer to the derivation where j2 is 36 and where the full dynamics of the quasiparticle motion is treated. The curves denoted C correspond to the case where j2 is reduced to 18. All curves for ls9Ho include the point due to the 547.4 keV 93 transition tentatively identified in ref. r3).
‘“gHo
327
One must also employ eq. (10) rather than eq. (12) when V,,I, is calculated. In this way it is found that a smooth continuation of eq. (11) can be obtained only if the levels having I++ even and I+$ odd are fitted separately. Hence the result of the analysis splits into two separate curves for spin larger than v. However, as seen in fig. 4 curves B, both curves indicate a very rapid increase of the moment of inertia at h202 z 0.095 MeV2. With the last analysis the curves for 1“Ho fall much closer to the ones for 16’Er for low spin expressing the improved treatment of the quasiparticle degree of freedom. However, for the larger spin values the calculations predict the back-bending to occur at a larger rotational angular velocity than in i6’Er and “‘Dy. The position of back-bending in 15gHo determined is remarkably stable with respect to the choice of parametrisation and other details of the fit. Thus in calculations where E2, eq and tcx are chosen in different ways and in calculations where levels up to spin si- are fitted using eqs. (11) and (12) the position of back-bending always falls within the limits 0.094 MeV2 5 h2m2 5 0.100 MeV2. From this study it is concluded that in our model l5 ‘Ho is predicted to back-bend at a significantly larger angular velocity than “*Dy and 160Er. This different position of back-bending in ’ 5gHo and in 16’Er and ’ 58Dy may reflect different properties of the core for the even and odd systems but could also reflect errors inherent in the present model and method of calculation. In the former case one may think that the h, protons to some extent take part in the back-bending process and that the blocking of an h, orbital through the short-range residual neutron-proton interaction effectively raises the excitation energies of the rotationaligned states. This will indeed move the point of intersection of the ground band and the rotation-aligned band to higher spin and angular velocity but at the same time the angle of intersection will decrease. This means that the bend is expected to become smoother. Such an effect is observed in “lrl g3Hg [ref. “)I and lg3Pt [ref. ‘“)I where the h+ orbital and the iY orbital probably are both responsible for back-bending. Here, such an explanation contradicts the experimental findings because the abrupt increase of the moment of inertia is essentially the same in ’ “Ho and in 16’Er. The blocking of an h, proton orbital could also effect the properties of the core in a more indirect manner e.g. through inducing a change of the nuclear deformation. Such effects are expected to show up also when back-bending is investigated as a function of proton number and since all doubly even nucleides around A = 160 back-bend for the same angular velocity such effects are believed to be small. As the second main explanation for the large value of the critical angular velocity found one may argue that the approximations and assumptions inherent in the present treatment of the motion of the odd particle are not justified. One of these assumptions is that the moment of inertia is the same for all levels of a given spin. Since the ground band, due to the rotation-alignment mechanism, has a very low vaiue of the rotational angular velocity one may argue that the angular velocity and
328
I. FORSBLOM et al.
of inertia the excited are larger. means smaller matrix elements smaller energy between the elements in energy matrix. effects of changes are and tend cancel. Therethe error by the of having common moment inertia for the bands believed to small and will be absorbed in adjustable parameters /? and The last may not when the exhibits back-bending here the of the to the of higher of inertia manifest themselves at average velocities below expected critical of 0.085 The fact our calculated behaves as in the below ti2w2 0.085 MeV’ a posteriori justifies the approximation of having the same fl for all levels of a given spin. In the present evaluation of H,,, the matrix elements of I *j were reduced while the ones ofj’ were not. This increases the rotational energies and means that #i202 may be overestimated in the calculations. This is shown by the curves C in fig. 4 which represent a calculation wherej2 is reduced from 36 to 18. Since the predicted rotational energies are all reduced by the same amount the quality of the fit to the rotational energy spacings is not affected and the reduction ofj2 essentially implies a linear translation of the curve in fig. 4 from right to left. The calculation shows that it is indeed possible to obtain a curve resembling the ones for the even isotopes in this way. However, the employed reduction of jz is larger than expected from the introduced reduction of the matrix elements of j. The second possibility of obtaining back-bending at hZo2 w 0,085 MeV2 in ’ 59Ho, namely to remove completely the artifical reduction of the Coriolis matrix elements, destroys the fit to the low-spin states. A possible way out of this dilemma is then of course to allow for the attenuation factors to be spin dependent. However, it seems premature to introduce such a hypothesis at the present time. 4.2. THE RELATIVE
ELECTROMAGNETIC
TRANSITION
RATES
In the present experiment the angular distributions of the y-rays depend on the E2/Ml mixing ratios in the cascade transitions 1“). Similar information on the relative strength of the E2 and Ml radiation is contained in the ratios of the intensities in the cross-over and cascade transitions. While these branching ratios normally give the most accurate information, the E2/Ml amplitude mixing ratios are useful in the sense that they determine the sign of the mixing. Since the amplitude mixing ratios are positive, both in the experiment and in all theoretical calculations, only the branching ratios will here be compared with theoretical calculations. Such theoretical calculations of branching ratios are described in ref. 27) for the special case when the rotational band is not perturbed, while the formulae for the calculation of nuclear electromagnetic transition probabilities in deformed nuclei in general may be found in refs. 2**29). Before the experimental and theoretical branching ratios are compared it is useful to remove the strong energy dependence of the branching ratio by multiplying
’ 59Ho
I,
I,
11 13 15 17 T-TY?TTTTTTTTT
329
I
I
I
I
I
I
t
I
I
19
21
23
25
27
29
31
33
35
I Fig. 5. Experimental
and theoretical branching ratios within the ground-state rotational band in ls9Ho. The errors in the experimental points correspond to the statistical errors and to an estimated uncertainty of w 10 % in the intensity calibration curve. The curve labelled A corresponds to the completely adiabatic case where the intrinsic wave function is assumed to be the p- 15231 Nilsson orbital and where Qc/(gx--gR) is adjusted to fit the point for Z = +. The curve labelled B is based on the results of the Coriolis calculations described in the text and uses g,, I = gOf = 0.6 g,, rrco = 3.35, gt = 1.0, gR = 0.36 and Q,, = 6.85 b. In the curve labelled C a further reduction of the matrix elements of g,*s,: is simulated by putting (KflIplK) =+-uKax+l * 0.5
Z,(Z --, Z-2)/1,(1 + I- 1) with (E,(Z + I- l))3/(E,(Z --, Z-2))5. The comparison of such reduced experimental branching ratios with four theoretical calculations is shown in fig. 5. In the first calculation, labelled A, the wave function for the intrinsic state is assumed to be 100 y0 s- [523] and the quantity (gK-g&Q0 is chosen as 0.092 which fits the branching ratio in the decay of the y- level. With Q, = 6.85 b [ref. “)I and gR = 0.36 this implies gK = 0.99. This value of gK is definitely smaller than the range of values 1.26 5 gK 2 1.51 suggested by the calculations of Lamm 31). Furthermore, with the parameters chosen this simple calculation predicts too large branching ratios for the high-spin states. One may try to improve on these matters by using the wave functions obtained in the present Coriolis calculations and by considering various assumptions about the longitudinal and transverse spin gyromagnetic ratios gS,z and gSk. The results of such calculations are included in fig. 5 (curves B-D) where also the parameters corresponding to each calculation are given. It turns out that these calculations correctly predict the branching ratios in the decay of the low-spin states but the disagreement for the high-spin states is not removed. Furthermore, these more sophisticated calculations tend to predict too large oscillations between the points where I+$ is an even integer and the points where Z++ is odd. These oscillations are strongly connected with the contributions to the total transition amplitude which
330
I. FORSBLOM
er al.
