IL N UOVO CIMENTO VOL. 60 A, N. 1 1 Novembrc 1980

The Positive-Parity States in 170 and 1717.

W. H. CHU~'G (*) and Y. ~I. StIIN

Saskatchewan Accelerator I, aborato~'y, University o/ Saskatchewan, Saskatoon, Canada S7N OWO

(riccvuto 1'11 Agosto 1980)

S u m m a r y . - - The intermediate-coupling unified model has been applied to the (~ closed shell plus one ~ nuclei, ~ 0 and 17F. The energy levels, the electromagnetic-transition rates and the static moments have been calculated and compared with the experimental results. A satisfactory agreement with the experimental data is obtained.

1 . - I n t r o d u c t i o n .

According to the simple shell model one would expect pos i t ive-par i ty spectra of the ld~, 2si and ld~ single-particle s ta tes in 170 and 17F. The spectroscopy of 170 and 17F looks fine in t ha t the pos i t ive-par i ty spins of the three lowest s ta tes are jus t wha t is expected f rom the simple shell model, i.e. these levels p r e sumab ly correspond in the simple picture to placing an odd ex t ra nucleon outside the 160 core in the l d : , 2s�89 and ld~ levels of the 2s-ld major shell. t towever~ the whole observed spectra (1) show m a n y levels beyond the second- excited s ta tes as well as m a n y negat ive levels be tween the normal -pa r i ty states. The group of three nega t ive-par i ty levels, �89 ~- and ~-, be tween 3 and 5 MeV of exci ta t ion would be easily produced b y breaking up the 1~O core. The ob- served E2 t rans i t ion ra tes (~.3) f rom the first-excited to the ground s ta te are also obviously too large to fit the simple shell model picture.

(*) Present address: Department of Physics, Busan National University, Busan, South Korea. (1) F. AJZI~NBERG-SELoVE: ~Vr Phys. A, 281, 1 (1977). (2) J. A. B~C~E~ and D. H. WILKInSOn: Phys. l~ev. Sect. B, 134, 1200 (1964). (3) S . J . SKORKA, J. H]~RT~L and T. W. R~Tz-ScHMII)T: ~Vq~cl. Data A, 2, 347 (1966).

27

2 8 W. II . C H U N G a n d Y. M. S H I N

To underst .md the bulk of the low-lying states we may need to take into

account multipartic]e-multihole configurations and core deformations. Recently, several theoretical calculations (4-s) have been performed on the ~( closed shell plus one )) nuclei, ~70 and ~TF. B~ow~ and GREE.~ ~ (4) have described the low-

lying positive-parity spectra of ~70 and ~TF by mixing the usual states in the spherical-shell model with deformed states obtained by exciting particles out

of a deformed core, i.e. 3p-2h and 5p-4h excitations in the Nilsson diagram. In this work they have tried to reproduce several experimentally observed �89 and 23+ states in 170 and 17F above 5 MeV of excitation which are not pro-

duced by the spherical-shell model. A weak-coupling calculation on ~70 has been reported by BElC~STEIN (5), using the low-lying multiparticle-multiho]c

states up to 5p-4h states. BERNSTEIN has interpreted some of the low-lying states in 170 as mixtures of spherical-shell model configurations and deformed multi-

particle-multiho]e configurations in a Weak-coupling approximation. ENGELAb~D and ELLIS (6) have calculated gamma-transi t ions and spectroscopic factors

for the low-lying states of ~70 and ~TF by using a weak-coupling model ap-

proach. In this work they have considered excitations from the l p to the

2s-ld shell, assuming tha t excitations involving the shells ]s and 1]-2p are of minor importance. There also exist a few calculations (7,s) on the negative- par i ty stas in 170 and ~TF.

In this theoretical situation it may be interesting to investigate the low- lying states in ~70 and ~TF based on the unified model. I n the present work

we have calculated level properties of the low-lying posit ive-parity states in ~70 and ~F within the framework of the intermediate-coupling approach in the unified model, assuming tha t the last odd nucleon is coupled to a quadru-

pole vibrational core. To our knowledge, there exists as yet no such a t rea tment

dealing with positive-parity states in these nuclei. The basic formulae used ~re briefly described in sect. 2. The calculated

results are given in sect. 3 and are compared with the recent experimental

results. Finally, conclusions and discussions are presented in sect. 4.

2 . - T h e m o d e l .

