Single-neutron states at high spins in ytterbium nuclei: Evidence for the quenching of static neutron pair correlations

Nuclear Physics A442 (1985) 509-546 ©North-Holland Publishing Company

SINGLE-NEUTRON STATES AT HIGH SPINS IN YT]'ERBIUM NUCLEI: EVIDENCE FOR THE QUENCHING OF STATIC NEUTRON PAIR CORRELATIONS J.C. BACELAR a, M. DIEBELb, C. ELLEGAARD, J.D. GARRETT, G.B. HAGEMANN, B. HERSKIND, A. HOLM, C.-X. YANG c and J.-Y. ZHANG a

The Niels Bohr htstitute, Unioersity of Copenhagen, Copenhagen, Denmark P.O. TJ~IM

Physics Institute, Unit,ersity of Oslo. Oslo, Norway and J.C. LISLE

Schuster Laborato~', University of Manchester, Manchester, UK Received 14 March 1985 Abstract: The decay schemes of 167'168'169ybare established to the largest angular momenta yet known in

stably-deformed rare-earth nuclei using the 124Sn(4SCa,3-5n) reaction. The systematics of the resulting spectrum of single-neutron states in these and neighbouring Yb isotopes at large and small angular momenta are discussed with regard to that expected in the presence and absence of static neutron-pair correlations. This comparison, together with cranking-model and "gauge-space" analyses, provides evidence for the effective disappearance of static neutron-pair correlations for the odd-N isotopes at h~o/> 0.38 MeV. A composite empirical spectrum of single-neutron states is constructed for the "unpaired" regime and is compared with cranking calculations.

E

NUCLEAR REACTIONS 12aSn(4SCa,3n), (4SCa,4n), (4SCa,5n), E = 201 MeV: measured Ev, y't-coin, "I,(8). 15*Sm0SO,4n), E = 7 8 MeV; measured y(8): deduced static t67yb, 16syb, 169yb deduced levels, J, ~r, rotational bands. Cranking-model, gauge-space analysis, neutron-pair correlations.

1. Introduction

The study of high-spin decay sequences in deformed odd-A nuclei has yielded considerable spectroscopic information on the spectrum of single-particle states in rotating deformed nuclei. Summaries of the type of information obtained prior to a Present address: Nuclear Science Division, Lawrence Berkeley Laboratory, Berkeley, California, USA. b Permanent address: AEG Software Technik, Darmstadt, West Germany. ~ Permanent address: Institute of Atomic Energy, Beijing, People's Republic of China. a Permanent address: Institute of Modem Physics, Lanzhou, People's Republic of China. 509

510

J.C Bacelar / trz-Ir°Yb

about two years ago are contained in e.g. refs. 1-3). Such data are the experimental basis of our understanding of independent-particle motion in rotating deformed nuclei in the presence of static pair correlations. For more than two decades it has been recognized 4) that the fictive Coriolis force acting in a rotating quantum system, such as the nucleus, quenches pair correlations. The sign of the Coriolis force for nucleons moving in the direction of rotation is opposite to that for nucleons moving counter to the rotational direction. Hence the spatial overlap of time-reversed orbits is decreased in the rotating system resulting in reduced pair correlations. This effect is largest for nucleons moving with, or counter to, the direction of rotation, i.e. for orbits in which the particle angular momentum is aligned with or counter to the rotational angular momentum. Indeed, rotational alignment or "backbending" s) is interpreted 6) as the breaking of one such highlyalignable pair of quasiparticles. The angular frequency of such band crossings or "backbends" corresponds to the rotational frequency at which the centrifugal plus Coriolis forces counter balance the pair correlations for a specific pair of highlyalignable particles. A cardinal question, however, remains: Does the rotationally-induced Coriolis force completely quench pair correlations? Indeed, at higher angular momenta there is mounting evidence 7-9) that static neutron-pair correlations are greatly reduced. The definitive answer to this question should be contained in the energy spectrum of. the "basic" single-particle, or single-quasiparticle, excitations, which is available from the experimentally-established level schemes. In the absence of pair correlations the spectrum of single-particle states should be greatly simplified, since quasiparticles need not be considered. When expressed in the rotating frame, where the independent-particle hamiltonian is expected to be valid, the observed decay sequences for unpaired odd-A nuclei should define a series of nondegenerate single-particle levels as a function of the angular frequency of rotation, h co. The rules for occupying such states should be the same as for any "shell model" with the exception that each level can be occupied by only a single nucleon. [The nondegeneracy condition is a result of the small number of symmetries present in a rotating deformed system. Time-reversal invariance is broken in a rotating system, thereby lifting the twofold degeneracy of Nilsson states ~0).] This paper presents new discrete-line data for 167yb, 168yb and 169yb. In t67yb and 169yb known decay sequences H.12) are extended to larger angular momentum. In 168yb two new decay sequences are established, and three known sequences are extended to larger angular momentum. Such systems are characterized by well-defined, stably-deformed shapes and a large frequency difference between the low-lying neutron band crossings and the proton band crossings, which are expected at relatively large rotational frequencies in these isotopes s.9). Therefore, rotational bands, corresponding to a variety of specific intrinsic configurations with a relatively small number of "aligned" particles can be studied to quite large angular momentum (or angular frequencies).

J.C. Bacelar / ter-te°Yb

511

The new data are compared with existing high-spin data for lighter odd-N Yb isotopes 8,11,13,14). The features of the single-particle spectra at large angular momenta both can be distinguished from those at smaller values of h60 and are characteristic of that expected for a nucleus without static neutron pair correlations. From these data, together with the existing large angular momentum data for 165yb, ref. 8), a composite spectrum of single-neutron states is constructed at large angular momenta for N = 95-100. The qualitative features expected for an unpaired system are observed, even though the observed spectrum of states is not exactly reproduced by self-consistent cranking calculations 15). When the complications of pair correlations are removed the spectrum of states should become more sensitive to other nuclear properties, e.g. the nuclear shape and the details of the nuclear potential. Some of the ideas and data presented in this paper have been given in preliminary form in refs. 16-21).

2. Experimental procedures and analysis 2.1. EXPERIMENTALPROCEDURES The decay sequences of 167yb, 168yb and 169yb were studied at large angular momentum (see figs. 1-3) by measuring yy coincidences corresponding to the appropriate evaporation residue populated by the ~243n(aSCa, 5n), t24sn(aSCa,4n), and 124Sn(48Ca, 3n) reactions, respectively. A 201 MeV 48Ca beam was provided by the Nuclear Structure Facility at the Daresbury Laboratory. The coincidence y-rays were observed in an array of six Compton-suppressed germanium detectors (TESSA II)22). A total of 45 million Compton-suppressed coincidence events were obtained. Gamma-ray multiplicities M r and total y-ray cascade energies also were recorded in an array of 50 bismuth-germanate crystals subtending a physical solid angle greater than 90% of 4~r. Specific evaporation residues, e.g. 4n, were enhanced relative to neighbouring channels, e.g. 3n and 5n, by imposing appropriate y-ray multiplicity restrictions. This technique also discriminates against other low multiplicity processes, e.g. Coulomb excitation and radioactivities from daughter nuclei. Typical spectra obtained by summing coincidence spectra derived from several gates are shown in figs. 4-7. Since TESSA has Compton-suppressed germanium detectors located at 30 ° , 90 ° , and 150 ° with respect to the beam axis, it is possible to extract limited angular distribution information, which can assist in the assignment of y-ray multipolarities. The ratios of the coincident y-ray yields at 30 ° and 90 ° are given in tables 1, 2, and 3 for ~67yb, ~68yb, and 169yb, respectively. The 150 ° data are included with the 30 ° data, since such angular distributions are symmetric about 90 ° . For ~68yb, in which new decay sequences have been established, full angular distributions also have been measured for y-rays following the *54Sm(*80,4n) r

512

J.C. Bacelar / t~z-J6°Yb

(73/~)

10644 I124 9zo8

(%-)

I 1032 8676 t ( % ) 962 7714 t (6~:-)

4497

4~2-

3837

4y;

3237

3~'z-

2683.6.

3~v'{

2158.3

2

1656,5 l,92.6 783Z

442.3

on

178.9

¢" ~

(,5./;) (6y2*)

939 6504 [ 6012

I

(sJ/~.)

878 5134 t (47/2") 829 4305 I (,,3/~) I

57/2*

869

5635 t

s~2*

8O2 4833 t 49/'j 742 4090.8 45/,~

773

3532

3~/~

3398.5

41/2÷

2817

35/~

2750.9

37/2+

.~2158.5

31/'2

2148.1

~1569.5

"/2~'--.1060.6 17//~-~ _ ~

8449 t 7443 t

6818 ~ ( % - ) I

7~'4 52L3 t ( ' ~ - ) 78

('~'j)

1006

896 831 5987 t ( ~ N )

9520

2~2 "~'-~1601.1 2~/~ "~--~1 ,2

""~644.3

,.7

,9/~ ~ 7 2 1 . 2

13/i~~r~ 15/,~ ~--~---_~407.8 9/2--""---~5~-~' } ,l;2~"~- ------~.185.9 9"2- 3 3 8 " - - ' 1 ~ 1 - - ' - - T / ~ 584 ,

zs/z~ Zl/2÷ 17/2" 13J2*

16"ryb Fig. 1. Decay scheme for 167yb. Gamma-ray transition energiesare given for new transitions established in the present work. The remaininglower-spintransitions are from ref. u ).

J.C. Bacelar / t ~ - / ~ q Y b t~

e

i

OJ

0,1

0,.I

O,J

513

OJ

0

4 0

o

i

OJ

I'~

03

0,1

0,,I

OJ E o

~

~

o

o

~

~

--

_

_

Q~

.

C~

&

D-

0 (kJ

Z

I ] -I -I ~1 ] -I ] .~

~

~

o ~

~ ~

co

o

~.

~

~.

-

--

--

J.C. Bacelar

514

//~7-t~°Yh

10962

( 73/~- )

I 1107

9855~

("/2-)

1047 5 8807.81

r 7771.8

9833 7824.5 ~ (61/2-)

(%*)

I

I

9197

896

6875.8 t (~") 829

6904.8 [

6046.8 t ( 5~/~*)

6049.5 ]

855.5

(4~2÷)

5116

I

786

4339 t (4~')

3782.8

4~

3075.8

37/2*

2426.7

3 ~ 4"

1845.2

29/'~

1336.8

25/z*

903.5 547. 3

17,/'2" ~

70.3

~3~-

791.8

t (%')

748.1 452510 4~ +

269.7

',~'2-

I

773.7. 5273.,

(~/~-)

3588

742

49/£

I

729.6 4528.1 45/~-

3857.3

4Yz-

3239.8

37/~-

2 6 69.2

3~-

2141.4

2~2-

1656.7

2~,~-

1218.8

zY2-

834.1 ,~12.3 264.2

13/~-

(~'~)

2929

35//2"

2258

31/2~"

1666

z7~ ÷

~_1156.9 19/2+

,~i6o.9

5257.7 I

404.3

. ~ '

1:/2-

169yb Fig. 3. Decay scheme for 169yb. Gamma-ray transition energies are given for new transitions established in the present work. The remaining lower-spin transitions are from ref. - ).