have 1dK.l = 1 and the curves become smoother when gSk is reduced. In principle it is possible to check this explanation by considering the magnetic moment which exhibits *‘) a different dependence on gS,z and gSrt. However, since the magnetic moment of the ground state of ’ 59Ho has not been measured, such a comparison cannot be done at the present time. It is an interesting fact that the theoretical branching ratios are too large for the higher spin states and that this discrepancy persists in all the calculations. Furthermore, it is not at all obvious how the Coridis calculations should be modified to give theoretical branching ratios in agreement with experiment. On the other hand, the collect.ive gyromagnetic ratio gR which is approximately given by
is expected to decrease with increasing spin since 3m is expected to increase faster than .Yb for spin 5 20 tt. In particular, if back-bending is a property of the neutron system a substantial reduction of gR should accompany the sudden increase of the moment of inertia.
Fig. 6. Experimental and theoretical branching ratios within the ground-state band in 13gffo. The theoretical curves labelled A and D are the same as the curves with the same labels in fig. 5 except that ga varies with spin according to eq. (15) and fig. 4.
In order to demonstrate the effect of the variation of gR we have made the extreme assumption that the increase of the moment of inertia observed in fig. 4 is entirely due to the neutrons and that gR decreases with spin according to eq, (15). With this variation of gR included the results of calculations analogous to the ones giving the curves A and D in fig. 5 are shown in fig. 6. Obviously the data are better described in this way. However, even on this rather extreme assumption a small
159Ho
331
discrepancy remains. This must be interpreted within the ambiguities of our Coriolis coupling calculations or, alternatively, be interpreted in such a way that the collective B(E2) value also varies. The very small values of the branching ratio which are observed for I = v and I = y could thus be due to the reduction of the B(E2) value expected ‘*“) m * t he re g’Ion where the moment of inertia changes very rapidly. The present data indicate that a reduction of about 50 % is conceivable. Unfortunately, the presence of events of other origin in the tiny photopeaks corresponding to the 31 and q to q transitions would have the same effect and cannot be 2 to 2A 2 excluded because in the coincidence spectra the error in the intensity values are of the order of 50 %. 5. Conclusions
It is possible to draw the following main conclusions from the present investigation: (i) The pattern of rotational energies in the ground-state band of 15’Ho is well described within a model where a particle is coupled to a rotor with a variable moment of inertia if the Coriolis matrix elements connecting the I- [523] orbital are reduced. Furthermore, the set of values deduced for the moment of inertia and angular velocity resembles the set which is found in the adjacent nucleides of even mass. This suggests that the employed method of subtracting out the angular momentum of the odd particle is essentially correct. The fact that this method, when applied to the neighbouring nucleides with an odd number of neutrons, yields moments of inertia which are quite regular far beyond the critical region of h2fD2 = 0.085 MeV’ [ref. ‘)I indicates that the observed anomalies in the moment of inertia 2*3) are due to the neutrons. (ii) The observation that the deduced values of the moment of inertia for ’ 59Ho start to rise sharply at a slightly larger rotational angular velocity than in the adjacent doubly even nucleides is not completely understood. Judging from the discussion in subsect. 4.1 the uncertainties inherent in the procedure of subtracting out the angular momentum of the odd particle are quite large and are probably the most likely explanation for the discrepancy observed. Thus the discrepancy is removed if the introduced ad hoc attenuation of the matrix elements connecting the 3- [523] orbital disappears at high spin or, more likely, if the matrix elements of j2 are also reduced. The second possibility of explaining the postponement of backbending to higher angular velocities namely as an effect of the blocking of an h9 orbital is believed to be less likely. Furthermore, due to the aforementioned uncertainties in the treatment of the odd particle the presence of small effects due to the blocking of one h, orbital cannot be established. (iii) The persistent disagreement between the trends of the experimental and theoretical branching ratios suggests that the collective gyromagnetic ratio gR or the rotational B(E2) value of the core is reduced with increasing values of spin. In
332
I. FORSBLOM
er al.
particular, by making a reasonable but somewhat optimistic assumption about the variation of gR the agreement between theory and experiment is greatly improved for spin values below 9. Beyond spin “$, e.g. in the region where the moment of inertia of the core rises very sharply, a 50 % drop in the trend of the branching ratio suggests that the B(E2) values are reduced here by roughly the same amount. The authors wish to thank the cyclotron reconstruction and operation groups under the directions of Prof. H. Atterling, Mr. K. G. Rensfelt and Mr. G. Printz for having completed the improvement program for our cyclotron and for the operation of the accelerator. References 1) F. S. Stephens and R. Simon, Nucl. Phys. A183 (1972) 257 2) 3) 4) 5) 6)
7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24) 25) 26) 27) 28) 29) 30) 31)
R. A. Sorensen, Rev. Mod. Phys. 45 (1973) 353 A. Johnson and Z. Symadski, Phys. Reports 7C (1973) 181 B. R. Mottelson and J. G. Valatin, Phys. Rev. Lett. 5 (196O) 511 E. Grosse, F. S. Stephens and R. M. Diamond, Phys. Rev. Lett. 31 (1973) 840 W. F. Davidson, R. M. Lieder, H. Beuscher, A. Neskakis and C. Mayer-BOricke, in Proc. Int. Conf. on reactions between complex nuclei, Nashville, 1974, vol. 1 (North-Holland, Amsterdam, 1974) p. 190 S. A. Hjorth and W. Klamra, University of Stockholm report USIP 74-21 (1974) L. L. Riedinger et al., Phys. Rev. Lett. 33 (1974) 1346 R. M. Lieder, H. Beuscher, W. F. Davidson, A. Neskakis and C. Mayer-Boricke, preprint 1975 M. Piiparinen, I. C. Cunnane, P. J. Daly, C. L. Dots, F. M. Bemthal and T. L. Khoo, MSUCL 150 (1975) R. M. Lieder, private communication C. Foin and D. Bam6oud, Phys. Rev. Lett. 33 (1974) 1049 E. Grosse, F. S. Stephens and R. M. Diamond, Phys. Rev. Lett. 32 (1974) 74 L. Harms-Ringdahl and J. Sztarkier, Nucl. Instr. 108 (1973) 557 T. Yamaxaki, Nucl. Data 3 (1967) 1 J. Bout&, private communication R. M. Diamond, in Proc. Int. Conf. on the properties of nuclei far from the region of beta stability, Leysin, Switzerland 1970, p. 65 J. Bout&, Thesis, University of Grenoble, 1973 L. Funke, private communication P. Vogel, Phys. L&t. 33B (1970) 400 F. S. Stephens, R. M. Diamond and S. G. Nilsson, Phys. Lett. 44B (1973) 429 S. 0. Nilsson et al., Nucl. Phys. A131 (1969) 1 M. A. J. Mariscotti, G. Scharff-Goldhaber and B. Buck, Phys. Rev. 178 (1969) 1864 0. Saethre, S. A. Hjorth, A. Johnson, S. Jiigare, H. Ryde and Z. Sxymslllski, Nucl. Phys. A207 (1973) 486 W. Ogle, S. Wahlbom, R. Piepenbring and S. Fredriksson, Rev. Mod. Phys. 43 (1971) 424 I. Alonso, H. Bakhru, F. M. Bernthal, J. Bout&, B. Olsen, I. Rezanka and J. 0. Rasmussen, Nucl. Phys. A160 (1971) 193 F. Boehm, G. Goldring, G. B. Hagemann, G. D. Symons and A. Tveter, Phys. Lett. 22 (1966) 627 S. A. Hjorth, H. Ryde, K. A. Hagemann, G. Lsvheiden and J. C. Waddington, Nucl. Phys. Al44 (1970) 513 S. A. Hjorth, A. Johnson and G. Ehrling, Nucl. Phys. Al84 (1972) 113 P. H. Stelson and L. Grodzins, Nucl. Data 1 (1965) 21 I. L. Lamm, Nucl. Phys. A125 (1969) 504