The intermediate-coupling unified model was first introduced by Bomr and MOTT~LSO~ (~) and further developed by C ~ O U D ~ Y (~o). Since detailed

(4) G. E. BgOWN and A. M. GRn~N: .Yucl. Phys., 75, 401 (1966). (5) A. ~-~. BERNSTEIN: Ann. Phys. (N. Y.), 69, 19 (1972). (~) T. ENGELAND and P. J. ELLIS: -Natl. Phys. A, 181, 368 (1972). (7) S.T. HSIEH, K. T. KN6PFLt:, G. MAIgL~ and G. J. WAGing: ~Yucl. Phys. A, 243, 380 (1975). (8) P. J. ELLIS and T. EN~ELAND: Nucl. Phys. A, 144, 161 (1970). (9) A. BOtI~ and B. R. MOTT~LSON : ,~lat. tZys. Medd. Dan. Vid. Selsk., 27, No. 16 (1953).

T I t E P O S I T I V E - P A R I T Y S T A T E S I N 1 7 0 A N D *7F 29

descript ions of the intermediate-coupl ing unified model have been repor ted elsewhere (~-n), we shall only outline the model and give the basic formulae

used in our calculations. The to ta l Hami l ton ian for the core-particle sys tem is given as

(1) H = H -b Hp -[- Hi= t ,

where H is the core Hami l ton ian , Hp the particle Hami l ton ian and Hi= t the core-part icle interact ion t Iami] ton ian :

(2) Htn ~ = - - ~ t / (o (~) ~ ~ (b. @ (--)'b+_,) Y~,(0, 99), /z

where b, and b~, are the annihi lat ion and creation operators, ~r is the spherical- ha rmonic opera tor and ~ is the dimensionless coupling pa rame te r :

(3) ~ = K ,

where the coupling constant K ~ <k(r)> is t aken to be 40 MeV and C is the nuclear deformabil i ty .

The basis eigenvectors of the coupled sys tem are denoted b y

[J; 2r IM> = ~, (jRmm'ljRIM) Ijm>INR' m'> , ~rt,m'

where ( jRmm'I jRIM) axe the Clebseh-Gordan coefficients, j the single-particle angula r m o m e n t u m , N the num ber of phonons of the core angular mo- m e n t u m R and I the to ta l angular m o m e n t u m of the nucleus.

Each energy mat r ix of spin -r I is diagonalized according to

(5) HIE; IM> = EIE; IM> ,

where E is the energy eigenvalue associated with H.

(6) ]E; IM> = ~ A~,w(E)Ij ; N R ; IM> i ,NR

is the wave funct ion represent.ing the s tate of the nucleus and A~,sa(E)* are the expansion coefficients of the eigenvector which are obta ined b y diagonalizing the to ta l Hami l ton ian H.

(10) 1). C. CIIOUDtIUI~Y: Mat. Fys. Medd. Dan. Vid. Selsk., 28, No. 4 (1954). (11) W. tI. (]Hu~r~: Z. Phys., 266, 1 (1974).

30 w . H . C~tUNG and Y. M. SHIN

The wave f u n c t i o n s o b t a i n e d b y d i agonu l i z ing t h e t o t a l H a m i l t o n i a n a r e

used t o ca l cu l a t e t h e r e d u c e d t r a n s i t i o n p r o b a b i l i t i e s a n d n u c l e a r m o m e n t s .

3 . - R e s u l t s .

3"1. E n e r g y levels. - I n t he p r e s e n t work , i t is a s s u n l e d t h a t t h e l a s t o d d

nuc l eons of 170 a n d 17F h a v e t h e ld+, 2s�89 a n d ld+ s t a t e s a v a i l a b l e to t h e m

a n d a re c o u p l e d to t h e q u a d r u p o l e su r face v i b r a t i o n s (up to N--~ 3) of t h e

d o u b l y even core, 160, of t h e nucle i . The e f fec t ive s p a c i n g p a r a l n e t e r s ,

e 1 = s�89 a n d e2- - - -d~--d~, t h e p h o n o n e n e r g y hw a n d t h e coup l ing p ~ r a -

m e t e r ~, wh ich e n t e r t h e ca l cu la t ions , a re c ons ide r e d to be a d j u s t a b l e p a r a -

me t e r s . T h e b e s t fi t was d e t e r m i n e d b y c o m p a r i s o n w i t h a l l t h e a v a i l a b l e

e x p e r i m e n t a l d a t a ( ' ) , as shown in t a b l e I . The n u m b e r of p h o n o n s is a s s u m e d

Lu

- - ' 0

Fig. 1. Fig. 2.