515

J . C . B a c e l a r / t ~ z - t*o Y b

TABLE 1 Experimental

summary

for 167yb

E~, a) [keV]

Lib)

I ~l: / l~)o "b

f27.5 178.9 204.4

3s 47 75

f.os (0.02) 0.96 (0.061 0.9o (o.o2l

221 9

100

1.00 10.02l

I +, ~l '.:' - ':~

263.4

53

0.93 10.051

i 15 .) I - . ~ I 2 ~ "-

313.4 ~ 314.2 / 341.2 400.5

184

1.00 (0.02)

27

1.24 (0.03) d)

4o9.1

lo5

1.55 (o.o71

416.3

92

1.44 (0.021

463.9

121

o)

479.4

1.00 (0.041 dl

d)

Assignment

I + , '~l ? ' - 2'" ( - , '~l ~ ~ ~ • ( +, - ~) ':" ~ '.,' "

I + ,.,t ) :!: . ~ 17.,• (+. _ i,) ,~.• ~ i~ . I • ':) ': ~ ': ( +, '.,l.,:* z} *

i - . ':) -'.2 ~ 7 ( +. - ': ) ~ " ~ ~ " I - . ~) 2: ~ 2; ( +. '.,)-'~" ~ " "

501.8

115

10510.021

I - , ~) :.,' ~

~':

509.0 525.3 547.0

'1 102 105

") 0.95 (0.03) 0.92 (0.02)

I+.

~ ~'" :* :

589

112

0189 (0.021

( + . -- '~) ') " __ 2: +

600

120f)

0.88t)(0.06l

(-,:)

602.8 647.6

121

0.91 (0.06l ~)

I + , '.,I ' : " -- 7 ' ( + 1 I] ~ ~ I " ~ ': '

93

1.22 (0.03)

( + ' - I'1 ~ " ~t I -.',). '~., ~ '~'., ( +, t) 7 " -- ~ "

658.5 660.0~ 692•3

')

121

0.98 (0.031

715 716

163

0.95 (0.04)

742

70

1.16 (0.051

773 774 802

82

1.29 (0.061

"~)::" ( _ L ). .: 2 ~ ( + . ~z) ~:" ~ i

,it

2~. 37

2

~ -.'-

I + . - '.,1 ~ ' ' ~ ' " :

(s " •

( - . ': ) ' :

( +, ':) '.,'" - ','" (+,

829 831 869

1.14 10.071

~,)'~" ,,. i ) ~3 -" *q : ~ : I +. '2) s~ • ~ ~, •

') a)

( + . - '.,) ~'" ~ 7 ' (-.~l~] --']

(-..

47 ~) d) 17

1.14(0.101

( + , ~ ) 7 " -- ' : "

87s

")

")

(+.-':)

896

'1 )

~)

{ I ' 2I ) ~1

939

21

0.86 (0.091

7 " ~'.,' " ~

':

( +. ~2) ~ " ~ s] ,

962

a)

d)

( _ . ~.,i (,.~ ~ I

1006 1032

dl d)

al aI

( + , ':)~' " ~ ; ' ( _ . ' ) ~" ~(','

1o71

")

")

( + . ' : ) 7 ' ~'.,'"

1124

dI

d)

i + . '., ) ~' " _ _ ~ ' '

~) Gamma-ray transition energy obtained from the l:4Sn('aSCa.3nl reaction. For some of the lower transitions these values may vary from those in fig. 1 which have been taken from previous studies. ~') Gamma-ray intensities obtained from average of 30 ° and 90 ° projection spectra normalized to the intensity of the 221.9 keV '] " ~ *] " transition. ") Ratio of 30 ° and 90 ° intensities obtained from projected spectra normalized to the value for the 221.9 keV ~7 • ~ ,] + transition. The uncertainties, shown in parentheses, are the larger of 0.02 or the value obtained from ganssian fits to the d a m d) Value cannot be extracted due to either proximity of another transition or too weak. c) Masked by transition in 169Yb. t) Possibly contaminated by nearby transition.

J.C. Bacelar / t~7-t~°Yh

516

TABLE 2 E x p e r i m e n t a l s u m m a r y for t 6 s y b

Ivb )

Ev a )

A 2/Ao ~ )

13o./I~:

Assignment

- 0.044(036) - 0.113(027)

0.97~ ) (0.02) 1.00 (0.02) 1.89 (0.24)

0.251(025)

0.105(026)

1.11¢)(0.02)

-0.562(243)

-0.189(325)

0.216(030)

- 0.092(032)

( + , 0 ) 4+--*2 + ( + ,0) 6+---, 4 ÷ ( - , 0 ) 10 ~ 8 (+,1)9+~7 + (+,0) 8+~6 + ( - , 0 ) i2---* 10cross-band 10 - --* 9 * (+,1) 11'~9' ( + ,0) 10 + ~ 8 + ( - , 0 ) 14 - - 1 2 cross-band 8 - - 7 + (+,1)13+~11 +

[keY]

154Sm(l"O,4n )

t24 Sn(48 Ca,4n)

198.8 298.7 325.8

121 100

99c) 1130 3

0.405(034) 0.265(026)

384.3 ~ 384.6 ) 398.6 423.7 441 455.4 470.0 482.2 488

104

113':)

2

r) 75

91 5

5

r) 104~ )

1.44 (0.14) 2.15 (0.11) 3.07 (0.30) r)

r)

37

15 3 4 9

5

1.17 (0.02) 1.21 (0.06) r)

1.00~ ) (0.002)

3 4 2 63

1.55 (0.04) 0.75 (0.09) f)

1.04 (0.06)

4

510.6 ~ 511.01 517 532.6 535 540 552.8 ~ 552.8) 582.5 584.7 594.4 597 613.9 62O.3 630.5 643.9 649.9 656.6 677 686.2 699.9 702.4 705.0 747.9 754 760.5 764.0 767.2

15 2

A4/Ao ~ )

93 13¢ ) 53 15 6 39 6 7 20

48" ) 10 6 6 21 15 12g ) 9" ) 3 23 24~ ) 14

0.171(033)

-0.082(0361

0.249(035)

- 0.094(036)

0.267(0821

-0.101(091}

- 0.252(106) 0.260(152) 0.129(054)

-0.039(100) -0.011(146) 0.034(060)

1.25 (0.02) 1.1V) (0.12) 1.10 (0.02) 1.15 (0.11) 1.12 (0.13) 1.13 (0.08) 1.15 (0.13) 0.56(0.04) 1.13 (0.03) 1.16 ~ ) (0.02) 1.22 (0.07) 1.25 (0.12) 0.67 (0.05) 1.26 (0.10) 1.22(0.11)

1.23~ ) (0.12) 0.507( 156P" )

0.336( 169p' )

0.7~ ~ ) (0.09) f) 1.14 (0.06) 1.25 g ) (0.05) 1.72 (0.151

( + ,o) 12 + ~ 1o

( - , 1 ) 17 - - 15 i+,1)15"~ 13 ~ ( - . 0 ) 16 ~ 14 (+,1)17"~ 15 + cross-band 6 ~ 5" ( + , 0 ) 14 + ~ 1 2 * ( - , 1 ) 19 4 1 7 ( - , 0 ) 18 ~ 1 6 ( + , 0 ) 1 6 ' ~ 14 + (-,1)21 419 ( + , 1 ) 1 9 - ~ 17 + ( + , 0 ) 1 8 + ~ 16 + ( - , 0 ) 20 ~ 1 8 cross-band 21 ~ 20 + (-,1)23 421

( + , 0 ) 20 + ~ 1 8 ' ( - , 0 ) 22 ~ 2 0 (+,1121'~ 19" cross-band 19 ~ 18" ( + , 0 ) 2 2 ' ~ 20 + (-,1)25 423 ( - . 0 ) 24 ~ 2 2 cross-band 17 ~ 1 6 ' (+.1)23'~21" ( + , 0 ) 24" ~ 22" ( - . 0 ) 26 ~ 2 4 ¢-,1127 425

J.C. Bacelar / I~r-I~°Yb

517

TABLE 2 - - c o n t i n u e d

E, ~ ) [keV] 822.4 826.5 827 828.5 836.3 892 893.0 896 907.4 910 958.3 977.2 1016 } 1016 1020.4 1032 } 1033 1045.7 1079.1 1110.3 1126 1169.7 1169

I, h ) t~a Sm(l~O, 4n) 6

12aSn(4SCa,4n) 8

A 2 / A oc )

-0.149(208)

A 4 / A o~ )

0.046(238)

15c )

13u*/ I~o

0.78 (0.04) 0.99 c ) (0.03)

9

1.13 (0.04)

14

0.95 (0.06)

5

") 12 8 3 6

0.018(197)

") 1.16(0.16) 0.98 (0.18) 1.85 (0.14) 1.29(0.11)

32 g)

3

-0.019(165) g)

1.36(0.10)

-0.220(157)

0.575(065) g) r)

7

7e ) 5 3 3 r) r) r)

f) - 0.255(150)

--

1.09 c ) (0.10) 1.56 (0.40) 1.30 (0.65) 1.30 (0.52) f) r) r)

Assignment

cross-band 15 4 144 • ( + , 0 ) 2 6 + 4 24 + ( + , 1) 2 5 ' --, 23 ~ ( - , 0 ) 28 - . 2 6 ( - . 1 ) 29 ~ 27( + , 1 ) 27 ~ ~ 25* ( + ,0) 28 ~ --* 26 + ( - , 0 ) 30 4 2 8 (-,1)31 429 cross-band 13 4 12 + ( + ,0) 30 + .--* 28 + (-,1)33 431cross-band 11 ~ 10" cross-band 5" 4 4 + ( + , 0 ) 32 + 4 30 + cross-band 9 + ~ 8 + cross-band 7 + --* 6 + ( - . 1 ) 35 ---* 33 ( + ,0) 34 * --* 32 ÷ ( - , 1) 37 --* 35 ( + , 0 ) 36 ~ 4 34* ( - , 1) 3 9 - ---, 37 ( + , 0 ) 38* --, 36 +

a) G a m m a - r a y transition energy obtained from coincidence data for the t24 Sn(48 Ca,4n) reaction. For some lower-energy transitions these values may deviate from those given in fig. 2 which have been taken from previous studies. b) G a m m a - r a y intensities obtained from singles angular distribution measurements for the tS4Sm('SO,4n) reaction and from the average of the 30 ° and 90 ° projected coincidence spectra for the l-'4Sn(4SCa,4n) reaction. The values are normalized to the 298.7 keV 6 ÷--* 4 ÷ transition. c) O b t a i n e d from analysis of the 154Sm(lSO,4n) angular distribution data. d) Ratio of 30 ° and 90 ° intensities obtained from projected coincidence spectra for the t24Sn(4SCa,4n) reaction normalized to the 298.7 keV 6 + ---*4 + transition. The uncertainties, shown in parentheses, are the larger of 0.02 or the value obtained from gaussian fits to the data. e) C o n t a m i n a t e d by another transition in 167yb or 169yb. r) Value c a n n o t be extracted due either to proximity of another transition or to being weakly populated. g) Probably c o n t a m i n a t e d by nearby transition.