@J North-Holland Publishing Co., Amsterdam
Not to be reproduced by photopriut or microfilm without written permission from the publisher
ANOMALOUS
ROTATIONAL
AND TRANSITION
ENERGIES
PROBABILITIBS
AT HIGH SPIN IN THE GROUND-STATE
BAND IN ‘s9Ho
I. FORSBLOMt , S. A. HJORTH and A. SPALEK tt Research Institute for Physics, Stockholm 50, Sweden Received 27 January 1975 (Revised 26 June 1975) Abetract:
Levels with spin values up to T in the ground-state band of 15gHo are populated when a lsgTb target is bombarded with 49 MeV a-particles. The observed pattern of rotational energies is discussed within a model where a quasiparticle is coupled to a rotor with a variable moment of inertia. The analysis suggests that the core exhibits back-bending similar to the behaviour in 160Er and lssDy. However, the angular velocity giving back-bending is larger than in the adjacent even nucleides unless the introduced ad hoc attenuation of the Coriolis force disappears at high spin, or more likely, unless the matrix elements of 1’ are also reduced. This uncertainty associated with the motion of the odd particle precludes the analysis of small effects on the back-bending mechanism due to the blocking of an h+ orbital. The relative y-ray intensities support the notion that g, decreases with increasing values of spin. Furthermore, in the region of back-bending an additional drop of the E2/Ml branching ratio suggests that the B(E2) values are here reduced by about 50 %.
E
NUCLEAR REACTION rsgTb(a, 4ny), E = 43-51 MeV; measured Er, $(E,, O), w-coin. lsgHo deduced levels, Z, rc, moment of inertia, rotational angular velocity, y-branching, Coriolis talc.
1. Introduction
According to a theoretical model proposed by Stephens and Simon ‘) the strong anomalies exhibited by the nuclear moment of inertia at high rotational angular velocities in doubly even nucleides 2*“) are due to a sudden decoupling of a pair of high-spin nucleons from the pair correlated state. The decoupled nucleons align their angular momenta in the direction of the collective rotation and one may in fact achieve a situation where the nuclear rotational angular velocity temporarily decreases slightly with increasing total spin of the nucleus. Because of the characteristic shape of the curve depicting the nuclear moment of inertia as a function of the square of the angular velocity the latter phenomenon is often referred to as backbending. It is an important feature of the Stephens and Simon model that backbending can only occur when intrinsic orbitals of high j and low K occur in the t Permanent Finland. tt Permanent
address: Accelerator
Laboratory,
address: Nuclear Physics Institute,
Department
of Physics, University
Praha, Czechoslovakia. 315
of Helsinki,
316
I. FORSBLOM et al.
nei~bourhood of the Fermi surface. In the classical region of nuclear deformations, i.e. for 150 2 A. 5 190, these conditions are met for the Nilsson orbitals emanating from the i+ neutron shell around A = 160 and for the orbitals emanating from the h, proton shell for A !.z 180. In addition, for nuclei around A = 190 there is evidence that back-bending is caused by i+ neutrons or h+ protons in connection with an oblate shape of the nucleus. In comparison with the more traditional approach describing back-bending as the result of a pairing phase transition from the superfluid ground-state band into a band where the pairing correlations have completely disappeared 2-4), the “decoupling model” may give similar predictions of rotational energies and nuclear ~a~ition probabilities. One possibility of deciding between the two models is then to compare situations where high-j orbitals of low K are available for low-lying pair excitations with situations where such orbitals are not available. A very direct test of the rotation-alignment (RA) model is thus to compare light deformed rare-earth nucleides of even and odd mass. Here, the i, neutron orbitals with low K-values are near the Fermi level and the doubly even nucleides exhibit back-bending at critical angular momenta of about 16 units. In the odd-N nucleides the strongly decoupled positive-parity band, based on the i, neutron intrinsic state, is strongly populated in (particle, m) reactions and has been followed as high as v units of spin ‘, 6). Because of the fact that one i, neutron now occupies and blocks a “decoupled” i, orbital a further decoupling and alignment of i, neutrons from the core is strongly hindered. Thus, in the model of Stephens and Simon back-bending will not occur in these decoupled bands of positive parity and, indeed, experimentally they are quite regular ‘-*). Of course the interpretation of the data for the odd-N nucleides depends to some extent on our treatment of the motion of the odd particle. It thus appears urgent to verify this treatment by studying rotational bands in odd-mass nucleides which are expected to exhibit back-bending. Since ispHo is situated in a region where the doubly even nucleides have a strong tendency of showing a back-bending behaviour and since high-spin members of the $- [523] ground-state rotational band should be strongly populated in (particle, xn) reactions it was decided to employ the ’ 5‘Tbfa, 4n) ’ 5‘Ho reaction to probe the properties of the bond-state band at high-spin values in ’ 59Ho. During the conrse of the present experiment similar studies of the properties of rotational bands in lgl* lg3Hg iref. ‘)I, ‘*‘, ls9* 191*193Pt [ref. ‘“)I, 181*1830~, islRe [ref. ““)I, 16’Lu [ref. ‘“)I, 165Yb [ref. “)J and 157*159*161H~ {ref. ‘“)I have come to our knowledge. With the exception of the i+ decoupled band observed in 187-193pt, 1830~ and 165Yb and the &-[541] band in 16’Lu and ‘*‘Re backbending is observed in all cases. These observations certainly support the idea that the i, neutrons and the h, protons play a critical role in the mechanism for nuclear back-bending. It is, however, somewhat surprising that the h, protons seem to play this role for A = 167.
