Ground-state rotational band in 181Ta

Nuclear Physics A270 (1976) 255-268; ( ~ North-Holland Publishino Co., Amsterdam

Not to be reproduced by photoprint or microfilm without written permission from the publisher

GROUND-STATE

ROTATIONAL

B A N D IN tSiTa

T. INAMURA Cyclotron Laboratoo', The Institute t~f Physical and Chemical Research, Wako-shi, Saitama 351, Japan and Department t~f Physics, Schuster Laboratory, The University, Manchester MI3 9PL, UK

and F. KEARNS, G. VARLEY and J. C. LISLE Department of Physics, Schuster Laboratory The University, Manchester MI3 9PL, UK

Received 24 May 1976 Abstract: The ground-state rotational band in tSaTa has been observed up to a spin of 22t+ using Coulomb excitation with 84Kr ions. The nuclear lifetimes of the band members have been determined from the Doppler-broadened lineshapes of the de-excitation y-rays, and the angular distributions of the 7-rays have been measured. It is found that the E2 transition rates from higher spin states (> T~7+)are retarded relative to the rotational model predictions. A suggestion is made that this retardation may be due to the Coriolis antipairing effect.

El

NUCLEAR REACTION aSlTa(S4Kr, S4Kr'7), E = 348 MeV; measured a(E; E~, 0), E I Doppler shift attenuation. ~SITa deduced levels. J, r m, ;'-mixing, ;'-branching, B(E2). Coriolis calc. Natural target.

1. Introduction

T h e nucleus 181Ta was first C o u l o m b - e x c i t e d in the p i o n e e r i n g e x p e r i m e n t o f H u u s a n d Zupan6i6 1). T h e y s h o w e d t h a t the l o w - l y i n g states possessed the r o t a t i o n a l p r o p e r t i e s p r e d i c t e d by B o h r a n d M o t t e l s o n 2). L a t e r m e a s u r e m e n t s c o n f i r m e d t h a t 18tTa h a d a p e r m a n e n t d e f o r m a t i o n a n d it was s h o w n t h a t the g r o u n d - s t a t e b a n d was b a s e d on the K ~ = ~+ [404] N i l s s o n o r b i t a l 3,4). It is t h e r e f o r e o f interest to see h o w well the r o t a t i o n a l m o d e l can describe the p o s i t i o n s a n d t r a n s i t i o n rates o f h i g h e r excited states in 181Ta" In p a r t i c u l a r it is i m p o r t a n t to m e a s u r e the n u c l e a r lifetimes which p r o v i d e t r a n s i t i o n p r o b a b i l i t i e s i n d e p e n d e n t l y o f a n y m o d e l . S e a m a n et al. o b s e r v e d the g r o u n d - s t a t e r o t a t i o n a l b a n d u p to J " =- ~ + by m e a n s o f C o u l o m b e x c i t a t i o n with t 6 0 ions s). O n the basis o f the r o t a t i o n a l m o d e l t h e y discussed M1 t r a n s i t i o n p r o b a b i l i t i e s b e t w e e n the b a n d m e m b e r s , i n d i c a t i n g t h a t IgK--gRI was c o n s t a n t for all states. H o w e v e r , they d i d n o t d e t e r m i n e B(E2) values o f the i n t r a b a n d t r a n s i t i o n s . W a r d a n d A d e r 6) p r o p o s e d a level scheme u p to J~ = ~-+ by o b s e r v i n g ?-rays f o l l o w i n g C o u l o m b e x c i t a t i o n with 35C1, b u t 255

256

T. I N A M U R A et al.

the position o f the ~ + state was uncertain. Recently, Mann et al. 7) proposed a level scheme for the ground-state band as far as J " = ~ + on the basis of y-rays observed in the ~S°Ta(n, y) reaction. Their measured branching ratios were in accord with the predictions o f the rotational model. In the present study we measured nuclear lifetimes of high spin rotational states up to J~ = ~ ÷ using the Doppler-broadened y-ray lineshape technique in multiple Coulomb excitation of t s ~Ta with S4Kr ions. We also measured angular distributions of de-excitation y-rays.