Fig. 1. - Plot of E as a function of the coupling parameter ~ for 170.

Fig. 2. - Plot of E as a function of the coupling parameter ~ for 1T.

TIIE P O S I T I V E - P A R I T Y STATES IN 170 AND 17F

TABLE I. -- Parameters used iu the calculations.

31

Nucleus e 1 = s�89 d$ (MeV) e 2 = d~ -- d~ (MeV) he) (MeV) $

170 2.10 6.20 6.20 1.250

17F 1.50 5.10 6.10 1.200

to be N < 3 . The total Hamil tonian is diagonalized for the final values of el, e2 ~ < I < - z �9 Table I I provides a tabulat ion of the computed and $ for each I , 1 1

and experimental energy levels in 170 and 17F. The energy eigenvMues are

plot ted as a function of the coupling parameter in fig. 1 and 2 and the predicted energy spectra are compared with the experimental data (~) and with other

theoretical results (~,5) in fig. 3 and 4.

9.0

MeV

8.0

7.0

6.0

5.0

4.0

3.0

2.0

1.0

3 2 7

2 - - 1 3 3 2 2

- - ' ~ - L 2

5 1 7 2

_ _ 2 2 5 2 "...3_ - - . $ 3 2 1

2 2

1 2 7 2 - - 2 2 3 2 5 / 9 3

2 2 2

3 3 2 - - ~ L . 5_ 3

2 3 2

3 2 2

1 1 1 2 2 2 1

--f.

5 5 5 5 2 2 2 2

exper imen t pres en t Bern stein B r o w n

Fig. 3. - Compar ison of t he p r ed i c t ed pos i t ive -pa r i ty levels in 170 wi th the exper i - m en t a l spec t rum, along w i t h o the r predic t ions .

32 w.H. CHU~-G and Y. ~. SH~N

J

10.0

MeV

9.0

8.0

7.0 L

6.0

5.0

4.0

3

2 7

5 - - 2 1

2 3

3 2 - - 2

- - 1

3

2

3 2

3 2 1 2--3

i ~ - 2 2 5

- - ' 7 )- 2 9 2

3 2

1

2 3

i ( -s 2

3 2

30 2.0

1 .0~_

0

1 1 1 2 2 2 5 5 5 2 2 2

exper/ment present 1grown

Fig. 4. - Comparison of the predicted positive-parity levels in I~F with the experi- mental spectrum, along with other predictions.

As can be seen in fig. 3 and 4, the agreement between the experimental and predicted energies in both 170 and 17F is relatively good compared to other theo- retical calculations. It is noted in the figures that the first- and second-excited states observed experimentally in 170 and 17F are well predicted theoretically, although the predicted second-excited states are located at slightly lower posi- tions. In the higher region in 170, the predicted levels with spins 23 at 7.035 MeV,

at 7.142 MeV, �89 at 7.430 MeV and 2 a at 8.272 3leV would correspond to the

Ttt]~ POSITIVE-PARITY STATES IN 170 AND 17F

TABLE I I . - E~ergy levels in 1~0 and I~F.

3 3

5 1_ ~ 9 7 2 2 2 2 2

Experi- Calcu- Experi- Calcu- Experi- Calcu- Experi- Calcu- Experi- Calcu- mental l~ted mental luted menta l luted mental luted mental luted

17 O

0 0 0.871 0.871 5.086 4.620 5.721 6.160

6.499 6.357 5.870 8.474

7.383 7.142 7.956 7.430 7.202 7.035

8.410 8.352 8.070 8.272

8.898

0 0.495 0.495 5.103 3.997 5.640 6.078

6.355 5.682 5.817 9.91 11.948 8.416 9.520

6.686 6.556 6.774 6.737

8.075 7.750 7.116 7.356

8.750 7.479 7.539

8.760

e x p e r i m e n t a l l y o b s e r v e d levels 7.202 MeV (~), 7.383 MeV (25), 7.956 5IeV (�89 a n d

8.070 MeV (~), r e s p e c t i v e l y . The s i t uu t i on in 17F is s imi la r , b u t t h e o rde r of

t h e leve ls is s o m e w h a t c h a n g e d . N e a r 5.8 MeV of e x c i t a t i o n , howeve r , t h e

e x p e r i m e n t a l l y o b s e r v e d 1 a n d ~ levels in b o t h nuc le i a re n o t p r e d i c t e d in t h e

t h e o r y , whi le t h e y a re r e p r o d u c e d b y B]~ow~ a n d GREE~ (4).