’ -Ho
317
The present investigation parallels the work on ’ “-’ 61Ho in ref. 13) to a considerable extent. However, in that investigation only levels having I++ equal to an even integer were reported and the theoretical model used to extract the moment of inertia was very crude. With the improved treatment of the particle-rotation coupling introduced in the present work it should be possible to verify the previous analysis and, furthermore, to look for evidence that the h+ protons in analogy with the h, protons are involved in the mechanism for back-bending. For these reasons it was felt to be both urgent and justified to continue and publish the results of the present investigation. 2. Experimental procedure and results A metallic terbium target was bombarded with u-particles accelerated to various energies between 43 and 51 MeV employing the 225 cm Stockholm cyclotron. At the same time the emitted y-rays were detected at an angle of 125” with respect to the incident beam using Ge(Li) detectors and a pulse sorting system. These y-ray spectra suggested that an a-particle energy of about 49 MeV was the most favourable for production of ’ 5‘Ho and beams of this energy were accordingly employed for the rest of the investigations. A two dimensional yy coincidence experiment was performed with two 40 cm3 Ge(Li) detectors viewing the target from angles of about +90” with respect to the beam. The coincidence events were stored on magnetic tape by the TRASK computer and the 4000x 4000 channel matrix was analyzed by introducing gates on one of the axes. A description of the electronic equipment employed in the coincidence experiment is given in ref. r4). In order to obtain information about the multipolarities of the observed transitions angular distributions of the emitted y-rays were measured by recording spectra at 56”, 90”, 120”, 135” and 149” with respect to the beam. The normalization of the spectra was based on the intensity of the K.# X-rays, the distribution of which was assumed to be isotropic. This normalization was checked by recording the number of cl-particles which were elastically scattered in the target through an angle of +40” by using two monitor detectors. Furthermore, the angle between the target normal and the detected y-rays was kept constant in order to minimize normalization errors due to absorption of photons in the target. The observed angular distributions were fitted with the expression w(e) = A, +A,P,(cose)
+A,P,(cose).
(1)
Conclusions about the initial and final spin and the multipolarity of the y-ray can be drawn from the ratios AZ/A, and A,/A, [ref. ‘“)I. The results of the present experiments are summarized in tables 1 and 2 and in figs. 1 and 2. Fig. 1 gives one example of the singles y-ray spectra observed and fig. 2 shows two coincidence spectra. The energies, the relative y-ray intensities and
318
I. FORSBLOM
et al.
TABLE 1 Summary of the y-ray data obtained in the lsgTb(a,
zvfor Z&=49MeV (65.0) 66.1 89.0 97.5 (107.2) 119.8
8.6 20.7 9.8 1oOb) 12.1 34.0
121.2
296.6
127.5 129.6 130.6 141.8 149.0 150.4 (162.0) 165.0 166.1 167.4 175.6 191.3 195.3 199.4 201.8 203.5 204.8 206.6 215.7 219.0 (225.0) 233.3 (241.6) (249.5) 251.3
9.7 8.6 13.4 10.6 17.3 114.1 6.5 8.6 39.0 91.1 12.2 8.4 14.2 10.4 13.8 90.9 54.9 18.0 18.6 23.2 7.1 29.5 8.6 11.0 22.1
253.0
166.5
255.0 257.2 258.7 267.5 269.3 271.9 283.1 284.5 289.4 292.5 295.2 297.0 302.5
17.8 20.7 8.7 6.9 16.6 52.4 5.8 7.4 5.6 6.4 34.2 42.6 8.2
Z,(E, = 20 MeV)
Z&E,= 49 MeV)
4n)i5gHo
reaction ‘) Assignments
Angular distribution coeaticients
AlAo
11 4
iAc/Ao
Zr x;
0.00~0.08 2.8 1.0’) 3.6
0.03 kO.03 0.19f0.10
1.4
0.06kO.03
Q %- 4 -H-
t+H-
8 b-;
radioactivity
0.42f0.10 (3.2) 2.5 0.45
3.2
1 1.6
6.7
(0.3)
1.7
0.9
0.45 3.4
2.4
0.16f0.06 I -0.08&0.11 0.03 &to.01 0.37&0.10 0.12f0.10 0.06f0.04 0.07zkO.12 0.28&0.15 -0.42&0.15 0.19~0.15 -0.11 rF_0.09 0.14po.11 -0.28f0.12 0.4o,f,o.o7 0.17hO.25 -0.02f0.15 0.04~0.04 0.24&0.12 -0.30*0.30
0.01 kO.01 0.02*0.09 0.26&0.10 0.29rfrO.16 -0.26&0.40 0.09f0.10 0.29 f0.02 0.02rjIo.20 0.07f0.20 0.28f0.30 -0.11 rto.40 0.07~0.04 0.08f0.04
-0.06&0.02
y
ic- J#
(3? 4’
H-;
ik St+)*
radioactivity q f- zj g-
part radioactivity 9 )- y )-0.04f0.03
-0.01 kO.05 -0.01 f0.05
y 4
%- aie %H- 6 f-
319
’ 5gHo TABLE 1 (continued) Zr for
Z,.(E, = 20 MeV)
Z% = 49 MeV
z,,(E, = 49 MeV)
Angular distribution coefficients AlAo
304.0 308.0 315.9
25.1 9.7 11.6
0.32
0.2950.03
317.8
72.7
0.26
0.24hO.03
323.2 349.3 353.0 (369.0) 370.8 (383.9) 385.6 389.0 408.2 (419.0) (425.0) 427.8 (441.3) 451.0 (455.8) 457.9 461.3 468.4 471.0 484.4 486.5 490.9 496.4 505.4 518.1 528.6 (534.4) 536.7 541.1 547.4 550.3 (554.5) 559.0 575.8 578.0 586.2 587.9 592.0
11.7 22.0 23.0 16.7 73.5 8.8 20.9 25.5 95.1 5.1 5.6 29.4 18.9 12.3 10.6 78.7 16.7 16.9 20.0 19.2 71.9 13.0 31.3 11.0 11.2 53.6 12.6 9.1 5.6 20.1 54.0 14.6 17.1 6.8 24.6 12.1 13.2 22.4
AlAo
Assignments Zt &
Zr Kr
-0.04*0.04
0.45i-0.25
0.49 1.4 0.27 0.46
-0.09~0.08 0.29f0.05 0.08f0.04 -0.30*0.50 0.42&0.10 0.41*0.15 0.30&0.07 0.37f0.05 0.31 f0.04 0.40f0.25 0.26&-0.20 0.30f0.05 0.15~0.08 -0.26&0.15 0.29f0.02 0.17&0.06 0.43 ho.08 0.35kO.06 0.41 kO.15 0.29hO.04 0.42&0.06 0.23 kO.05 -0.27f0.27 0.04&0.24 0.38f0.04 0.34*0.20 0.27f0.20 0.46hO.30 0.25&0.10 0.32kO.04 0.23f0.14 -0.10*0.12 -0.21 f0.14 0.42+0.07 0.96kO.23 -0.40~0.30 0.13*0.13
-0.06~0.04 -0.07*0.07 0.01*0.05
-0.09*0.07 -0.11 *to.06
-0.07&0.08 -0.13*0.11
-0.11*0.05 -0.10*0.07
-0.05*0.05
-0.14*0.14 -O.OS=kO.O5
‘) Transitions assigned to other final nuclei and transitions with relative intensity less than five are excluded from this table. Parentheses around energy value indicates that assignment to 159H~ is judged to be uncertain. s) Used for normalization.