J.C. Bacelar

518

/ 1*7-1°yb

TABLE 3 E x p e r i m e n t a l s u m m a r y for 169 Y b Assignment

[keY]

Ivb )

I3o./19o.': )

160.9 199.4

19 d)

1.06 (0.08) d)

243.4 248.1 277.6 321.8 331.3 356.2 384.7

69 100 82 110 85 113 d)

0.97 (0,03) 1.00 (0.02) 1.05 (0.03) 1.04 (0.03) 1.05 (0.03) 1.10 (0.03) %

421.3 433.3 437.9 484.7 508.4 509.1

71 125 *) 95 d) d)

0.97 (0.05) 1.08 (0.03) c)

( + - b ~++ - ': "

1.04 (0.03) d) d)

(-,~)~2Q -

527.8 570.6

80 143

581.5

d)

1.02 (0.05) 1,09 (0.04) d)

592 617.5 649.1

~)

~)

84 d) 42

1.34 (0.09) d) 1.38 (0.10)

(+,-b

~ + ~ ,+

(+ _ ! ) ~ + 4 2 2 -

H÷ 2 9

(+,b7+4'-,. (_

,+

,)~,-4,,,.+

( + , - ,:) ,:+ ~ ? +

(+,,)

2s+ 4 2 ~ +

I

33

4

~-

2~

-

29

-

~ (-,~)~ l 33 + 4 ~ 9 + (+,.~) I

37-43

( + , - ~ ) 31, + - , .27 + ( - , ~ )l 41, ~ : ~7 (+,~) ~;+~ 7 + -

+ , - ~ ) 3q+4~+ ,

659 670.8 } 671

170

1.44 (0.08)

+,_,;) ,:+.++,.

707.0 729.6

15(/) 152

1.19 (0.05) 1.14 (0.10)

1 41 + .+) ,

_

441-

t)4s

2

"2

I

-.2)

49

2

+ -

4

~

37 .

+

4S

+. _ ~) 7 + + -'++'.,+

742 } 742.2 748.1

97

773.7 786 791.8 829

41 91 31 d)

1.07 ~0.11) 1.24 ~0.11) 1 . 1 0 ~ 0.O6) d

(+.~)

855.3 896

26 d)

1.23,0.30) d

(-.b~,-47

919.7 983.3 1047.5 1107

14 10 ~) ~)

2.17,0.16) 1.29 0.25) ~) ")

(_.,;) ++,- + ++,-

1.35 (0.11)

d)

d

5+441+ (+,~) 4.~

(+,t2) o:

+

4

4S +

:

( + , - : i)4"t+ ~

+J+

(_,~) ~, 4 7 (+.b

~ " + . ?-

4++, <-.b '+ 4 7 ( -, ~ ) ',+ - 4.," (_.,+) +,+

") Gamma-ray transition energy obtained from coincidence data for the t24Sn(48 Ca, 5n) reaction. For some lower-energy transitions these values may deviate from those given in fig. 3 which have been taken from previous studies. b) Gamma-ray intensities obtained from the average of 30 ° and 90 ° projected spectra normalized to the intensity of the 248.1 keV Tt3- 4~9 - transition, ¢) Ratio of 30 ° and 90 ° intensities obtained from projected spectra normalized to the value for the 248.1 keV ~ - ~ -~- transition. The uncertainties, shown in parentheses, are obtained from gaussian fits to the data. d) Masked by transition in 167yb or t6syb. c) Value cannot be extracted due either to proximity of another transition or to being weakly populated.

J.C. Bacelar / 167-tr°Yb

519

1249n (48Co, 5n)167Yb T

t

T

i

I000

x2 s3/z-

(_..I/z)

500 O3 Z

:::3

0

0 1500

(*. * I/z)

(D I000

5~g t I I 6~4 65/~

500

•

69/2+

0 .25

.50

.75

1.0

1.25

ENERGY (MeV) Fig. 4. Gated y-ray spectra for the ( + , ~ ) and ( - , ~ ) decay sequences of t67yb populated by the 124 5n(48 C a , 5n) reaction. The composite spectra for the various decay sequences are obtained by summing individual spectra associated with a number of gates within the band. The following gates, denoted by their parent states were used: ( + , ~) decay sequence-'~+, 5} +, 57+., and 6~ +: ( _ . ~,) decay. s e q u e n c e - 3, 3 - , ~ , and 429 A multiplicity gate of fourteen or more of the BGO detectors in TESSA II also were required for these spectra.

reaction. The results of these measurements, made using the 78 MeV tSo beam of the Niels Bohr Institute Tandem Accelerator, are given in table 2.

2.2. DECAY SCHEMES OF 167yb AND 169yb

The (~r, a ) = ( + , ½), ( + , - ½), and ( - , - ½)t decay sequences in 167yb have been established n ) to I " = ~ + , ~ + , and ~ - , respectively, in previous studies of the 1 5 4 5 m ( 1 7 0 , 4n) reaction. Similarly, the ( + ,_~), ~ ( + , - ~_), 1 and ( - , - ½) sequences in 2 5 - , respectively, using the 167Er(ot, 2n) 169yb were established 12) to ~ +, -'_3w+, and w. reaction. In the present study these sequences have been extended as indicated in t In accordance with established convention, see e.g. ref. n ), the various decay sequences are labeled by the conserved quantum numbers of the "axially-symmetric" intrinsic system, parity ~r, and signature a. Signature is the quantum number associated with the rotation of 180 ° about an axis perpendicular to the nuclear symmetry axis. As defined 23), signature is related to angular momentum by 1 = a mod 2.

520

J.C. Bacelar / l~7-tnVYh

124Sn 148C(], 4000

-

-

5rl)169yb

T

T

x2

(+,+~z)

49/2+ 2000

" 53//" 5r/

Or) I'0 Z :E) 0 t..) 4 0 0 0

(-,+l/z)

-x2 49/£ 55/2.

57/2- e5/2-

2000

0 .25

.50

ENERGY

.75

1.0

1.25

(MeV)

Fig. 5. Gated "t-ray spectra for the (+,~) and ( - , ~ ) decay sequences of 169yb populated by the 124Sn(4SCa,3n) reaction. The following gates were used: ( + . ~) decay sequence- ~5 + t] + ~ +, ~-223+, ~ ÷ , a n d ~ + ; ( _ . ~)decavsequence_ 9- , 123-, 22 -, 29. ~ _ ,3} - , 37 s~ - ,and-257 -. See also caption to fig. 4.

figs. 1 and 3. Typical coincidence spectra for the ( + , ½) and ( - , ½) sequences of t 6 7 y b and 169yb are shown in figs. 4 and 5, respectively. Spin and parity assignments for new states in 167yb and 169yb are based on the angular correlation information available from the 30 ° to 90 ° ratios obtained with TESSA, see tables 1 and 3, as well as the relative intensities of "rotational-like" intraband transitions. The systematics of 30 ° to 90 ° v-ray intensities for a large number of transitions of known multipolarity from similar (heavy ion, xn) studies are summarized in figs. 3 and 8 of ref. 62).

2.3. DECAY SCHEME OF t68yb E v e n t h o u g h o n e of the earliest " b a c k b e n d s " was o b s e r v e d 24) in 168yb, very little is k n o w n a b o u t the level s c h e m e of this n u c l e u s at large a n g u l a r m o m e n t u m . T h e

J.C. Bacelar / t~z-1ngYb

521

yrast sequence has been established z4"25) to I " = 20 + using the 167Er(a, 3n) and the 16°Gd(12C,4n) reactions. A recent (a, xn) study 26) also has identified the lower portions of an odd-spin, positive-parity and an even-spin, negative-parity decay sequence. (The conserved quantum numbers labelling these sequences are (~r, a ) = ( + , 1) and ( - , 0), respectively.) In the present work these three known sequences are extended and two new sequences have been established, see fig. 2. Typical summed coincidence spectra corresponding to these five sequences are shown in figs. 6 and 7. Relative intensities and the ratios of 1,-ray yields at 30 ° and 90 ° in the 124Sn(48Ca, 4n) reaction together with A 2 and A 4 values from the ~54Sm(180, 4n) reaction are given in table 2. The spin assignment for the odd-spin, negative-parity sequence, which becomes yrast at the highest angular momentum, is based on stretched dipole transitions connecting this sequence with the yrast sequence. The dipole nature of the five

fe4Sn (4eCo, 4 n)fssyb 6000

4000

2000 O0 I-Z Z) 0(..)

0 6OOO

4000

2000 0 • 25

.50

.75

1.0

1.25

ENERGY (MeV) Fig. 6. Gated 3,-ray spectra for the lowest ( + ,0) and ( - , 1) decay sequences of 16syb populated by the 124Sn(48Ca,4n) reaction. The following gates were used: ( + . O) decay sequence- 22 +, 24 +, 26 +, 28 +, 30 + and 32+; ( - , 1) decay sequence - 23-, 25 -, 27-, 29-, 31 - and 33-. See also caption to fig. 4.

J.C. Bacelar / Ih7-1agYb

522

i

4001 I

"-- .. I I

kZ 0

.

2oo L:o I:I: 300

tat)

~

o

kl

0

~N

I

I

0

I00

• '

~

•

_ !

[1111t,

600 -

iIIlp

m i

~

~ ,_

_

"!!"111

. . . . .

O'l I'M I'.-

400 0

200~ 0

w

.25

i

.50

.75

1.0

I

1.25

ENERGY (MeV) Fig. 7. Gated "pray spectra for the ( + , 1), upper, ( - , 0), middle, and unassigned, lower, decay sequences of ]6syb populated by the 124Sn(4SCa,4n) reaction. The transitions in the unassigned sequence are labelled with their energies. This spectrum contains several transitions associated with the decay of 169Yb. The following gates were used: ( + , 1) decay sequence- 11 + and 21 +; ( - , O) decay sequence- 14 ; unassigned decay sequence- 680, 791, and 854 keV transitions. See also caption to fig. 4.

intersequence transitions is founded both on the negative A 2 values for the 21 - ---, 20 +, 15---* 14 +, and 13---* 12 + transitions (at 630.5, 822.2, and 910 keV, respectively) obtained from the (XSO, 4n) reaction and on the small ratios of the 30 ° to 90 ° yields measured for all these transitions in the (4SCa, 4n) reaction. In the singles spectra of the oxygen-induced reaction the 686.2 and 747.9 keV 17---* 16 + transitions were contaminated making the extraction of reliable A 2 and A 4 values impossible.