2. Experimental procedure A thick Ta foil (85 mg/cm 2) was bombarded with a 348 MeV S~Kr beam from the Manchester HILAC, and de-excitation y-rays from Coulomb-excited states in ~S,Ta were observed using two O R T E C Ge(Li) counters placed at 0 ° and 90 ° with respect to the beam direction. The counter at 0 ° was a 70 cm 3 true coaxial Ge(Li), the front surface o f which was 5 cm from the target. Its energy resolution ( F W H M ) was 3.0 keV at 1274.5 keV. A y-ray spectrum observed at 0 ° is shown in fig. 1. This spectrum was used to extract nuclear lifetimes from the Doppler-broadened y-ray lineshapes. The counter placed at 90 °, which had a volume o f 50 cm ~, was used to deduce the energies of y-rays. Angular distributions o f de-excitation y-rays were also measured using the 70 c m 3 Ge(Li) counter, measurements being made at six angles between 0 ° and 90 °, with respect to the beam direction. In this case the counter was placed 7 cm from the target. Corrections were applied for dead time losses and for small anisotropies associated with incorrect centering and ~-ray absorption in the apparatus. The centering correction was obtained by observing target produced K X-rays. "~To ( ~ K r . e 4 K r ÷

~) " T o ' .

Ex, = 3 4 8 M e V

+

+

"~

Io'

id

8

10 3

I I00

,

I 200

a

I 300

,

400

, CHANNEL

I SO0

~

I 600

i

I 700

,

i 800

,

I 900

NUMBER

Fig. I. Singles de-excitation ),-ray spectrum observed at 0 ° with respect to the beam direction. Errors in ~,-rays energies are less than 0.5 keV.

tS~Ta

257

3. Method of analysis 3.1. DOPPLER-BROADENED y-RAY LINESHAPE

The method used to analyse Doppler-broadened y-ray lineshapes has been published previously a). It is now briefly summarised. In Coulomb excitation with a thick target the observed number of de-excitation y-rays, with a Doppler shift AEv, associated with transitions from the state N to the state M between time t and t + At can be written as AN(AE~) oc e ( ~ )

d2tr(N ~ M ) A E u, P At2pAQ~At, dQpdQ~ dEp/dx

(1)

where Ep is the projectile energy and dEp/dx is the stopping power of the target material for the projectile, e(t2~) is the differential detection efficiency of the y-ray detector, and d2a(N -~ M)/dt2pd[2~ is the appropriate differential cross section for exciting the state N followed by y-decay to the state M. The energy shift 3E~ of y-rays emitted in the direction n~ from recoiling nuclei with velocity v(t) is given, to first order in v/c, by AE~ = E.l(v(t)" n~)/c,

where E~ is the unshifted y-ray energy. The time t is related to the velocity of the recoil by ('v(t)

t = M1

Jv~o(dE/dx)-ldv,

(2)

where v(0) is the initial recoil velocity induced by the projectile, M t is the atomic mass of the recoil ions, and d E / d x is the stopping power for them in the target material. The fraction of unshifted (AE~ = 0) y-rays is given by the integration of eq. (1) over the time t from tr to infinity; t r is the time at which the recoils with the initial velocity v(0) come to rest, tr = M t

( d E / d x ) - ldv. (0}

In semiclassical Coulomb excitation theory the differential cross section can be written in the form d2a(N ~ M) _ dO'RutherforddWN~M(Ep, 0p, 0r, q~r,t) df2pdf2~ dt2p df2r

(3)

According to Winther-de Boer 9) the angular distribution WN~M for the de-excitation y-rays is given by

OWN~M(Ep,0p, 0),, ~b~, t) = df2~

(4rt)_ ~

~

k=0,2,4

-k<=x<=k

AkK(Ep '

Op,t)Fk(I M, Ix)YkK(Or, q~r),

(4)

258

T. INAMURA

et al.

in the coordinate system with the z-axis in the beam direction. The coefficients Ak~(Ep, Op, t) are given by

Ak~(Ep, Op, t) = ~RK(N)TffI exp ( - t / r N ) + Ctk~(N+ 1)Gk(Is, I N+1) X

{

1

exp(-t/~N)+

1

--exp(-t/rN+~) TN+I'--T'N

T.M__q~N+ 1

t + ....