I. FORSBLOM
320
ef al.
TABLE 2 Qualitative results of the coincidence measurement Coincident prays
Energy of gate y-ray (keV)
97.5 121.5 1so.4 167.4 203..5+204.8 215.7 219.0 233.3 253.0+255.0 271.9 295.2 304.0 317.8 349.3 370.8 389.0 408.2 457.9 448.4 486.5 490.9 528.6 534.4f536.7 541.1 550.3 578.0 586.2+ 587.9 592.0
122 98 98 550 98 122 208 150 541 122 458 116 272 98 110 208 98 253 122 540 207 122 122 216 122 253 150 253 216 122 592 150 150 550 150 529
(keV)
150 150 122
167 167 167
204 204 204
253 233 219
272 253 233
318 295 253
371 318 295
408 371 371
458 408 408
487 458 458
529 487 487
550 529 496
550 529
122 150 253 167 550 150 550 122 295 167 122 211 122 257 150 JSO 216 150 150 253 IS0 2.57 167 257 253 150
150 167 304 204
204 204 389 233
219 233 468 240
233 253 541 253
253 272
272 408 29.5 318
458 371
487 458
529 487
550 529
550
295
297
304
318
371
408
487
529
167
204
208
219
253
272
276
280
295
318
371
408
130 304 204 150 216 204 428 204 578 253 167 167 304 167 349 204 349 304 167
142 318 233 167 233 219 491 219
150 349 239 204 253 253 537 233
167 371 242 216 257 267
204 389 253 284 389 269
208 408 295 329 468 295
216 428 371 358 541 408
219 487 408
233 491 484
249 529 487
253
257
529
550
458
487
529
550
592
253
272
295
353
408
441
458
487
529
304 233 204 389 204 386 253 388 389 204
468 541 253 272 233 255 541 604 25.5 272 428 496 255 272
295 272
318 318
406 371
471 529
487 550
J18 J78
529
550
295
318
371
408
550
578
318
371
458
578
167 167 559 167 544
204 204 578 204 550
233 233 592 219 559
468 233
253
272
318
323
371
408
458
487
586
253 253
269 269
318 295
458 323
487 371
529 408
588 458
487
529
544
233 586
253
267
272
295
318
371
408
458
487
the angular distribution coefficients for all the lines one may assign to 159Ho are tabulated in table 1. Included in this table is some information derived from an experiment in which l5 ‘Ho is produced in a (p, 2n) reaction ’ “). The occurrence of a high value of the ratio I,@$, = 20 MeV)/f,(E, = 49 MeV) indicates that the transition is associated with the decay of low-spin levels in r5gHo. The qu~itative summary of all the coincidence relations observed is given in table 2.
lsgHo __, X103
321
_r-?--__r--_.----_?~~-~~-,-i-r-
15’Tb + 49 MeV
I-2035
6012i.5.R
60 1
1 1.99J i( I
R'i
CHANNEL
1
/ /
*2OL.6 253.0*R
I
a-particles 0=125”
I
I
NUMBER
Fig. 1. Spectrum of y-rays recorded during the bombardment of lsgTb with 49 MeV a-particles. The notations (a, a’), R and B indicate that the peak is assigned to the target nucleus, to the radioactive decay of rS9Ho, and to the background, respectively. 3. The level scheme of 159Ho
The rotational been established
levels belonging to the ground-state band in ’ 5 ‘Ho have previously q by Diamond t ‘) and Bout& “). With the aid of the up to spin
present experimental information it is a straightforward procedure to extend our knowledge of this band up to the member with spin y as shown in fig. 3. For the levels where I++ is an even integer these assignments agree with the recent findings
I. FORSBLOM
322
et af.
200
100 v) I-
z 3
0 ”
00
LL 0 CT
w
r” 200 2
300
-T-v-m-500 CHANNEL
7009
NUMBER
Fig, 2. Examples of the coincidence spectra obtained.
of Grosse et a2. ‘“) who in addition tentatively give the energy of the y level as 3707.3 keV. In addition to the ground-state band the experimental data (tables 1 and 2) suggest the existence of two additional cascades of E2 transitions namely the 253.0, 349.3, 427.8,490.9 and 536.7 keV cascade and the 215.7, 304.0, 389.0, 468.4 and 541.1 keV cascade. Due to the low energy of the low-spin transitions and the isomerism of the 4, f’ [41 l] level good coincidences with lower lying transitions are lacking and it has not been possible to establish without ambiguity the nature of these two cascades. Most probably they are associated, as suggested by Funke et al. 19) with the 3’ [41i J and $- [541] rotational bands. 4. Discussion 4.1. THE ROTATIONAL
ENERGIES
At low-spin values the energy of the E2 cross-over transitions in the ground-site band increases with increasing angular momentum roughly in agreement with the I(I+ 1) rule. At higher spin values this trend is broken and the data suggest that the transition energy may here even decrease a little as the spin of the nucleus becomes higher, The absence in the singles spectrum of prompt peaks with energies
l=Ho
323
‘-Ho
4865
233.3
21/2-. 19/2-
I 2530 4Y7Q
.L I 204.8 / ---+-370.6 203.5
4m2
t712m/213/2-
317.8 *01
7/2-
l6z4
1 2h.Q 1
do.4 12J.5
1
sg.5
1198.1 945.0 7403 5368 36ql 219.0 97.5 0
7/2- Et5231 Fig. 3. Level scheme of lsgHo showing the ground-state rotational band.
larger than 592 keV and of sufficient intensity (> 2 %) is a beautiful verification of this statement. Thus the y-ray spectrum resembles very closely the spectra that are found in (‘particle, xny) reactions leading to the neighbouring doubly even nucleides. The anomalous properties of ground-state bands in doubly even nuclei are perhaps best exposed when the moment of inertia is plotted as a function of the square of the rotational angular velocity 2P3). For such bands the formulae 24 -= R2
M-2
(2)
E(I)-E(I-2)’
tr202 = 4(P--1+1)
(E(I)-E(I-2))2 (4f-2j2
’
(3)
offer an approximate but reasonably accurate possibility of deriving the quantities desired. For the odd nuclei these formulae are not immediately applicable since the odd particle makes a contribution to the total angular momentum of the nucleus which must be subtracted out before the desired relationship can be derived. In a first
324
I. FORSBLOM
et al.
rough approximation one can find the moment of inertia and the angular velocity for the odd nucleus by assuming the validity of the rotation-aligned coupling scheme 20*21)* In this scheme the angular momenta of the particle and the rotation are completely aligned for the levels where I-j is an even integer and one has the relation R = I-j, (4) with R being the angular momentum of the core. For the other levels (where I-j is odd) the angular momentum of the core is R = I-j+l.