J.C. Bacelar / trz-t~°Yb

523

Unfortunately, the intensity of the transitions connecting the ( - , 1 ) and yrast sequences is too weak for the measurement of conversion coefficients. The intensity is fragmented over five interband transitions, of which one of the most intense, the 747.9 keV 1 7 - ~ 16 + transition, is contaminated by a transition from 169yb. Consequently, the parity assignment for the ( - , 1) sequence is less straightforward. It is based on several arguments: (i) The small values of the 30 ° to 90 ° ratios of the y-ray yields in the 124Sn(48Ca,4n) reaction indicates a pure or nearly pure dipole transition. The 30 ° to 90 ° ratios for the 630.5, 686.2, 747.9, and 822.4 keV transitions may be compared to that of the known 1 0 - ~ 9 ÷ E1 transition of 423.7 keV and to the known mixed M1-E2 1033 keV transition. (ii) If the sequence in question had positive-parity, then it decays to the yrast sequence in a very different manner from that of the known odd-spin, positive-parity sequence. (It would decay by nearly pure M1 transitions between spins 13 and 21 in contrast to the stronglymixed M1-E2 transitions between spins 5 and 11 for the known positive-parity, odd-spin sequence.) (iii) The analogy with the low-lying negative-parity, odd-spin sequence in the neighbouring even-even isotone 17°Hf, which decays to the yrast sequence between spin 11 and 17 [ref. 27)]. (iv) The assignment of positive-parity to the band in question would give two low-lying positive-parity, odd-spin decay sequences and no negative-parity, odd-spin sequences for 168yb. In contrast, it is the positive-parity, odd-spin sequence which is rare and the negative-parity, odd-spin sequence that is common in the near-yrast spectrum of even-even deformed rare-earth nuclei, see e.g. refs. 9.27-31). The observed negative A 2 values for the 1033 keV transitions (9÷--* 8 ÷ and 7 ÷---, 6 ÷ interband transitions connecting the ( +, 1) and ( + , 0) bands) are consistent with the previously assigned 26) dipole multipolarities. The intense yield of the 1016 keV transition in the (180, 4n) reaction indicates that this transition is contaminated in these singles spectra. The 30 ° to 90 ° y-ray ratio for the 1033 keV transitions in the ~2aSn(48Ca, 4n) reaction is intermediate between that of the 423.7 keV 10---, 9 + E1 transition and the intraband E2 transitions indicating that the decay between the positive-parity, odd-spin and the yrast sequences is by mixed M1-E2 transitions. Similarly, both the negative A 2 and the observed ratio of the 30 ° to 90 ° yields for the 423.7 keV 1 0 - ~ 9 ÷ interband transition between the ( - , 0 ) and ( + , 1) bands are consistent with the previous spin and parity assignments26). The large yield for the 482.2 keV 8 - ~ 7 + transition in the singles angular distribution measurement indicates that this transition is contaminated. The 540 keV 6---, 5 ÷ transition is too weak to be observed in singles measurements. Because of the very weak yield of the 1076 and 1092 keV transition connecting the remaining decay sequence, shown in fig. 2, with the 18 + and 16 ÷ members of the yrast configuration, it is not possible to assign spins and parities to this sequence from the present measurements. Except for I < 20 in the yrast sequence and I < 23 in the ( - , 1) sequence, for which angular distribution information is available - see table 2, the spin and parity

J.C. Bacelar / t~z-1ogYb

524 .. .......

(,, ~/,) (-, Y',, ) 0.,- Yz) .... ~---- ( - : 7 = ) I

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0.5

'~3yb

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0

0.5

>

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I

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I

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Q5

0

(D

<3 I "o

I

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ISlyb

1.0

I

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p-0.5

0.5

0.5

0

0

-0.5 0

t 0.1

t 0.2

i 0.3

I 0.4

165yb

-0.5 0

I 0.1

,ol d"

,o,"

I 0.2

I 0.3

i 0.4

I 0.5

h~ (MeV) Fig. 8. Rotating-frame excitation energies e' as a function of rotational frequency hto for decay sequences in le'l'163"165'lbT"169yb. The experimental data are from refs. s,H.13,1a) and the present work. The values of e' are defined as in ref,23) except they are referred to a reference configuration with a moment of inertia J = 0.01252A s/3 MeV i.

assignment within a decay sequence is based on the 30 ° to 90 ° intensity ratios and the relative intensities of "rotational-like" intraband transitions in the coincidence measurements. The consistent values of the 30 ° to 90 ° ratios for the large number of intrabands measured in 167'168'169yb lend credibility to this criterion for multipolarity assignments. 2.4. T H E T R A N S F O R M A T I O N OF T H E LEVEL S C H E M E I N F O R M A T I O N TO T H E ROTATING FRAME

The level scheme information (E x and I) for 167yb and 169yb is shown in fig. 8 expressed in the rotating intrinsic frame (i.e. rotating-frame excitation energies or routhians, e', and angular frequency of rotation, h~o [see e.g. refs. 23'32)]). For comparison experimental routhians corresponding to t61yb [ref. 13)], 163yb [ref. 14)], and X65yb, [refs. 8.11)] also are contained in this figure. Routhians corresponding to the 168yb level scheme are shown in fig. 9. All these routhians are referred to a

J.C. Bacelar / la7-t~°Yb 2.0

-

~

t

525

i

t

I

--1

Issyb _ ~ : (+,,) "t ~ ) ( - , o ) _1

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0.0

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o.z

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03

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hw(MeV) Fig. 9. Rotating-frame excitation energies e' as a function of hto for decay sequences in 16sYb. See caption to fig. 8.

TABLE 4 Average moments of inertia a) for the high-spin region of some N = 95-99 nuclei Nucleus

Data source

Config.

(~r,a)

j(l) a) [MeV l]

j(2) ~) [MeV l]

j(2)/A5/3 b)

45 ~ - 61~ ,7 ~9 i -3 41 2 _67 2 22-32 ~-~!

66.68

61.04

0.01230

64.58 66.80 61.66 65.46

61.35 65.30 62.03 64.28

0.01236 0.01290 0.01213 0.01244

165 Yb95

c)

167yb97 16syb,~s 169yb,) 9

d) d) d)

167H f95

e)

( - , ~)

~-~

67.06

63.86

0.01261

c)

( _ _ ~) (+,0) (+,0)

~2_ 5~ 22-32 24-30

64.75 61.66 64.20

64.88 62.03 65.08

0.01281 0.01213 0.01248

168Hf96 17° W96

f)

( - , 12) ( _ , - ~) 2 (--" ~) ( + , O) (--, ~)

I range

a) Defined by eqs. (2) and (3). In this definition the total angular momentum, I, is not corrected for K. b) Units of M e V - l . amu-5/3. c) Ref.9). d) Present work. c) Ref.32). t) Ref.2s).

526

J.C. Bacelar / 167-t~°Yb

configuration with m o m e n t of inertia J = 0.01252A 5/3 M e V - t.

(1)

This reference configuration has the average m o m e n t of inertia of the highest spin portions of several nuclei in this mass region s'9,2s'33) - s e e table 4, and is in a c c o r d a n c e with the expected mass dependence for m o m e n t s of inertia. Such a reference m o m e n t of inertia is also consistent with that obtained for 168yb at large rotational frequencies from the width of the equal-energy valley in y-ray energy correlation studies~7"ls). Except at low values of h~0 this reference is nearly equivalent to the " y r a s t reference" which has been used previously [see e.g. refs. t1.32)]. T h e choice of K-values, i.e. the projection of the intrinsic single-particle angular m o m e n t u m on the nuclear symmetry axis, also must be considered in the transform a t i o n of the data to the rotating intrinsic frame. For example, if K = ½, rather than g = 7 ~, had been used for the ( + , I) decay sequence in 169yb the routhian for this configuration at the largest values of hto would be nearly degenerate with that of the ( - , I ) decay sequence, see fig. 8.

3. The disappearance of static neutron pair correlations 3.1. HISTORICAL BACKGROUND Naively, the most obvious signature of the disappearance of static pair correlations t would seem to be the observation of a rigid-body m o m e n t of inertia. The nucleus, however, is a quantal system; therefore, it is necessary to consider the microscopic nuclear structure. Microscopically rigid-body moments of inertia can be o b t a i n e d for marginally paired systems and reduced moments of inertia can be o b t a i n e d for unpaired systems [see e.g. ref. 37)]. N o t only does the m o m e n t of inertia vary nonlinearly as a function of the magnitude of pair correlations, but fluctuations b e c o m e increasingly important with decreasing static pair correlations. Therefore, m o m e n t s of inertia, which for t67'168'169yb at large angular m o m e n t u m reach about 75% of the rigid-body value - see fig. 10, are not definitive evidence for the loss of n e u t r o n pair correlations. Neither is the observed near equality of the kinematic and d y n a m i c m o m e n t s of inertia3S), j t t ~ and ,/t2~, in these, see fig. 10, and an increasing n u m b e r of other [see e.g. refs. 8'9'28"3°'33"39A°)] rapidly-rotating systems * The term pair correlations as used throughout this paper refers to a static equilibrium deformation of the pair field. Such correlations are characterized by a sizeable pair-gap parameter d (in comparison with the average single-particle level spacing) and an associated quasiparticle coupling scheme. Of course, pair correlations, e.g. the pair-vibration mode in near closed-shell nuclei lo.3a), are important in nuclei without a stable deformation of the pair field. Indeed, such "'dynamic" pair correlations are also expected 35.36 ) to modify the spectrum of independent-particle excitations. An analysis of these effects, however, is beyond the scope of the present work, which is to establish that static neutron-pair correlations are greatly reduced at large rotational frequencies and to indicate the features of the single-neutron spectrum of states at such frequencies.

J.C. Bacelar / tn7-tn~Yb '

\

I

\

70

~.

527

168vk 70t u98

~(2)-

\ >

60

50 -

-

.2

.3

.4

.5

~loJ(MeV) Fig. 10. Kinematic and dynamic moments of inertia, ..¢(1) and ,~(2~ respectively, for the (+,0) decay sequence in 16syb, see fig. 2. The moments of inertia are defined 3~) by eqs. (2) and (3). In this definition the total angular momenta 1 are not corrected for K.

p r o o f of such a transition to an unpaired system: j(t)_

l(d E/d I)-1 = I/w,

j (2) = (d 2 E / d 12 ) - l = d I / d oo.

(2) (3)

Indeed, such classical-like behaviour for a wide range of nuclei is very difficult to u n d e r s t a n d in either " p a i r e d " or " u n p a i r e d " microscopic systems 33.41,42). Estimates 37"43)'based on the assumption of equidistant twofold degenerate levels and the average density of Nilsson states ~ ) indicate that the excitation of three to four quasineutrons in rare-earth nuclei is sufficient to destroy neutron-pair correlations. However, such average estimates, which ignore the effects of rotation, underpredict the average residual interactions between the excited quasineutrons by nearly a factor of two 1,45). T h e existence of band crossings based on the alignment of the second pair of it3/z q u a s i n e u t r o n s in the positive-parity decay sequences of o d d - N isotopes and the negative-parity sequences of even-N isotopes, see figs. 8 and 9, indicates 7"43) that some neutron-pair correlations must be present* for three- and four-quasineutron * Though the existence of "large-frequency" band crossings in the positive-parity, single-quasineutron configurations of odd-N rare earth nuclei and in the negative-parity, two-quasineutron configurations of even-N nuclei is normally interpreted 7"43) as evidence for the existence of a static neutron-pair field, the detailed crossings between one- or two-quasineutron bands and bands based on nonpaired configurations are not sufficiently investigated to completely exclude them as the basis of such crossings.

528

J.C. Bacelar / t~7-t6°Yb

configurations at h~ > 0.3 MeV. Other high-spin studies s'9) seem to indicate that neutron-pair correlations in stably-deformed, rare-earth nuclei are greatly reduced at h ~ > 0.38 MeV. Furthermore, the band crossings, based on the alignment of a pair of negative-parity quasineutrons, are not observed in the positive-parity decay sequences of 167"168'169ybup to the largest angular momentum studied 7). The various theorists do not agree on the transition from a "paired" to an " u n p a i r e d " system. For example, in an adjacent Hf isotope the neutrons are predicted 46) to be "unpaired" in the yrast configuration for h ~ > 0 . 3 8 MeV. Similarly, the 16Syb high-spin data can be reproduced in a reasonable manner by calculations in which both protons and neutrons are unpaired 47). In contrast, other theorists 4s'49) find that neutron pair correlations continue nearly indefinitely. Much of this uncertainty is associated with the difference in the definition of pair correlations in particle number projected and unprojected calculations 50).