(5)

where the quantities ~k~(N) are the statistical tensors of the state N. The coefficients FR(IM, IN) and Gk(IM, IN) are as defined by Winther and de Boer 9) and, in evaluating them, experimentally determined branching ratios and E2/M1 mixing ratios were used. To determine the lineshape we integrate eq. (1) over Ep, £2p, ~2~, and t (or v). The statistical tensors of the Coulomb-excited states for each Ep and 0p are calculated using an extended Winther-de Boer multiple Coulomb excitation program. We make use of rotational model reduced E2 matrix elements in this computation. It has been shown that the calculated lineshape is only weakly dependent on the matrix elements used s). A least squares method is employed to fit the experimental lineshape. In the fitting procedure a feeding correction is made by taking higher excited states into account a). 3.2. GAMMA RAY ANGULAR DISTRIBUTION Angular distributions of de-excitation y-rays in multiple Coulomb excitation can be calculated by integrating eq. (4) over the projectile energies Ep, scattering angles of the projectile 0p and the time t (from 0 to infinity). In order to obtain a conventional distribution function, we also integrate it over the y-ray trajectories which include the geometry and relative differential detection efficiency of the Ge(Li) detector. In the case of a true coaxial Ge(Li) we have

WN~(O~) = 1 + G2A2PE(COS0~)+ G,,AgPg(cOs 0~),

(6)

where the angular distribution coefficients A i (k = 2, 4) are given by

f~foako(Ep'Op) dORutherf°rdsinOpdEpdOp Fk(IM, IN) ~ p dEp/dX d~p , Ak= 2kx/2~F°(IM'IN) fEp fop A°°(Ep'OP)dEp/dxd°Rutherf°rdsinOpdEpdOvd~p and the geometrical attenuation factors f

Gk =

(7)

Gk (k = 2, 4) are given by

p t t ~(O'~)Pk(COS0~) sin 0~d0~

r

e(O's)sin O'~dO'y

;

(8)

~81Ta

259

the primed system refers to the coordinate with the z-axis along the centre axis of the detector. The detection efficiency e(0)) was assumed to be t t ! e(0~) oc exp ( - ~tz(0~,))[1 - e x p (/~x(0~))],

(9)

where/~ is the total linear absorption coefficient, x is the path length of photons in the active region of the detector, and z is the path length in the inert core. For cross-over transitions, it is only necessary to compare the calculated distributions for pure E2 transitions with the experimental ones, while for stop-over transitions it is possible to determine fractional M1 intensities 1/(1 + 6 z) from the best fits to the observed angular distributions.

4. Results 4.1. M U L T I P L Y

COULOMB-EXCITED

ROTATIONAL

S T A T E S I N 181Ta

Since the projectile energy is well below the Coulomb barrier (370 MeV in the lab system) the stronger y-rays observed are considered to be de-excitation y-rays from Coulomb-excited states in Ta only. From y-ray energies and intensities we propose a level scheme for the ~+[404] ground-state rotational band as shown in 1539.7

21/2* 1.09-+ 0.14 574.3 /

1239.5 274.(

[ / 522.7

965.4 248.4 716.8 221.5

13/2+ 9.05 :t 1.13 11/2+(22.9 ± 4.8) 9/2*

181 .;3Taloe 2. A

17/2+ 2.78 + 0.3,',, 15/2+ 4.32 +- 0.54

z,95.5 193.8 35 301.6 165.61 I I ,36.5 ~3011"61

Fig.