(5)
By replacing I with R in eqs. (2) and (3) and by assuming j = q- (the s- [523 J orbital originates from the h9 shell-model state) the analysis becomes identical to the one employed for the doubly even nuclei. The results are presented in fig. 4 (curves A) together with the curves for the surrounding doubly even nucleides ’ s8Dy and 16’Er. Because the assumption of complete decoupling is a poor approximation for the low-spin states in 159Ho, the curves for 1-j even and I-j odd (= 1+$ even or odd) differ and show too low moments of inertia for low angular velocities. However, in the region where the curves for ’ “Dy and r6’Er rise sharply, the curves for I5 ‘Ho follow the ones for the even nuclei amazingly well. It is possible to improve on the previous analysis by treating the full dynamics of quasiparticle motion in a deformed field which is assumed to have a variable moment of inertia. Such calculations are described in ref. 7). Very briefly the present calculations imply an approximate diagonalization of the Hamiltonian H = Hqp “i-H,, .
(6)
Here, the quasiparticle Hamiltonian is chosen in the spirit of the procedure developed in ref. ““). This means that the eigenstates of the Nilsson Hamiltonian are found for several values of e2 and .sq using ttcu, = 7.5 MeV, K = 0.0637 and p = 0.6. These single-particle states are then transformed to the quasi~rti~le scheme by means of the Bogo~ubov-Valatin transforn~ation using the BCS approximation with separate blocking, G = 0.1281 MeV and 32 levels included in the gap equation. The rotational Hamiltonian is written
ee*= 2$(I-j)”
+ v,,,,,
(7)
where the moment of inertia of the core, 3, is the constant relating core angular momentum I-j to core angular velocity 0 i.e. h(l-j)
= $a.
(8)
In the following the moment of inertia of the core is allowed to vary with spin or angular velocity as suggested in the VMI model 2*3*23).Hamilton’s equations then
159H~
325
lead to the condition ‘*“) d
do = odl,
/s
(9)
which after some trivial algebra ‘) may be integrated to give
vcore
=
l
-
2 9(&=0)
co* d9.
For low and moderately high rotational angular velocities the success of the classical VMI model 23P24)tells us that the expansion 9 = &)+$uP,
(11)
Vcore = 3h4,
(12)
with gives an amazingly good description of nuclear rotational energies in doubly even nuclei. Only the orbitals emanating from the h, state are strongly connected through H,,,. Therefore, only these six orbitals are needed in the rotational energy matrix. Following an estimate given in ref. * “) a common value of 36 is chosen for j* for the 11, orbitals. It is also assumed that V,,,, is diagonal in the representation chosen and that V,,,, and X are the same for all states of a given spin. In practical calculations the values of a*, 9 and V,,,,, are found through an iterative procedure using eqs. (lo)-(12) and the equation
(13) where Q1, o is the wave function of the lowest level with spin I. When the data for r 59Ho are analysed within the present model it is found that the Coriolis matrix elements connecting the ground-state band have to be reduced while the matrix elements connecting the excited bands must be enhanced if the transition energies in the low-spin part of the band shall be fitted. Such a need for reducing the theoretical Coriolis coupling strength is generally encountered and is here defined as
=
(14
On the other hand, the need for enhancing these couplings is judged to be unphysical and is believed to be an artifact of the calculation. This may be explained as follows. Since the energies of the low-K h, orbitals decrease very strongly with increasing deformation and since these orbitals are hole states in the present calculation one may expect that the deformation is here smaller than in the ground state. With decreasing deformation the importance of the couplings to the high-lying low-K orbitals is rapidly increasing as discussed by Alonso et al. *“). Our actual
326
I. FORSBLOM
et al.
calculations also show that a very good fit to the transition energies below spin J$ can be obtained with all CL~S 1 provided the quadrupole deformation e2 is reduced from the recommended value of z. 0.23 [ref. ““)I to 0.215. In comparison, a variation of sq between 0 and -0.04 has little influence on the calculated energies. Therefore, the values of &2 = 0.215 and E,, = -0.02 [ref. ‘“) J are chosen for the rest of the calculations. By following the aforemention~ assumptions and the approximations implied by eqs. (11) and (12) the experimental transition energies below spin 9 are fitted within an average error of 0.1 keV. In this fit the values of the adjustable parameters are 24,/fr2 = 45.6 MeV-‘, /?,&” = 107.1 MeVW3 and 4 = 0.65 (all cc, with K # 5 are assumed to be unity) in addition to the previously chosen value of s2. Beyond spin y the theoretical transition energies are systematically smaller than the ones experimentally observed and one may assume that these deviations are primarily due to the approximations implied by the use of eqs. (11) and (12). A set of “experimental” values of 2Y/fi2 and h2w2 is here found by maintaining the parameter values s2 = 0.215 and R%= 0.65 while varying the moment of inertia along other straight lines X = ~~(~)~~~(~)~2.
fll’)
The new parameters g&(1) and /?‘(I) are subject to the constraint that the previously calculated moment of inertia of the preceding level f-2 is reproduced.
3
lsoEr ‘Y!?? ,,,’ ,,,_....---,! .f”
Id
I.
*
*
*
1..
on5
*
a
I
I
L
al0 #%?
Fig. 4. Moment of inertia as a function of the square of the rotational angular velocity for ’ sgHo and for lsaDy and 160Er. The curves for rs9Ho labelled A correspond to the derivation assuming the validity of the rotation-aimed coupring scheme while the curves labelled B refer to the derivation where j2 is 36 and where the full dynamics of the quasiparticle motion is treated. The curves denoted C correspond to the case where j2 is reduced to 18. All curves for ls9Ho include the point due to the 547.4 keV 93 transition tentatively identified in ref. r3).