3.2. SINGLE-QUASIPARTICLE(AND SINGLE-PARTICLE)SYSTEMATICSOF LEVELSIN SYSTEMS WITH (AND WITHOUT) PAIR CORRELATIONS Experimental evidence for the disappearance of pair correlations usually has been discussed in terms of alignments 23) i = - de'/dc0

(4)

or moments of inertia, see eqs. (2) and (3). Since these quantities involve the differentiation of energy with respect to either angular momentum or rotational frequency, the relative spacing of the energy levels is lost. The maximum information, however, is available from the spectrum of single-particle states, which are the basic excitations of the independent-particle models. Therefore, in the present work the details of the experimental spectrum of single-particle states are investigated. The effects of neutron pair correlations on the single-neutron spectrum of states are illustrated in fig. 11. Pair correlations increase configuration mixing in two ways: (i) In the presence of pair correlations the low-lying spectrum of single-particle states is compressed; compare the left-hand and center portions of the top part of fig. 1'. (ii) The presence of both particle and hole components in the quasiparticle wave function provides a greater possibility for mixing. For fight rare-earth nuclei such pairing-induced mixing is especially important in the positive-parity quasineutron states. The highly-alignable, low-~2 components of the i13/2 neutrons are just below the Fermi surface, see fig. 12. Thus in a paired system the wave function of these configurations will contain sizeable particle, as well as hole, components. When such a paired system is rotated, the most highly-alignable components then mix into the lowest positive-parity configurations, which are a nearly equal mixture of particle and hole. This mixing is enhanced by the large Coriolis matrix elements between adjacent f2 and ~2 + 1 components of the

J.C. Bacelar / It';'-1agy b

529

- A/p',p'! IEv- kl

-

Ev

=j; e'v(~w)

I ¢J b.I

0

.4

.2

h~ (M~)

h. : s0i N

,

,..,!;--::=-_...." '~

I/

510~O

"='~

•

• ....

i

- -

--"

.2' ' ~w (MeV}

.4L 1

Fig. 11. Comparison of single-particle states in rotating systems with (top) and without (bottom) static pair correlations. In the top portion the calculated spectrum of Nilsson states (left), Nilsson-plus-pairing states (center), and independent-particle states in a rotating system with pair correlations (right) are shown. In the bottom portion the Nilsson spectrum of states is repeated to the left, and the independentparticle spectrum of states in a rotating system without pair correlations is shown to the right. In the right-hand portions of the figure configurations with (~r, t~) = ( + , -~), ( + , _ ~),t ( - , ~ ), and ( - , - i~) are denoted by solid, short-dashed, dot-dashed, and long-dashed curves, respectively. The hamiltonians for each calculated spectrum of states are indicted. These spectra were calculated assuming e2 ~ 0.242, e4 = 7 = 0.0, A n = 0.87 MeV, • = 0.0637, and p. = 0.42 and are appropriate for single-quasineutron and single-neutron states in 165Yb.

J.C. Bacelar / 1n7-t~oYb

530

UV 0

0.25

05

4 .Q7

[Nnza ]

X(N) It/2" [615]

52

X ( 115)~ X (113) ~ ), ( III ) ~

3/2 ~1/2~"-,/2 ~7,,2

[5~2 ] [510] * [ 651] -[ 5031

3

X(109)_ 2 X(t07)_ ), (1057 ~ ~

50

48

r,'z - [514 ]

k(103)_ ),(101)~

~u

S/2"[624]

~ 5/Z -[512] ~,,2-[521] 7,,z" [ C33]

),(99)~

v

k (97)~

..< i ~O

k (95)~ X (93)4.

S/:,- [ 5 2 5 ] 5/2" [6421 3/;,- [521 ]

I

> 0

- I

x (91)~ )- (89) ~ X(87)--

312"[651] It/z- [5051 I/2 ° [660"1 3/z- [ 532 ] J/z- [ 5301

-2

1.0 MeV -3

3.5 MeV

-4

Fig. 12. Spectrum of Nilsson orbits calculated assuming the same deformations and modified-oscillator parameters given in the caption of fig. 11. The position of the Fermi level ~ for J , = 0.90 MeV is also indicated for odd-neutron numbers, as are the asymptotic q u a n t u m numbers for each orbital. To the right the product of the occupation and nonoccupation amplitudes, UV, is given as a function of e,, - ~, for A = 0.5 and 1.0 MeV. Such a distribution is a measure of the effective pairing gap, since the probability of scattering a pair of particles from one orbit to another is proportional to the occupation of the initial orbit and the emptiness of the final orbit.

i13/2 shell-model configuration. Thus low-lying positive-parity states are favoured relative to negative-parity states. (The low-O, high:/" negative-parity components are not near the Fermi surface.) Such a favouring of the lowest positive-parity state is illustrated in the right-hand portions of fig. 11. On this figure cranked neutron and quasineutron spectra are compared for the same Fermi level, ~,, appropriate to N = 95. The mixing of the ~2 = ½ and ~ component of the i~3/2 neutron states into the lower-lying, positive-parity configurations, described in the preceding paragraph, is observed for the paired

J.C. Bacelar / Jn7-tn°Yh

531

calculation, but not in the unpaired calculation. The favouring of low-lying, positive-parity states relative to the low-lying, negative-parity states is especially pronounced for rotational frequencies up to that of the i13/2 quasineutron alignment (hto = 0.22 MeV in fig. 11). At this frequency the highly-alignable, positive-parity components are absorbed into the quasiparticle vacuum. That is, an "aligned pair" of i13/2 quasineutrons built from the low-~2 i13/2 components becomes occupied. An increase of the Fermi level corresponding to the addition of two neutrons alters the wave function of the low-lying quasineutron levels by increasing the components lying above the Fermi level and decreasing those below the Fermi level. However, if the pairing is sizeable, the features of the low-lying spectrum of states evolve slowly as a function of neutron number and are not characteristic of the details of the two single-particle states lying just above the Fermi level that would be occupied in the absence of pair correlations in going from the N to the N + 2 isotopes. Indeed, for h~0 < 0.37 MeV the observed spectra of states in, for example, 161-169yb, s e e fig. 8, are typical quasineutron spectra expected for such nuclei in the presence of sizeable neutron pair correlations, see the upper right-hand portion of fig. 11: (i) The yrast sequence is positive parity. (ii) There are band crossings based on the alignment of various pairs of i13/2 quasineutrons. (The highly-alignable components of the i13/2 neutrons would be occupied for these neutron numbers in a system without pair correlations). (iii) Except at the lowest rotational frequencies, the level spacing is sizeable. (iv) The quasineutron spectrum of states in neighbouring odd-N isotopes is quite similar. The gradual reduction both in the signature splitting of the positive-parity sequences and in the spacings between the positiveand negative-parity bands as a function of neutron number reflects the motion of the Fermi surface away from the low-I2 i13/2 orbitals, see fig. 12. In contrast, a distinctly different spectrum of states is observed at the largest rotational frequencies, i.e. hto > 0.38 MeV. For each isotope the spectrum is unique. For example, in 165yb and 169yb the levels are closely spaced with the negative-parity configurations occurring lowest in excitation energy. Conversely, for 167yb there is a large gap between the positive-parity yrast sequence and the excited negativeparity sequence. Such spectra are characteristic of single-particle motion in a rotating unpaired deformed nucleus. Each level scheme is uniquely characteristic of the appropriate sequence of single-neutron states occupied to give the appropriate neutron number. Such systematics are a necessary, but not sufficient, condition to conclude that the neutron pair correlations have effectively disappeared.

3.3. THE SINGLE-QUASINEUTRON AND SINGLE-NEUTRON SPECTRUM IN " G A U G E SPACE"

Another illustration of the contrast between the quasiparticle spectrum of states at small angular momentum and single-particle spectrum at large angular momentum is

532

J . C . Bacelar / I * r - t * g Y b i

i

i

~o= IEMeV

98

•

o (+,Vzl • 1-//=1

o

/.

N 94

I

-9

(+,0)

i

I

I

-8

I

i

(+,~'2)

.Z.~. -°

(+, V=) ./.~" (- '/,).~,~"

/ •

./ 90

i

fl~=.38MeV

/~/~

I+,O)

•

i

o 38 0 .45

O~y ;I

//

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-9

I

I

-8

I

-9

I

-8

X (MeV) Fig. 13. "Gauge space" alignment plots for the lowest (+,0), (+,-~), and (-0,~) configurations of ytterbium nuclei at hto = 0.12 (left) and 0.38 MeV (center). In the right-hand portion the same type of plot for the (+, ~) configuration of ytterbium nuclei is compared for various large values of hto. The Fermi levels X are defined as in ref. s~), except that our definitions are at constant ho0 instead of constant L Physically, )~ is half the two-neutron separation energy between A N = 2 neighbours with the same (~r, a) and at the same hoJ.

provided by " g a u g e space" alignment plots t. In such a plot the - ~ N rotating-frame hamiltonian 23,5~) h ' = h 0 - ~Q - ~ ( / ' + + / ' - )

- ~)x - a N

term in the

(5)

is treated in an analogous way to the centrifugal-plus-Coriolis term, -o~jl. The plot of h versus N shown in fig. 13 then can be identified as the " g a u g e space" analogue of the alignment plot, h ~ versus I x, in configuration space. In this case, )~ (=- - d E/d N, where E is the total binding energy of the nucleus in the configuration of interest) is simply half the two-neutron separation energy. The actual definitions used for N and )~ are given in refs. 20,51), except that in the present work )~ is defined at constant h ~ instead of at constant I as previously. This change of definition, which is within the spirit of the cranking model, only alters the details.of the curves shown in fig. 13. [Compare, these curves with the corresponding curves shown in fig. 10 of ref. 20).] T h e change in binding energy due to the addition of a pair of neutrons is seen directly from the "gauge space" alignment plots. As expected for a quasineutron coupling scheme, the two-neutron separation energies for small rotational frequencies change smoothly with the addition of neutrons, see the left-hand portion of fig. * "Gauge-space" alignment plots were introduced 51'52) to show the analogy between the configuration- and gauge-space lagrangian terms, i.e. -~}l and - h fi/ respectively, in the single-particle hamiltonian. These plots have been used to study the angular momentum dependence of the shape changes between spherical and deformed nuclei. Until recently2°), this technique has not been applied to odd-mass or non-yrast configurations at large angular momentum.