1

19/24 1.62 + 0.20

7/2 + l~

,L,'exp( ps )

level scheme for the ~+[404] ground-state rotational band in ZSZTa.

fig. 2. This is in good agreement with a recent proposal based on the y-rays observed in the 18°Ta(n, V) reaction 7). The level energies were found to be fitted very well by the expression

E(I) = E r + A [ I ( I + 11- K 2] +B[I(I+ 1 ) - K 2 ] 2, where A = 15.19 keV, B = - 4 . 7 3 eV and E x = - 5 2 . 8 0 keV; these coefficients were obtained by a least squares fit to the levels up to ~ ÷

260

T. I N A M U R A et al.

Spin assignments were confirmed by angular distributions of de-excitation ~-rays as described in subsect. 4.3. 4.2. NUCLEAR LIFETIMES OF ROTATIONAL STATES

Fig. 3 shows the Doppler-broadened v-ray lineshape fits used to extract nuclear lifetimes. For de-excitation 7-rays from higher excited states such as 2~+, 17+2, 19+2 and ~2+ we integrated eq. (1) over Ep from the initial energy of 348 MeV down to 188 MeV in steps of 20 MeV. For lower excited states, ~ + and ~ + , the integration was made from the initial energy down to 68 MeV in steps of 35 MeV because the

5 2 104 5 2 103

'10~rev~

'

0.607keV/CHANNEL 5

11/2+-)7/2+ E~=301"6key z =22.9ps "! (X2=4"07)

~]OkeV ~ . . . . . .

.

O.607keV/CHANNEL

2

~

17/2+->13/2+

2

.

. iI Eir=469.9keV 10' ~ . ~=2.78ps ~ " x2~1"15 5 " i ~ _ ~'t, l 2 ~"~ T 10 ~T {

Az2 k v

ILl]iT*

10" A

r "=1.62ps

10;'

21/2+--'17/2+ t~' [ [ ' ~

,

s

~(~ "r=l.09ps ;(2=1.12

It till11

iltit

.t

2 10~ 5 2

10 20 30 40 CHANNELS Er--)

10~

10 20 30 40 50 CHANNELS El.'-->

Fig. 3. Least squares fits to Doppler-broadened lineshapes of cross-over E2 transition ~,-rays. Best-fit lifetimes together with the reduced X' values are presented in the drawings.

la*Ta

261

excitation probabilities for these low-lying states were no longer negligibly small at lower projectile energies. For the ~2 + state, both integration procedures give an almost identical result. F o r each energy value the integration over scattering angle 0p was made from 0 ° to 180 ° (c.m.) in steps o f 10 °. The feeding correction was made as described in ref. s), the E2/M1 mixing ratio o f the stop-over transitions being taken into account. The extracted lifetimes are given in fig. 2. The errors quoted include an error due to uncertainties in the stopping power and other probable systematic errors in the fitting procedure as well as the statistical error. The methods used to derive the errors are discussed in ref. s). Values of the reduced ~(2 are presented in fig. 3 as a guide to the goodness o f fit. The 482.2 keV y-ray line from the interband decay of the ~+ 2 [402] state to the ~+ [404] state appears in the middle of the Doppler broadened 469.9 keV line from

9/Z-~/2÷gnd

1.1- 11/2+-'~?/~+-ndLg14 - ~ -

0.9

)'9I lV2%9/f 4-

I.£

1.1~ 0.89M1+O.11E2 -

~

{ O.78M1+O.22E2 1 . 0 ~ - ~

+

13/2 ~

0.91 0.8

13/2+-.lt,2+ 1/ 0.78M1+O.22E2

1.0 0.9

~o.8

l

"~ 15/2 "-'13/2 ~: I O~"0"82M1+O'18E2

1.0 0.9 0.8

o.st-~ 1.01 _ . ~ + 0.9 0.8

1.G o.s

0.~ o.~

! -

0.7 17/2 -"15/2 0.90M1+O.10E2 O.6

O.E I

J

J

I

~

I

0"15"35"55"75'90" e

a) CROSS OVERE2

"1's'~'~ ~o" o

b) STOP OVER MI+E2

Fig. 4. Angular distributions of de-excitation 7-rays from multiply Coulomb-excited states in 181Ta. Thick lines are least squares fit curves, and thin lines are theoretical distributions computed using the Winther-de Boer multiple Coulomb excitation code. For the stopover Ml +E2 transitions, the M1 fraction of the total was treated as a free parameter and determined so as to make the computed A2 value equal to the experimental one.