‘“gHo
327
One must also employ eq. (10) rather than eq. (12) when V,,I, is calculated. In this way it is found that a smooth continuation of eq. (11) can be obtained only if the levels having I++ even and I+$ odd are fitted separately. Hence the result of the analysis splits into two separate curves for spin larger than v. However, as seen in fig. 4 curves B, both curves indicate a very rapid increase of the moment of inertia at h202 z 0.095 MeV2. With the last analysis the curves for 1“Ho fall much closer to the ones for 16’Er for low spin expressing the improved treatment of the quasiparticle degree of freedom. However, for the larger spin values the calculations predict the back-bending to occur at a larger rotational angular velocity than in i6’Er and “‘Dy. The position of back-bending in 15gHo determined is remarkably stable with respect to the choice of parametrisation and other details of the fit. Thus in calculations where E2, eq and tcx are chosen in different ways and in calculations where levels up to spin si- are fitted using eqs. (11) and (12) the position of back-bending always falls within the limits 0.094 MeV2 5 h2m2 5 0.100 MeV2. From this study it is concluded that in our model l5 ‘Ho is predicted to back-bend at a significantly larger angular velocity than “*Dy and 160Er. This different position of back-bending in ’ 5gHo and in 16’Er and ’ 58Dy may reflect different properties of the core for the even and odd systems but could also reflect errors inherent in the present model and method of calculation. In the former case one may think that the h, protons to some extent take part in the back-bending process and that the blocking of an h, orbital through the short-range residual neutron-proton interaction effectively raises the excitation energies of the rotationaligned states. This will indeed move the point of intersection of the ground band and the rotation-aligned band to higher spin and angular velocity but at the same time the angle of intersection will decrease. This means that the bend is expected to become smoother. Such an effect is observed in “lrl g3Hg [ref. “)I and lg3Pt [ref. ‘“)I where the h+ orbital and the iY orbital probably are both responsible for back-bending. Here, such an explanation contradicts the experimental findings because the abrupt increase of the moment of inertia is essentially the same in ’ “Ho and in 16’Er. The blocking of an h, proton orbital could also effect the properties of the core in a more indirect manner e.g. through inducing a change of the nuclear deformation. Such effects are expected to show up also when back-bending is investigated as a function of proton number and since all doubly even nucleides around A = 160 back-bend for the same angular velocity such effects are believed to be small. As the second main explanation for the large value of the critical angular velocity found one may argue that the approximations and assumptions inherent in the present treatment of the motion of the odd particle are not justified. One of these assumptions is that the moment of inertia is the same for all levels of a given spin. Since the ground band, due to the rotation-alignment mechanism, has a very low vaiue of the rotational angular velocity one may argue that the angular velocity and
328
I. FORSBLOM et al.
of inertia the excited are larger. means smaller matrix elements smaller energy between the elements in energy matrix. effects of changes are and tend cancel. Therethe error by the of having common moment inertia for the bands believed to small and will be absorbed in adjustable parameters /? and The last may not when the exhibits back-bending here the of the to the of higher of inertia manifest themselves at average velocities below expected critical of 0.085 The fact our calculated behaves as in the below ti2w2 0.085 MeV’ a posteriori justifies the approximation of having the same fl for all levels of a given spin. In the present evaluation of H,,, the matrix elements of I *j were reduced while the ones ofj’ were not. This increases the rotational energies and means that #i202 may be overestimated in the calculations. This is shown by the curves C in fig. 4 which represent a calculation wherej2 is reduced from 36 to 18. Since the predicted rotational energies are all reduced by the same amount the quality of the fit to the rotational energy spacings is not affected and the reduction ofj2 essentially implies a linear translation of the curve in fig. 4 from right to left. The calculation shows that it is indeed possible to obtain a curve resembling the ones for the even isotopes in this way. However, the employed reduction of jz is larger than expected from the introduced reduction of the matrix elements of j. The second possibility of obtaining back-bending at hZo2 w 0,085 MeV2 in ’ 59Ho, namely to remove completely the artifical reduction of the Coriolis matrix elements, destroys the fit to the low-spin states. A possible way out of this dilemma is then of course to allow for the attenuation factors to be spin dependent. However, it seems premature to introduce such a hypothesis at the present time. 4.2. THE RELATIVE
ELECTROMAGNETIC
TRANSITION
RATES
In the present experiment the angular distributions of the y-rays depend on the E2/Ml mixing ratios in the cascade transitions 1“). Similar information on the relative strength of the E2 and Ml radiation is contained in the ratios of the intensities in the cross-over and cascade transitions. While these branching ratios normally give the most accurate information, the E2/Ml amplitude mixing ratios are useful in the sense that they determine the sign of the mixing. Since the amplitude mixing ratios are positive, both in the experiment and in all theoretical calculations, only the branching ratios will here be compared with theoretical calculations. Such theoretical calculations of branching ratios are described in ref. 27) for the special case when the rotational band is not perturbed, while the formulae for the calculation of nuclear electromagnetic transition probabilities in deformed nuclei in general may be found in refs. 2**29). Before the experimental and theoretical branching ratios are compared it is useful to remove the strong energy dependence of the branching ratio by multiplying
’ 59Ho
I,
I,
11 13 15 17 T-TY?TTTTTTTTT
329
I
I
I
I
I
I
t
I
I
19
21
23
25
27
29
31
33
35
I Fig. 5. Experimental
and theoretical branching ratios within the ground-state rotational band in ls9Ho. The errors in the experimental points correspond to the statistical errors and to an estimated uncertainty of w 10 % in the intensity calibration curve. The curve labelled A corresponds to the completely adiabatic case where the intrinsic wave function is assumed to be the p- 15231 Nilsson orbital and where Qc/(gx--gR) is adjusted to fit the point for Z = +. The curve labelled B is based on the results of the Coriolis calculations described in the text and uses g,, I = gOf = 0.6 g,, rrco = 3.35, gt = 1.0, gR = 0.36 and Q,, = 6.85 b. In the curve labelled C a further reduction of the matrix elements of g,*s,: is simulated by putting (KflIplK) =+-uKax+l
Z,(Z --, Z-2)/1,(1 + I- 1) with (E,(Z + I- l))3/(E,(Z --, Z-2))5. The comparison of such reduced experimental branching ratios with four theoretical calculations is shown in fig. 5. In the first calculation, labelled A, the wave function for the intrinsic state is assumed to be 100 y0 s- [523] and the quantity (gK-g&Q0 is chosen as 0.092 which fits the branching ratio in the decay of the y- level. With Q, = 6.85 b [ref. “)I and gR = 0.36 this implies gK = 0.99. This value of gK is definitely smaller than the range of values 1.26 5 gK 2 1.51 suggested by the calculations of Lamm 31). Furthermore, with the parameters chosen this simple calculation predicts too large branching ratios for the high-spin states. One may try to improve on these matters by using the wave functions obtained in the present Coriolis calculations and by considering various assumptions about the longitudinal and transverse spin gyromagnetic ratios gS,z and gSk. The results of such calculations are included in fig. 5 (curves B-D) where also the parameters corresponding to each calculation are given. It turns out that these calculations correctly predict the branching ratios in the decay of the low-spin states but the disagreement for the high-spin states is not removed. Furthermore, these more sophisticated calculations tend to predict too large oscillations between the points where I+$ is an even integer and the points where Z++ is odd. These oscillations are strongly connected with the contributions to the total transition amplitude which
330
I. FORSBLOM
er al.
have 1dK.l = 1 and the curves become smoother when gSk is reduced. In principle it is possible to check this explanation by considering the magnetic moment which exhibits *‘) a different dependence on gS,z and gSrt. However, since the magnetic moment of the ground state of ’ 59Ho has not been measured, such a comparison cannot be done at the present time. It is an interesting fact that the theoretical branching ratios are too large for the higher spin states and that this discrepancy persists in all the calculations. Furthermore, it is not at all obvious how the Coridis calculations should be modified to give theoretical branching ratios in agreement with experiment. On the other hand, the collect.ive gyromagnetic ratio gR which is approximately given by
is expected to decrease with increasing spin since 3m is expected to increase faster than .Yb for spin 5 20 tt. In particular, if back-bending is a property of the neutron system a substantial reduction of gR should accompany the sudden increase of the moment of inertia.