J.C. Bacelar/ 1n7-t6gYb

533

13. Furthermore, this behaviour is independent of the intrinsic nuclear configuration. These smooth changes reflect the correlated nature of the intrinsic quasiparticle configurations, which remains constant for neighbouring even- or odd-N isotopes and indicates that a static deformation of the pair field exists [see e.g. pp. 392-5 ref. lo)]. In contrast, at large rotational frequencies the binding energies do not change smoothly with neutron number, see the center and right-hand portions of fig. 13. The average slope of such curves, which is a measure of the average density of single-neutron states t, remains about the same as for small rotational frequencies. The sizeable observed variations in the "gauge-space alignment plots" at large rotational frequencies are not simply associated with uncertainties in the nuclear masses which for 16°'161'162a64yb are obtained from systematics 53), since the plots for small frequencies are smooth. Instead, the fluctuations indicate that the two-neutron separation energies at large hto are dependent on the configuration of the pair of neutrons added or subtracted to produce the A N = 2 neighbour. This signifies that the neutron-pair field no longer has a static deformation.

3.4. CALCULATED SPECTRAOF SINGLE-NEUTRON STATES In order to establish definitively that static neutron-pair correlations are quenched, it must be demonstrated that the observed spectrum of single-particle states both (i) cannot be explained in the presence of pair correlations, and (ii) can be reproduced in realistic calculations with no pair correlations. From the observed difference between the low- and high-frequency portions of the spectra, shown in fig. 8, and the two-neutron separation energies, shown in fig. 13, naively it might be expected that these conditions are easily exhibited. However, in the presence of neutron pair correlations at large rotational frequencies, the lowest pair of positive-parity quasineutrons already is aligned in all low-lying configurations. (In the calculations shown in the upper right-hand portion of fig. 11 this occurs at hto -- 0.36 MeV). As a result, a large portion of the hole components of highly-alignable i13/2 neutrons, which existed in the vacuum at frequencies below the backbend, is now occupied. Therefore, the appropriate comparison is between the lowest negative-parity quasineutron states and the "second lowest" pair of positive-parity states. Thus the difference between the alignments of the low-lying positive- and negative-parity configurations in the paired and unpaired cases is less than would be the case for a similar comparison below the band-crossing frequencies. 3.4.1. Calculated spectrum of single-neutron states as function of A and A. To investigate the sensitivity of the detailed spacing of single-neutron levels to pair * The slope of the "gauge space" plot, dN/d)~, can be thought of as an effective density of single-particle states 2°'Sl). It, however, is not a direct measure of the density of the states, as it also contains contributions from the change of the radius with neutron number.

J.C. Bacelar / t6r-la°Yb

534

correlations, the single-neutron spectrum of states for hto = 0.45 MeV has been calculated in the cranking model 23) as a function of the neutron pair gap A and the Fermi level X. Such simple calculations of e'(A,, h), shown in figs. 14 and 15, are not expected to be correct in detail. They, however, demonstrate the effect of neutron-pair correlations on the single-neutron configurations for the variety of neutron level spacings expected in the midshell region. Both the maximum favouring of the positive- and negative-parity configurations are represented for this rotational frequency and choice of deformations 54) and Nilsson model parameters 55).

7.0

--

,i.,11' (Nnz A1 q t (u) I1~" 1~15] -3,1 3/'Z" [512) - I . 2 1/2" 1510) - I . I I/2" [651] 8.1 7t2" (503] - 4 . 8

@ 6.8

~Z" t 6 2 4 )

-O.5

b2" [5141

-I.7

@ ~,~" (5121

66

4= v

-L.5

I/2- [,5211

2.9

rr~. [633]

2,0

5t2- 15231

1.3

s,'2" I642.1

4 I

3,~- [521]

1.6

®

6.4--

~i~" [5051

~'

[651]

5.9 -5.0

t,,~, (660)

7.0

1~2" [5301

4.9

6.2-~/2t 14001 - 3 . 6

6.0--

9/2" (5141

-2.2

I,'z" [541]

7.1

0

.04

.08

faro (f~Wo)

Fig. 14. The calculated spectrum of N = 4 - 6 single-neutron states in a rotating nucleus in the absence of static pair correlations is shown in the center portion. (*r, cO = (+, ~2), ( + , - t), ( _ , 1.,), and ( - , ~) states are denoted by solid, short-dashed, dot-dashed, and long-dashed curves, respectively. To the left the asymptotic q u a n t u m numbers I2"[ Nn.. A] and quadrupole moments q2 (v) are given for each orbit. These values, of course, are only valid at h ~ = 0. To the right three calculations of e' are shown as a function of A . These calculations correspond to h ~ = 0.06ho~0 ( = 0.45 MeV for A = 165) and to the three choices of the Fermi level h indicated by the arrows. The calculations of e'(A n) are referred to the energy of the configuration nearest to the Fermi surface and, therefore, are labelled e'r¢I. Calculations of e ' ( A , ) for additional choices of h labelled by the numbers in this figure are contained in fig. 15.

J.C. Bacelar / 167-t~°Yb I

.04 I--I

I 8

.O~I

535

L~._'ll"

"~

'

I

I

i

i

otL_I

I

i

I

I

,

\\ -

.O4

I

~

I

6 \

04 ½

3 v

I

9 ~.~.~_.~ ~_.__~~.~

0

0 , 4 0 4 I- . ~ - . . . _ ~ - OI-t

.04

O,

0t--I

2

t

r

_-

I

041-

I ~._~....'

I

~ ~''-

02 0

ol0

.06

.12

1 0

,

I .06

I .12

Fig. 15. Calculated rotating-frame excitation energies, routhians, corresponding to low-lying single-neutron configurations are shown as a function of the neutron pair-gap parameter, i n . A value of hoJ = 0 . 0 6 h ~ o ( = 0.45 MeV for A = 165) and the parameters contained in fig. 14 were assumed for these calculations, appropriate for odd-N nuclei. Configurations of (~', et) = ( + , ~ ), ( - , 12), and ( - - t_,) are denoted by solid, dot-dashed, and dashed curves respectively. In the left- and right-hand portions of the figure the Fermi level is chosen to favour negative- and positive-parity configurations, respectively. The position of the Fermi levels and the distribution of nearby single-neutron orbitals at J n = 0 are indicated for each calculation by the corresponding numbers in fig. 14. The calculated values are shown relative to the single-particle level nearest to the Fermi surface and, therefore, are labeled ere I.

These calculations indicate: (i) When the Fermi level favours a positive-parity orbital*, including pair correlations does not change this preference, see right-hand side of fig. 15. (ii) When the negative-parity orbital is favoured by the position of the Fermi surface, the excitation energy of the lowest positive-parity orbital is decreased with respect to the negative-parity orbital with increasing pair correlations, see the left-hand portion of fig. 15. That is, [e'(~ = + ) - e ' ( ~ r = - ) ] decreases with in+ Note that a positive-parity level also is lowest, or is preferred, in an unpaired odd-N nucleus when the Fermi level lies between the two signatures of a positive-parity orbital, see e.g. calculation number 5 of fig. 14. In this situation it is the lowest signature of the positive-parity level that is unpaired even though this level may not be nearest to the Fermi level. Of course, an analogous favouring of a negative-parity level also can occur in the region of large splitting for negative-parity orbitals. Similarly, even more complicated cases can be imagined, with the Fermi level intermediate between opposite signatures of more than one strongly-split configuration.

J.C. Bacelar / Y~z-t~oYb

536

TABLE 5 Summary of "average worst" case values of A n h a)

d e~etd )

( A . [MeV] ~ )

e' (Tr= -1)

[h~0o]

Calc'b)

NO)

dA.

= e ' (~r= + )

6.377 6.411 6.580 6.634

2 3 7 8

91 93 99 101

0.67 0.55 0.37 0.33

0.30 0.36 0.54 0.61

e' (~r = - )

=e' (tr= +)-50

keV

0.22 0.27 0.40 0.45

~) Position of Fermi level. h) Key to calculations shown in figs. 14 and 15. ¢) Approximate neutron number. a) Average derivative of e'~t---e' (or= + ) - e ' (~r= - ) w i t h respect t o A n at small A see fig. 15. ¢) Value of A necessary to make the lowest positive- and negative-parity routhians degenerate and to give a 50 keV preference to the negative-parity routhian at hto = 0.45 MeV. These "average worst" case values are based on an average positive-parity level spacing of 400 keV and the locating of the Fermi level at a negative-parity level, see fig. 14.

creasing A n- Even though the most alignable pair of quasineutrons already is aligned at this rotational frequency, the calculations still indicate a preference for t h e positive-parity configurations with increasing A n. It is possible, on the average, to make a quantitative estimate o f the limiting magnitude of the neutron-pair correlations consistent with the observation of negative-parity yrast configurations in two, 165yb and t69yb, of the three cases observed at the largest rotational frequencies, see fig. 8. (The negative- and positiveparity levels also are degenerate in 161yb.) The value of A n consistent with a negative-parity yrast configuration depends on the derivative of [e'(~r= + ) e'(~r = - ) ] with respect to A n, and the relative excitation of the nearest positive-parity orbital, see fig. 15. An "average worst" case estimate then can be obtained from the calculated values of d[e'(~r = + ) - e'(~r = - )]/dA by taking the average spacing of positive-parity levels and a choice of the Fermi level to give a maximum favouring of the negative-parity configuration. Such estimates for the four cases depicted in the left-hand portion of fig. 15 are collected in table 5. In these estimates an average distance to the nearest positive-parity level of 200 keV, corresponding to an average spacing of positive-parity levels of 400 keV, see fig. 14, is taken for h~0 = 0.45 MeV in this mass region. For the positive- and negative-parity configurations to be degenerate this analysis leads to an "average" upper limit for A of about 300 keV at N = 91 and 550 keV at N---99. The "true average value" probably would be somewhat less. The corresponding "average worst" case values of A n to produce a 50 keV preference for the negative-parity level, as observed at high spin in 165"169yb and 167Hf [ref. 33)] are about 220 and 400 keV for N = 91 and 99, respectively.

J.C. Bacelar / t~z-tr°Yb

537

T h e s e a r g u m e n t s indicate that the static neutron-pair correlations in such nuclei at large angular m o m e n t u m are reduced to a magnitude, which is less than, or of the o r d e r of, the single-particle level spacing. Thus the "static deformation of the pair field", p r o d u c e d by the enhanced scattering of pairs of neutrons between closelyspaced, time-reversed orbitals, is essentially nonexistent, and the effect on the single-neutron spectrum of states is minimal 56). T h e s e arguments, of course, are valid only on average. It is possible to imagine situations, in which the distance to the nearest positive-parity level is much larger than average giving an added preference to the negative-parity level. It is not realistic, however, to have a large n u m b e r of such cases, as the i13/2 neutron levels are k n o w n to be evenly distributed in rare-earth nuclei 57). Therefore such arguments would be strengthened by the observation of additional cases showing a preference for negative-parity single-neutron states. 3.3.2. Self-consistent calculations of the spectrum of single-neutron states. In order to establish that neutron-pair correlations have effectively disappeared at the highest spins, it is also necessary to show that realistic calculations with vanishing pair correlations agree with the data. Cranked Strutinsky calculations 15.58), therefore, are c o m p a r e d with the data. A Strutinsky minimization was m a d e for each b a n d in e 2, e 4 and 3'. In these calculations the Bengtsson-Ragnarsson modified oscillator p a r a m eters 47) were used. These parameters, which differ for each major shell, are given in table 6. In the region of large rotational frequencies, i.e. h~0 >/0.30 MeV, the Strutinsky minimization gives in general, see table 7: (i) a decrease in e 2 with increasing hto; (ii) an increase in e 4 with increasing hto; and (iii) slightly negative values of 3'. In all the calculations the proton m o m e n t of inertia was assumed to be half the rigid-body value. The calculated and experimental spectra of states are

TABLE 6

Modified oscillator (Nilsson model) parameters a Oscillator shell ( N )

K

/t

neutrons

4 5 6

0.07 0.062 0.062

0.39 0.43 0.40 b)

protons

3 4 5 6

0.09 0.065 0.06 0.054

0.30 0.57 0.65 0.69

a) Except for #,(N = 6) these parameters are from the recent work of Bengtsson and Ragnarsson 47) and are defined as in ref. 55). b) For the CHFB calculations discussed in subsect. 3.4.2 and shown in fig. 16 the BengtssonRagnarsson value of 0.34 was used. See discussion in sect. 4.