262

T. I N A M U R A et al.

the decay of the ~z+ level. Fortunately, the lifetime of the intruder is comparatively long [z½ = 10.6 ns, ref. 1o)] and its effect on the 469.9 keV line is consequently localized to a small number of channels which were neglected in the fitting procedure. It can be seen from fig. 3 that, for the decay of the ~a+ and ~ + states, the lineshapes just above the stopped peak are not well fitted. This is probably due to the fact that the Blaugrund prescription, which is used in the lineshape fitting procedure, slightly overestimates the effects of multiple nuclear scattering a). For the ~ + --* 29-÷transition five channels in this region of the spectrum were omitted from the fit. The error, so introduced, is small compared with the quoted errors in the lifetimes. The occurrence of a Compton edge due to the 469.9 keV 7-rays and an intruder peak (~+ ---, ~ + ) in the region of the ~t+ ~ ~+ lineshape made it impossible to determine the lifetime of the ½t+ state using the conventional fitting procedure. However, in fig. 3 we show a theoretical curve corresponding to z = 22.9 ps, which is in accord with the value derived from B(E2)I' [ref. 4)]. 4.3. A N G U L A R D I S T R I B U T I O N S OF D E - E X C I T A T I O N ~,-RAYS

In fig. 4 angular distributions of de-excitation 7-rays from multiply Coulombexcited states in X81Ta are shown normalized to unity at 0 °, i.e. W(O)/W(O°).Least squares fits to the observed distributions of y-rays which were assigned as cross-over TABLE 1 Angular distribution coefficients for cross-over transitions Transition

Az

A4

Pi

Pr

Er(keV)

exp

theor

exp

theor

,~+ J-~-+ ~+ 1.2+

z+ 2+ ~+ x3+ xs+

301.6 359.0 415.2 469.9 522.7

0.020 +0.013 0.063 + 0.015 0.153 + 0.021 0.162 + 0.041 0.187+0.063

0.020 0.060 0.106 0.139 0.175

0.023 +0.020 0.040 + 0.022 0.006 + 0.029 0.036 + 0.059 0.012+0.101

0.001 0.002 0.008 0.006 0.006

2 2 2

29+ 2

2

2

2

----

---

----

---

TABLE 2 Angular distribution coefficients and fractional MI intensities for stop-over transitions li"

9+ 2

I~'

Er (keV)

A2

A4

Fractional M1 intensities

7_+

136.5

-0.028+0.011

0.007+0.017

+0.04 0.89_o.0s

2

l_t+

--

--

9+

165.6

0.026+0.012

0.025+0.019

0.78_o.2s+°'x2

xa+

1.1.+

193.8

0.082+0.016 --

0.002+0.024

+0.06 0.78_o.oa

15 +

J.a +

221.5

0.127 + 0.029

0.022 + 0.046

0.82 + 0.06

__

x~_+

248.4

0.103+0.068

0.021 +0.108

0.90_o.os

2

2

2

1~+ 2

2

2

2

2

--

--

--

--

--

--

--

--

+0.05

917.4_+117.8 b)

ed_+2

1)

1156.4

693.7

192.3 385.5

28.4 82.7

5.7

rotor c)

4.89 +1.58

2.31 +0.20 -3.22 +0.41 --

0.681 +0.049 -1.53 -+0.12 -

exp

6.14

4.78

2.40 3.53

0.621 1.43

rotor c)

Branching ratio (cross-over/stop-over)

Derived from T 1/2 = 40 ps given in ref. 4). Uncorrected for ~r. Qo = 7.1 b and {g~-gRI = 0.482 were used (see text). Derived from the mixing ratio, otherwise from the branching ratios.