Fig. 6. Experimental and theoretical branching ratios within the ground-state band in 13gffo. The theoretical curves labelled A and D are the same as the curves with the same labels in fig. 5 except that ga varies with spin according to eq. (15) and fig. 4.
In order to demonstrate the effect of the variation of gR we have made the extreme assumption that the increase of the moment of inertia observed in fig. 4 is entirely due to the neutrons and that gR decreases with spin according to eq, (15). With this variation of gR included the results of calculations analogous to the ones giving the curves A and D in fig. 5 are shown in fig. 6. Obviously the data are better described in this way. However, even on this rather extreme assumption a small
159Ho
331
discrepancy remains. This must be interpreted within the ambiguities of our Coriolis coupling calculations or, alternatively, be interpreted in such a way that the collective B(E2) value also varies. The very small values of the branching ratio which are observed for I = v and I = y could thus be due to the reduction of the B(E2) value expected ‘*“) m * t he re g’Ion where the moment of inertia changes very rapidly. The present data indicate that a reduction of about 50 % is conceivable. Unfortunately, the presence of events of other origin in the tiny photopeaks corresponding to the 31 and q to q transitions would have the same effect and cannot be 2 to 2A 2 excluded because in the coincidence spectra the error in the intensity values are of the order of 50 %. 5. Conclusions
It is possible to draw the following main conclusions from the present investigation: (i) The pattern of rotational energies in the ground-state band of 15’Ho is well described within a model where a particle is coupled to a rotor with a variable moment of inertia if the Coriolis matrix elements connecting the I- [523] orbital are reduced. Furthermore, the set of values deduced for the moment of inertia and angular velocity resembles the set which is found in the adjacent nucleides of even mass. This suggests that the employed method of subtracting out the angular momentum of the odd particle is essentially correct. The fact that this method, when applied to the neighbouring nucleides with an odd number of neutrons, yields moments of inertia which are quite regular far beyond the critical region of h2fD2 = 0.085 MeV’ [ref. ‘)I indicates that the observed anomalies in the moment of inertia 2*3) are due to the neutrons. (ii) The observation that the deduced values of the moment of inertia for ’ 59Ho start to rise sharply at a slightly larger rotational angular velocity than in the adjacent doubly even nucleides is not completely understood. Judging from the discussion in subsect. 4.1 the uncertainties inherent in the procedure of subtracting out the angular momentum of the odd particle are quite large and are probably the most likely explanation for the discrepancy observed. Thus the discrepancy is removed if the introduced ad hoc attenuation of the matrix elements connecting the 3- [523] orbital disappears at high spin or, more likely, if the matrix elements of j2 are also reduced. The second possibility of explaining the postponement of backbending to higher angular velocities namely as an effect of the blocking of an h9 orbital is believed to be less likely. Furthermore, due to the aforementioned uncertainties in the treatment of the odd particle the presence of small effects due to the blocking of one h, orbital cannot be established. (iii) The persistent disagreement between the trends of the experimental and theoretical branching ratios suggests that the collective gyromagnetic ratio gR or the rotational B(E2) value of the core is reduced with increasing values of spin. In
332
I. FORSBLOM
er al.
particular, by making a reasonable but somewhat optimistic assumption about the variation of gR the agreement between theory and experiment is greatly improved for spin values below 9. Beyond spin “$, e.g. in the region where the moment of inertia of the core rises very sharply, a 50 % drop in the trend of the branching ratio suggests that the B(E2) values are reduced here by roughly the same amount. The authors wish to thank the cyclotron reconstruction and operation groups under the directions of Prof. H. Atterling, Mr. K. G. Rensfelt and Mr. G. Printz for having completed the improvement program for our cyclotron and for the operation of the accelerator. References 1) F. S. Stephens and R. Simon, Nucl. Phys. A183 (1972) 257 2) 3) 4) 5) 6)
7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24) 25) 26) 27) 28) 29) 30) 31)
R. A. Sorensen, Rev. Mod. Phys. 45 (1973) 353 A. Johnson and Z. Symadski, Phys. Reports 7C (1973) 181 B. R. Mottelson and J. G. Valatin, Phys. Rev. Lett. 5 (196O) 511 E. Grosse, F. S. Stephens and R. M. Diamond, Phys. Rev. Lett. 31 (1973) 840 W. F. Davidson, R. M. Lieder, H. Beuscher, A. Neskakis and C. Mayer-BOricke, in Proc. Int. Conf. on reactions between complex nuclei, Nashville, 1974, vol. 1 (North-Holland, Amsterdam, 1974) p. 190 S. A. Hjorth and W. Klamra, University of Stockholm report USIP 74-21 (1974) L. L. Riedinger et al., Phys. Rev. Lett. 33 (1974) 1346 R. M. Lieder, H. Beuscher, W. F. Davidson, A. Neskakis and C. Mayer-Boricke, preprint 1975 M. Piiparinen, I. C. Cunnane, P. J. Daly, C. L. Dots, F. M. Bemthal and T. L. Khoo, MSUCL 150 (1975) R. M. Lieder, private communication C. Foin and D. Bam6oud, Phys. Rev. Lett. 33 (1974) 1049 E. Grosse, F. S. Stephens and R. M. Diamond, Phys. Rev. Lett. 32 (1974) 74 L. Harms-Ringdahl and J. Sztarkier, Nucl. Instr. 108 (1973) 557 T. Yamaxaki, Nucl. Data 3 (1967) 1 J. Bout&, private communication R. M. Diamond, in Proc. Int. Conf. on the properties of nuclei far from the region of beta stability, Leysin, Switzerland 1970, p. 65 J. Bout&, Thesis, University of Grenoble, 1973 L. Funke, private communication P. Vogel, Phys. L&t. 33B (1970) 400 F. S. Stephens, R. M. Diamond and S. G. Nilsson, Phys. Lett. 44B (1973) 429 S. 0. Nilsson et al., Nucl. Phys. A131 (1969) 1 M. A. J. Mariscotti, G. Scharff-Goldhaber and B. Buck, Phys. Rev. 178 (1969) 1864 0. Saethre, S. A. Hjorth, A. Johnson, S. Jiigare, H. Ryde and Z. Sxymslllski, Nucl. Phys. A207 (1973) 486 W. Ogle, S. Wahlbom, R. Piepenbring and S. Fredriksson, Rev. Mod. Phys. 43 (1971) 424 I. Alonso, H. Bakhru, F. M. Bernthal, J. Bout&, B. Olsen, I. Rezanka and J. 0. Rasmussen, Nucl. Phys. A160 (1971) 193 F. Boehm, G. Goldring, G. B. Hagemann, G. D. Symons and A. Tveter, Phys. Lett. 22 (1966) 627 S. A. Hjorth, H. Ryde, K. A. Hagemann, G. Lsvheiden and J. C. Waddington, Nucl. Phys. Al44 (1970) 513 S. A. Hjorth, A. Johnson and G. Ehrling, Nucl. Phys. Al84 (1972) 113 P. H. Stelson and L. Grodzins, Nucl. Data 1 (1965) 21 I. L. Lamm, Nucl. Phys. A125 (1969) 504