J.C. Bacelar / to7-ta°Yb

538

TABLE 7

M i n i m u m energy d e f o r m a t i o n s a) hto = 0 b) E2b )

(9'/',~)

hto = 0.30 MeV

e4b)

F2

hto = 0.35 MeV

E4

Y

F2

$4

~2

E4

hto = 0.45 MeV Y

e2

~4

Y

165yb95(+,~)

0.243

-0.002

0.229

0

-5

0.224

(+

_t)

0.243

-0.002

0.231

-0.002

-2

0.222

(-,~)

0.243

0.003

0.225

0.006

-3

0.221

0.243

0.003

0.225

0.005

-3

167YbvT(+,~) I

0.250

0.011

0.242

0.015

( + , - 2t-)

0.256

0.011

0.241

0.014

(-,~)

0.255

0.010

0.244

0.010

-3

= 0.264

0.020

0.257

-- 0.023

- 7

0.256

= 0.023

- 7

0.255

= 0.023

- 7

0.250

= 0.023

- 7

---0.264

0.020

0.263

=0.020

-5

0.262

=0.021

-7

0.260

=0.022

-7

0.257

=0.023

-7

=0.264

0.020

0.253

~0.022

-5

0.248

=0.024

-6

0.245

=0.025

-7

0.240

--0.025

(-,-

~)

169yb99 ( + , ~) (+,-~) (-,~)

0.003

h~0 = 0.40 MeV Y

-5

0.222

0.005

-5

~0.219

0.007

-3

=0.218

0.005

-4

=0.218

0.007

-5

0.010

-3

---0.218

0.012

-3

=0.215

0.013

-4

0.222

0.008

- 3

= 0.218

0.012

=0.216

0.013

- 1

-5

0.238

0.019

-7

0.236

=0.023

-8

0.233

= 0.025

-8

-5

0.238

0.016

-6

0.235

0.017

-8

0.233

= 0.021

-8

0.238

0.012

-6

0.235

0.014

-7

0.230

0.019

-8

0

0

-5

8

°) Deformation of the minimum in the potential energy surface calculated using the Strutinsky-Nilsson procedure as described in refs. l 5.ss ). b) y assumed to be zero.

compared in fig. 16. The agreement for t65yb and 167yb at h~o >/0.37 MeV is qualitative. The splitting between the yrast negative-parity configuration and the lowest positive-parity configuration in 169yb, however, is greatly overpredicted. The calculated slope for the (rr, a) = ( + , - ½) configuration, which has not been established as high in spin as the other experimental configurations, is not in agreement with the data. This suggests that the quasineutron alignment has not yet occurred for this configuration and that appreciable pair correlations remain. Likewise, the agreement for the other configurations is poor at frequencies below the quasineutron alignments. 4. Empirical spectrum of unpaired single-neutron states In the preceding section the experimental evidence for the effective disappearance of static neutron-pair correlations in odd-mass ytterbium isotopes at h~ >/0.38 MeV is presented. Thus it is both possible and desirable to construct an empirical spectrum of single-neutron states for N = 95-100 from the large angular frequency experimental single-neutron routhians for 165yb, 167yb, 168yb and 169yb. Such an empirical spectrum of states, shown in fig. 17 for h¢o > 0.38 MeV, is constructed from relative energies in the various nuclides. The use of relative energies avoids the uncertainties associated with, for example, the choice of a reference configuration and the addition or subtraction of energies between adjacent isotopes. [Differences

J.C. Bacelar / t~z-H'vYb

539

I

I

/'# / /

/ / /

/

1.0

I

'

I

,

,

I

l

1.0 ~)

(i)

~

0.5

I

1.0

I

'

#"

I

I

I

0.4

C~

0.:3

s°

0.5

0.4

0.5

h ~ (MeV) Fig. 16. Comparison of the experimental spectra of single-neutron configurations at large h~ (left) with self-consistent, unpaired CHFB calculations (right) for 165Yb (bottom), 167yb (middle), and 169Yb (top). The experimental values are those contained in fig. 8. The calculations are as described in subsect. 3.4.2 and in refs. ~5.5~) with A, = 0. Experimental values with (Tr,a) = ( +, ~), ( +, - ~), ( - , 2~), and ( - , - 12) are designated by filled dots, open dots, filled triangles, and open triangles, respectively. Heavy solid and dashed curves denote (or, a ) = (+, ~) and ( + , - ~) calculated configurations, and light solid and dashed curves denote ( ~r, a) = ( - , t ) and ( - , - ~) calculated configurations, respectively.

in c o r r e l a t i o n s for n e i g h b o u r i n g o d d - a n d e v e n - m a s s isotopes are well e s t a b l i s h e d for l o w e r a n g u l a r m o m e n t u m t'45"59).] O f course, the e x t r a c t e d level s p a c i n g s are exp l i c i t l y valid o n l y for the i s o t o p e f r o m which it was o b t a i n e d . F o r e x a m p l e , the e x t r a c t e d e n e r g i e s c o n t a i n the r e s i d u a l q u a s i p a r t i c l e i n t e r a c t i o n s [ i n c l u d i n g c o n f i g u r a t i o n - d e p e n d e n t c h a n g e s in d e f o r m a t i o n s , pair c o r r e l a t i o n s (if a n y r e m a i n ) , etc. 1,ts,32,43)] associated with the isotope f r o m which the e n e r g y d i f f e r e n c e was o b t a i n e d . U n t i l recently, s u c h effects were expected to b e m i n i m a l at large a n g u l a r m o m e n t a i n these nuclei (165-169yb), w h i c h are a m o n g the m o s t s t a b l y - d e f o r m e d r a r e - e a r t h n u c l e i ~4,55). T h e effects of d y n a m i c f l u c t u a t i o n s in b o t h the shape 60) a n d

540

J.C. Bacelar / #~7-t6°Yb I

0.4

-

I

I

I

(-,- 'A) "-"

...-'" ..""'~+,-'A)

p fP

'~,,

/""

""~. (-, '/l)

,sj

o

,p.

®

O.E

m

m

~(-,-V,) '~...~.

(+ ,'/,)

\ . \ (-/A) 0.4

0.5

fl(~ (MeV) Fig. 17. Composite empirical spectrum of single-neutron states at large rotational frequencies constructed from the observed level spacings in 165A67'I68'I69yb, see figs. 8 and 9. The levels are referred to the ( +, ~) configuration observed in 165yb. The gap in the single-neutron states at N = 97 is denoted.

p a i r 35,36) degrees of freedom, w h i c h o n l y r e c e n t l y h a v e b e e n i n v e s t i g a t e d , r e m a i n to be determined. T h e r e l a t i v e s p a c i n g of the three lowest e m p i r i c a l levels is t a k e n f r o m the large hw s i n g l e - n e u t r o n s p e c t r u m of states s h o w n for t 6 5 y b in fig. 8. T h a t is, the c o n f i g u r a t i o n s of t h e o b s e r v e d large a n g u l a r f r e q u e n c y d e c a y s e q u e n c e s in 165Yb95 are c o n s i d e r e d to b e a n N = 94 core with ~r = + a n d a = 0 t a n d the n i n e t y - f i f t h n e u t r o n o c c u p y i n g t h r e e s e p a r a t e o r b i t a l s with ( ~ r , a ) = ( - , ½ ) , ( + , ½) a n d ( - , ½ ) - see fig. 18. * The assumption of a (+,0) N = 94 core is supported by the close spacing of these three levels in 165yb95 and the large spacing between the lowest two levels in t67yb97. Other possible descriptions of such level spacings with these spins and parities are difficult in the rare-earth region, where large and small signature splittings are expected for positive- and negative-parity configurations, respectively, and where the spacing between the various components of the unique-parity, i]3/2 configuration is quite large. Such a pattern, which is independent of the details of the calculations - e.g. see fig. 14, is dependent on the general validity of the shell model and a general prolate nuclear shape.

J.C. Bacelar //67-1~gYb

541

( 7r, cl )

<-,-'/2) <+,-

t/z)

(-, '/~) (+, ~/2)

(-,- ,/2) (-, ~,)

•

165yb95

I

I

•

167yb97

I

I

169 Yb 99

Fig. 18. Pedagogical figure showing the assumed configurations of the various decay sequences observed at large rotational frequencies in 165.167.169Yb.

These three orbitals should be occupied in the yrast configuration of 167yb. The parity and signature of such a configuration would be ( +, 1), as observed. Since the ( - , - ½) and the ( + , ½) configurations are observed at nearly the same excitation, see fig. 17, two possible combinations exist for the excited ( - , 3) configuration of 167yb. Either the ( - , - 3) neutron is excited to a ( + , - ½) neutron orbital, or the ( + , 3) neutron is excited, to a higher ( - , ½) orbital. The ( - , 1) yrast decay sequence in 168yb96 for ht~ > 0.42 MeV (see fig. 9) indicates that at large angular frequencies the ( - , 3) neutron orbital is lower in energy than the ( + , - ½) orbital. Therefore, the second option, involving an excited ( - , 3) neutron with a ( + , 3) hole, as indicated in fig. 18, appears more likely and is suggested as the configuration corresponding to the excited decay sequence in ~67Yb. Thus, the energy difference between the excited and yrast configurations in 167yb is shown in the composite spectrum fig. 17, as the energy difference between the ( + , 3) and the ( - , 3) orbitals. The two low-lying ( - , 1) and ( +, 0) decay sequences in 168yb98 can be ascribed to the configurations shown in fig. 19. The crossing of these two configurations, however, is very "steep", see fig. 9, indicating the presence of some type of frequency-dependent correlations. (In a "paired" picture the ( + , 0) configuration corresponds to only two valence quasineutrons.) The relative energy of these two configurations then defines the separation between the ( + , - l) and the ( - , 3 ) neutron orbitals given in fig. 17. The fact that the ( - , 3) orbital is lower in energy than the ( + , - ½) orbital also is in agreement with the 169yb99 spectrum. The observed excited state of 169yb can be associated with the excitation of a neutron from the ( + , - ½) orbital to the ( - , - 3) orbital, shown in the composite spectrum, taken from the 169yb spectrum. There is a qualitative agreement between the empirically constructed spectrum of single-neutron states (fig. 17) and that calculated (fig. 14) from the cranking model 23) with no static pair correlations, frequency-independent deformation and

J.C. Bacelar / t~7-1~°Yb

542 ('rr, a )

(+,'/2) c-,-'/~) (+,-'/2)

(-,'4)

•

((% - , -~/ 2i).)