585.6+ 72.5 --

L9+ 2

") b) c) a)

201.1+ 25.2 -329.8+ 40.4 --

~-+ 2 17+ 2

(sec-

(29.0+ 6.1) -87.2+ 10.9 --

6.85 a)

exp

X 109

Transition probability

~2 + 13+ 2

~+ 2

State

0.49 +0.07 -0.12 0.33 +o.14 -0.10

0.53 +0.47 -0.20 0.53 -+o.t2 0.09

0 " 352 -+0.084 0.078

exp

Mixing ratio ~ = E2/MI

0.409 0.404

0.420 0.414

0.457

rotor c)

"

15 +0.20 -0.09

0.56+0.22 1" 33 +1'14 -0.79

0 " 32 -+°"13 0.10

0

exp

0.74 1.07

0.46

0.21

rotor c)

B(E2; I ---, 1 - 2 ) B(E2 ; I ~ 1 - 1 )

Physical quantities related to the transition m o m e n t s of the ~+ [404] ground-state rotational band in 181Ta

TABLE 3

0.068 +0.013

0.0728 _+0.0053

0.0633 + 0.0031

0.0655 + 0.0029

0.083_+o:o~6 a)

Oo

gK -- gR

oo

264

T. INAMURA et al.

E2 transitions in subsect. 4.1 agree well with the theoretical distributions. This confirms the spin assignments assumed in subsect. 4.1. The experimental angular distribution coefficients A 2 and A 4 together with the corresponding theoretical values are listed in table 1. In computing the theoretical distributions the same integration procedures for the projectile energy were used as in the lineshape analysis. For the stop-over M 1 + E2 transitions, least squares fits were made, which resulted in the distribution coefficients A 2 and A4 listed in table 2. The values of A 2 were used to determine fractional M1 intensities for the transitions. Because of large uncertainties in their experimental values, A 4 coefficients were not taken into account. Extracted fractional M1 intensities are given in the last column of table 2. 4.4. PHYSICAL QUANTITIES RELATED TO THE TRANSITIONS MOMENTS G a m m a transition probabilities, branching ratios, E2/M 1 mixing ratios and values of B(E2; I--. I - 2 ) / B ( E 2 ; I ~ I - 1 ) and (gK--gR)/Qo are summarized in table 3 together with the rotational predictions. Internal conversion coefficients, which were needed in estimating v-transition probabilities from the observed lifetimes, were obtained from the tabulation of Hager and Seltzer 11). In calculating rotational values we used Q0 = 7.1 b, the value adopted by Ellis 4), and IgK--gR[ = 0.482 n.m. from the publication of Seaman et al. 5). The signs of the mixing ratios 6, given in table 3, are determined from the cross terms in the function Fk(I~, IN) and follow the phase convention of Winther and de Boer 9). The positive sign of t$ implies that ( g r - gR)/Qo is also positive. The measured B(E2; I ~ I - 2) values and the predictions of the rotational model are compared and discussed in the following section.

5. Discussion 5.1. A RELATIONSHIP BETWEEN LEVEL ENERGY AND E2 TRANSITION PROBABILITY It was noted in subsect. 4.1 that the energy levels were well described by a rotor formula with an additional term proportional to [I(I + 1)] 2. It is, however, interesting to observe that as illustrated in fig. 5, a similar deviation from the simple rigid rotor predictions exists in both experimental cross-over B(E2) values and level energies; B(E2) and level energies as a function of Ii(I i + 1 ) - If(If + 1) begin to deviate significantly from the rigid rotor values at the ~z+ state and the difference seems to increase with the spin value. The level energy plot versus I~(I~+ 1 ) - I f ( I f + 1) implies that in 181Ta the moment of inertia increases with the spin value. There are two possibilities for such a change of the moment of inertia: one is a centrifugal stretching effect 12.13) which induces a larger deformation with increasing the spin value, and the other is a Coriolis antipairing (CAP) effect 14). Centrifugal stretching is likely to enhance the E2 transition moments 15,16). This is in contra-

18 iTa I

r

I

1.4 A

265 I

I

rotational

1.2

'~o 1.0 x

}/

0.8 eq

0.6

i/

0A

~

0.2

/"

ational

=1519

~x~

A 0.5 =E

~ 0A o.3 i

~ - 0.~

0~,~ 4,', 0.1

t

011

I

20

t

~

t L

~,~

t ~

t I

30 li(li+l ) - If ( I f + l )

t ]

40

Fig. 5. The B(E2) values and level energies as a function of li(li + 1) - If(If + 1). Cross points connected with dotted lines are rotational values: Qo = 7.1 b (8 = 0.24), and h2/2~¢o = 15.19 keV.

diction with the observation that the cross-over E2 transitions from higher spin states such as ~ + , ~ + and ~ ÷ are retarded. However, the CAP effect can cause such a retardation in E2 transition rates 17,1 s), and it therefore seems possible that this effect may play an important role in 1s iTa" An alternative explanation could involve an accidental cancellation in the transition matrix elements due to the Coriolis band mixing. We have carried out Coriolis band mixing calculations for the ground-state rotational band to see whether such a cancellation is possible. 5.2. C O R I O L I S B A N D M I X I N G C A L C U L A T I O N

F O R T H E }+[404] G R O U N D - S T A T E

BAND

The method of the calculation is similar to that described by the Stockholm group 19). However, we neglect the blocking effect because it is not expected to contribute to a retardation of the transition rates. The Hamiltonian is written as H = HNilsso n -4- Hpairins q-- H r o t q-- ncoriolis ,

and the diagonalization is carried out in three consecutive Hamiltonian we follow the method of Gustafson et al. 20) parameters 52 = 0.236 (fl = 0.24), 54 = 0 and o~o = 41/A* 52 was derived from the average value of Qo = 7.1 b. Some

(10)

steps. For the Nilsson using the deformation [ref. 21)]; the value of single particle energies

266

T. I N A M U R A et al.

of the N = 4 and 5 shell orbitals have been modified according to the semiempirieal single particle scheme given by Ogle et al. 22). If this is not done, the calculated lowest state is not ~+ [404] but ~+ 2 [402], and ½+, ~+ 2 and ~+ 2 intrinsic states which have already been known are not reproduced. In the second step the pairing force is treated in the BCS approximation with the G-value given by G = 24.0/A [ref. 23)]; 24 levels have been included, i.e. 12 above and 12 below the Fermi level. The Hamiltonian Hot is the diagonal part of the rotational Hamiltonian. The eigenvalues are assumed to be given by

-rotF'"""=

+ B ( I ( I + I ) - K 2)

I(I+I)--Ke+(--I)~+½a(I+½)~K ½],

(li)

where a is the decoupling parameter, and the parameter B is introduced to account for a variable moment of inertia. Off-diagonal terms are given by the Coriolis matrix elements ~12

Vx+ 1,x = - 2,~o x / ( I - K X I + K + 1 ) ( K + llj+IK), where ( K + llJ+lg) = ( K + llJ+lg)s.p.(Ux+ , U~c+ VK+, V~c),

(12)

( g + llj+lg)s.p. = ~ C~+af~x/(j - K ) ( j + K + 1). jz

The calculation was made by treating the values of h 2 / 2 J o and B as free parameters, the values h2/2Jo = 15.90 keV and B = - 6 . 0 eV being found to give a reasonably good fit to the observed level energies up to the ~ + state. The results are summarized in table 4, where only significant components of the wave functions are quoted. As can be seen from this table, the configuration 7+ [404] is dominant in all members of the ground-state rotational band. Hence, it is most unlikely that accidental TABLE 4

Results of Coriolis band mixing calculation for the ground-state rotational band in 1s ITa State I" 2+ 2 "9-+ 2 11+ 2 13 2+ 1.l+ 2 LT_+ z x~+ z z.t+ 2

Expansion coefficients E°'~c(keV) 3+[422] 0.0 136.9 302.9 497. I 718.7 966.5 1239.3 1535.7

0.0020 0.0040 0.0065 0.1)(193 0.0127 0.0163 0.0209 0.0251

~+[411] - 0.0027 -0.0053 -0.0085 - 0.0123 -0.0167 -0.0216 - 0.0270 --0.0331

3+[402] 0.0 0.001)1 0.0001 0.001)2 0.0002 0.111)t)3 0.0 0.00114

3+[413]

3+[402]

0.0417 0.0630 0.0818 0.0995 0.1167 0.1335 O.1501 0.1664

- 0.0304 -0.0459 -0.0594 - 0.0720 -0.0840 .0.1)955 - O.1067 -0.1176

~+[404] 0.9987 0.9969 0.9948 0.9923 0.9894 0.9861) 0.9822 0.9781

~+[-404] 0.0 -0.0025 -0.0038 - 0.0048 --0.0058 -0.111)67 - 0.0076 -0.01)84

lSlTa

267

TABLE 5 The B(E2) values for the cross-over transitions in the ~+ [404] ground-state band in lSlTa State

B(E2; I ~ 1 - 2 )

I" .