II

II

II

•

~'

(_,l~)

•

•

•

•

•

(-,I)

(+,0)

:

(-,0)

:

II •

(+,I)

Isaybse Fig. 19. Pedagogical figure showing the assumed configurations of the various decay sequences observed at large rotational frequencies in t6syb. Two possible configurations each are indicated for the ( - ,0) and the ( + , 1) sequences of this nucleus.

modified-oscillator parameters. The agreement, however, is not detailed. For example: (i) the observed signature splitting between the lower energy negative-parity orbitals is greater than predicted; (ii) the observed signature splitting between the positive-parity orbitals is less than predicted; (iii) the negative-parity orbitals are depressed in energy relative to the positive-parity orbitals; and (iv) the upper ( - , ½) and ( - , - ½) orbitals cross at hto = 0.40 MeV. It is not surprising that such a simple model does not reproduce the details of the single-neutron spectra. It is instructive, however, to consider the effects of the variations of the various parameters on the spectrum of single-neutron states. The negative-parity levels are depressed relative to the positive-parity levels by (i) a decrease in e 2, (ii) a more positive e4, (iii) a negative y, and (iv) a decrease in both the spin-orbit and the /-squared terms in the modified-oscillator potential. Indeed, smaller values of e 2, larger values of e4, and a slightly negative 3' are predicted for the high-spin minimum-energy deformations of these nuclei, see table 7. Calculations based on these parameters, give improved agreement with the experimental spectra of single-neutron states, see fig. 16. Such calculations apparently also are sensitive to the different proton contributions for the various deformations. Indeed, standard cranked shell-model calculation 23) based on the same deformations and modifiedoscillator parameters systematically predict the negative-parity levels too low in energy. Within the cranked shell model reasonable agreement for the single-neutron spectrum of states is obtained if the value of the spin-orbit strength for the N = 6 shell is taken to be/.t = 0.40. Such a calculated spectrum of states is shown in fig. 20 for average values of the minimum energy deformations for the low-lying configura-

J.C. Bacelar / ~n7-I~'°Yb

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Fig, 20, Calculated spectrum of N = 4 - 6 single-neutron states in a rotating nucleus in the absence of a static neutron-pair field. The deformations used correspond to the average values of the minimum energy deformations for the low-lying configurations of 1~5167'169Yb at hto = 0.45 MeV, see table 7, and are given in the upper right-hand portion of the figure, The modified oscillator parameters of ref.47) have been slightly modified in these calculations, see table 6, to reproduce the observed relative spacing of negative- and positive-parity orbitals, The predicted large rotational frequency gaps at N = 94, 97, and 106 are indicated. The general }ayout of this figure is as described in the caption to fig, 14.

544

J.C Bacelar / trr-t~°Yh

tions of 165"167'169yb at h~0 = 0.45 MeV (e 2 = 0.23, e4 = 0.018 and ~, = - 7 ° ) . Besides the observed gap in the single-neutron levels at N = 97, see fig. 17, these calculations also produce gaps at N = 91, 94, and 106. The N = 94 gap is insinuated by the sizeable space between the lowest two configurations observed in the highestfrequency spectrum of 163yb, see fig. 8. A systematic gauge-space analysis of the whole mass region also indicates 6~) the presence of the N = 91 and 106 gaps. The "steep" crossing of the upper ( - , ½) and ( - , - ½) orbitals at hto = 0.40 MeV in the composite spectrum of states (fig. 17) is not reproduced in the calculations. If this feature is physical, it indicates a change from f7/2 or P3/2 domination of the wave function of these states at lower hto to h9/2 or f5/2 domination at large h~o. This crossing, however, probably is the result of using the "partially-correlated" (0, + ) configuration of 168yb in constructing the composite spectrum of states. Recent calculations36), which include the effects of fluctuations of the neutron pair gap in the random phase approximation, indicate (i) that sizeable "dynamic" pair correlations exist for the ( + , 0 ) configuration, and (ii) that these correlations decrease with increasing rotational frequency. Such correlations thus explain the "steep" observed crossing between the ( + ,0) and ( - , 1) routhians of 168yb, which results in the ( - , ½) - ( - , - ½) orbital crossing in the composite spectra.

5. Summary The 167"168"169yb data presented in this work report the highest angular momenta yet established in stably-deformed rare-earth nuclei. Evidence for the effective disappearance of a static neutron pair field is obtained from systematic configuration- and gauge-space analyses of these data at h~o >/0.38 MeV. A composite empirical spectrum of single-neutron states is constructed for the "unpaired" regime at large frequencies from these data together with additional high-spin data for 165yb [ref. s)]. In the stably-deformed, "unpaired" system the sensitivity to spectroscopic properties other than static neutron pair correlations [e.g. deformations, the parameterization of the nuclear potential, dynamic fluctuations in both the shape and pair degrees of freedom, mixing with continuum states, and other interactions ~)] should be greatly enhanced. In the "paired" regime, i.e. at smaller rotational frequencies, the details of these nuclear structure properties are masked by the effects of static pair correlations. Hence only gross variations of these properties can be studied. In the "unpaired" regime, where the spectrum of single-particle states is characteristic of individual orbitals, not a mixture of all orbitals of a particular signature and parity within the "pairing gap", it should be feasible to obtain detailed information concerning such spectroscopic problems. The present work is an initial step toward such a detailed study of independent-particle motion in rotating unpaired deformed nuclei.

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a c k n o w l e d g e d , as is the f i n a n c i a l s u p p o r t of the D a n i s h N a t u r a l S c i e n c e R e s e a r c h Council,

the Danish

Ministry of

Education,

the U n i t e d

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Science and

E n g i n e e r i n g R e s e a r c h C o u n c i l , a n d the N o r d i c C o m m i t t e e for A c c e l e r a t o r B a s e d Research.

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36) 37) 38) 39) 40) 41) 42) 43) 44) 45) 46) 47) 48) 49) 50) 51) 52) 53) 54) 55) 56) 57) 58) 59) 60) 61) 62)

J.C. Bacelar / t~r-I^°Yb

R.S. Simon, P.A. Butler, P. Colobani, F.S. Stephens, R.M. Diamond, R.M. Ronnigen, R.D. Hichwa, J.H. Hamilton and E.L. Robinson, Phys. Rev. C21 (1980) 2366 J. Simpson, M.A. Riley, J.R. Cresswell, P.D. Forsyth, D. Howe, B.M. Nyako, J.F. Sharpey-Schafer, J. Bacelar, J.D. Garrett, G.B. Hagemann, B. Herskind and A. Holm, Phys. Rev. Lett. 53 (1984) 648 R.M. Lieder, G. Sletten, J. Borggreen and J. Pedersen, Nucl. Phys..4375 (1982) 291; C. Fahlander and G.D. Dracoulis, ibid. p. 263 S. Frauendorf, Proc. Int. Workshop on nuclear physics, ICTP, Trieste 1981. ed. C. Dasso et al. (North-Holland, Amsterdam, 1982) p. 111 E. Paul, J.C. Lisle, R. Chapman, J.N. Mo, J.C. Willmott, J.C. Bacelar, J.D. Garrett, G.B. Hagemann and B. Herskind, to be published R.A. Broglia, O. Hansen and C. Riedel, in Advances in Nuclear Physics, vol. 6 (Plenum, New York, 1973) p. 287 R.A. Broglia, M. Diebel, F. Barranco and S. Frauendorf, Proc. XXIII Winter Meeting on nuclear physics, Bormio, Italy, January 1985, ed. I. Iori (Ricerca Scientifica ed Educazione Permanente, Milan, 1985) R.A. Broglia, M. Diebel, M. Gallardo and S. Frauendorf, to be published I. Hamamoto, Phys. Scripta '1"5 (1983) 10 Aa. Bohr and B. Mottelson, Phys. Scripta 25 (1981) 71 A. Pakkanen, Y.H. Chung, P.J. Daly, S.R. Faber, H. Helppi, J. Wilson, P. Chowdhury, T.L. Khoo, I. Ahmad, J. Borggreen and Z.W. Grabowski, Phys. Rev. Lett. 122B (1982) 1530 H.G. Price, C.J. Lister, B.J. Varley, W. Gelletly and J.W. Olness, Phys. Rev. Lett. 51 (1983) 1842 H. Sagawa and T. Dossing, Phys. Lett. 96B (1980) 238 J.D. Garrett, Proc. 1984 INS-RIKEN Int. Heavy Ion Phys., Mt. Fuji, Japan, J. Phys. Soc. Japan, Suppl. II (1985) 456 S. Frauendorf, Nucl. Phys. A409 (1983) 243c J.D. Garrett and S. Frauendorf, Phys. Lett. 108B (1982) 77 S. Frauendorf, L.L. Riedinger, J.D. Garrett, J.J. Gaardhoje, G.B. Hagemann and B. Herskind, Nucl. Phys. A431 (1984) 511 Y.K. Agarwal, J. Recht, H. Hiabel, M. Guttormsen, D.J. Decman, H. Kluge, K.H. Maier, J. Dudek and W. Nazarewicz, Nucl. Phys..4399 (1983) 199 T. Bengtsson and I. Ragnarsson, Nucl. Phys. A436 (1985) 14 U. Mutz and P. Ring, J. of Phys. GI0 (1984) L39 R. Bengtsson and H.B. Hakansson, Nucl. Phys..4357 (1981) 61 W. Nazarewicz, J. Dudek and Z. Szymahski, Nucl. Phys. A436 (1985) 139 R. Bengtsson, J.-Y. Zhang and S. ,/kberg, Phys. Lett. 105B (1981) 5 R. Bengtsson, J. Ragnarsson, J.-Y. Zhang and S. ,h,berg, contributions to IV Conf. on nuclei far from stability, Helsingor, Denmark, June 1981, p. 509 A.H. Wapstra and G. Audi, Nucl. Phys. A432 (1985) 1 R. Bengtsson, J. de Phys. Colloq. 41 (1980) C10-84 S.G. Nilsson, C.F. Tsang, A. Sobieczewski, Z. Szymahski, S. Wycech, C. Gusstafsson, I.L. Lamm, P. Mi~ller and B. Nilsson, Nucl. Phys. AI31 (1969) 1 R. Bengtsson and J.D. Garrett, in Collective phenomena in atomic nuclei, International review of nuclear physics, vol. 2 (World Scientific, Singapore, 1984) p. 194 M.E. Bunker and C.W. Reich, Rev. Mod. Phys. 43 (1971) 348 M. Diebel, Proc. XXII Int. Winter Meeting on nuclear physics, Bormio, Italy, January 1984, ed. I. Iori (Ricerca Scientifica ed Educazione Permanente, 1984, Milano) suppl, no. 35, p. 646 J.D. Garrett, Phys. Scripta "I"5(1983) 21 I. Hamamoto and N. Onishi, Phys. Lett. I50B (1985) 6 J.C. Bacelar, J.-Y. Zhang, J.D. Garrett and A. Holm, to be published J.F. Sharpey-Schafer, in Proc. XXIII Winter Meeting on nuclear physics, Bormio, Italy, January 1985, ed. I. Iori (Ricerca Scientifica ed Educazione Permanente, Milan, 1985)