.

.

.

.

.

.

. . . . . . . .

11+

~~3 ÷ 2 ~-+ 2 1~+ 2 1_9+ 2 z~+ 2

exp

rotor

. . . . . . . . . . . . . . . . . . . . . . . . . .

(0.39-+0.08 0.72 + 0.09 0.93+0.12 0.90 + 0.11 1.01 +0.14 1.00+0.14 a)

Coriolis calc

--0~3-65 0.676 0.909 1.082 1.211 1.310

0.366 0.678 0.912 1.085 1.214 1.314

a) Rotational model branching ratio was used to derive this value from the observed lifetime; a probable error in the ratio was assumed to be + 3 0 %, referring to the experimental error in the branching ratio for the ~ + state.

cancellations in E2 transition matrix elements will occur. In table 5 experimental B(E2) values are listed for the cross-over transitions, together with rigid rotor values and the ones based on the Coriolis band mixing calculation. As is customary intrinsic transition matrix elements were neglected in the band mixing calculation of B(E2) and a value Qo = 7.1 b was assumed. 5.3. C O R I O L I S A N T I P A I R I N G E F F E C T

In the light of the above discussion it appears that the CAP effect plays a major part in the behaviour of the ground-state rotational band in 18XTa. This effect is associated with the core, so that the variable moment of inertia due to the CAP effect should be similar to that of 18°Hf, since the nucleus t81Ta may be regarded as 18°Hf plus a single proton. This is confirmed by comparing the values of hZ/2Jo and B parameters for the ground-state rotational bands in a81Ta and aS°Hf, and their intrinsic quadrupole moments Or; it is seen in table 6 that there is a striking similarity between their respective parameters. The data on tS°Hf are from a recent compilation by Greenwood 14). It is important that measurements of E2 transition probabilities for higher spin states in t8°Hf be made. TABLE 6

Parameters h2/2~ o and B, and intrinsic quadrupole moments Qo for the ground-state rotational bands in lSlTa and lS°Hf

h2/2Jo (keV) lStTa IS°Hf

B (eV)

Qo (b)

LSF

CC

LSF

CC

15.19 15.57

15.90

-4.73 -7.34

-6.0

7.1 6.8

Here LSF stands for least squares fit and CC for Coriolis calculation. Data on lS°Hf were obtained from a recent compilation by Greenwood 2,).

268

T. INAMURA et ai.

6. S u m m a r y The level scheme for the ground-state rotational band in I s iTa has been constructed up to the 22~+ state by measuring energies, intensities and angular distributions of de-excitation y-rays from multiple Coulomb excitation of XSlTa with S4Kr ions. In addition nuclear lifetimes of the band members were extracted from the Dopplerbroadened y-ray lineshapes. Gamma transition probabilities, branching and mixing ratios, B(E2; I --, I - 2 ) / B(E2; I -, I - 1), (gK--gs)/Qo and B(E2; I -* •-2) values have been examined. The reduced transition probabilities for the cross-over transitions, B(E2; I -, I - 2 ) , are compared with the rotational predictions (Qo = 7.1 b) and a retardation is found for the transitions from the ~+2 , 19+2 and ~2+ states. It is also noted that the moment of inertia deviates from the rigid rotor prediction for these higher spin states. This variable moment of inertia and the retardation in E2 transition rates in is tTa could be accounted for by the CAP effect. We would like to thank Prof. J. C. WiUmott for constant encouragement during the course of this study, and Dr. M. Wakai for help in making Coriolis band mixing calculations. It is a pleasure to express our thanks to the staff of the Manchester HILAC for the excellent operation of the machine. One of us (T.I.) would like to acknowledge the award of a Fellowship by the Science Research Council. References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24)

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