3"2. Elec t romagne t i c - t rans i t i on rates and nuclear m o m en t s . - The r e d u c e d

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 2 ) a n d B ( M 1 ) , were c a l c u l a t e d b y u s i n g t h e w a v e

f u n c t i o n s o b t a i n e d a b o v e b y d i a g o n a l i z i n g t h e e n e r g y m a t r i c e s . The ca lcula-

t i o n s were p e r f o r m e d w i t h t h e f r e e -pa r t i c l e gs f ac to r , i.e. gs ~- 5.5856 for p r o t o n

{17F) a n d g~ z - 3.8263 for n e u t r o n (170), a n d a n ef fec t ive cha rge for t h e

o d d nuc l eon in each nuc leus , i .e. e: u ~ - e ~ - 0.5 e, as u sed for a p o l a r i z a t i o n

cha rge in 170 a n d 17F b y ENGELAS~D a n d ELLIS (6). Va lues of o t h e r p a r a m e t e r s

u sed in t he ca l cu l a t i ons a r e g~ ~ 1 a n d gR ~ Z / A ~ 0.5294 for 17F; g~ ~ 0.0

a n d g , ---- Z / A ~- 0.4706 for ~70; K ---- 40 MeV a n d R0 --~ 1 . 1 8 A t f m .

T h e ca l cu ln t ed r e su l t s for t h e t r a n s i t i o n r a t e s a r e c o m p a r e d w i th a v a i l a b l e

e x p e r i m e n t a l d a t a in t a b l e I I I . U n f o r t u n a t e l y , a t p r e s e n t t h e r e is n o t enough

e x p e r i m e n t a l d n t a a v a i l a b l e on r e d u c e d t r a n s i t i o n p r o b a b i l i t i e s of 170 ~nd 17F

w h i c h can be c o m p a r e d w i t h t h e t h e o r e t i c a l va lues . I t o w e v e r , t a b l e I I I shows

t h a t t h e p r e d i c t e d B(E2) v a l u e of 12 .9e2 fm ~ for t h e �89 g.s. iu ~70 is up-

3 - II Nuovo Cirne~to A.

~ W . H . CIIUNC- a n d Y. M. SHIN

T a b l c I I I . - Reduced transition probabilities in ~70 and I~F.

T r a n s i t i o n s B(E2) (e ~ fm 4) B ( M 1 ) (n.m.) ~

E xpe r i - Calcu- Expe r i - Calcu- m e n t a l (a) l a t e d m e n t a l l a t e d

17 0

�89 g.s. 6.5 12.9 0.0

~1--~ g.s. 3.25 1.68 -+�89 7.34 0.013

~ - + g.s. 2.59 0 .00004 --~ �89 2.69 0.059

~a--~ g,s. 0.014 0.014 -+ �89 7.01 0.0

�89 g.s. 2.93 0.0

~3--~ g.s. 0.29 0.48 -~ �89 1.39 0.022

17 F

�89 g.s. 65 13.5 0.0

2al--~ g.s. 2.99 2.64 --~ �89 7.50 0.006 5

~ - -~ g.s. 3.12 0.011 --~ �89 3.06 0.047

TABLE IV. - :~lixing ,ratios and lifetimes in JTO and 17F.

Mixing r a t i o s L i fe t imes

T r a n s i t i o n s E x p e r i m e n t a l Ca lcu la ted Sta tes E x p e r i m e n t a l ( 1 ) Ca lcu la ted

170 xT0

~ l -+g . s . - - 0.054 �89 (258.6 4- 2.6) ps 126 lOS --~�89 + 0.540 31 0.34 fs

~ - -~ g.s. -~- 14.7 22a 0.018 ps --~�89 - - 0.35 ~a 0.011 ps

.~.~--~ g.s.' - - 0.059 �89 0 .012ps

~a--~g.s. - - 0 . 0 5 4 2 3 3 0.21 fs --~ 11 ~- 0.49

17 F 17 F

3 .~x-~g.s. + 0.035 �89 (412 4- 9) ps 2030 p s -+�89 - - 0.99 eal 0.34 fs

~2-~g.s . -- 0.94 ~2 8.83 fs --+ �89 -+- 0.42

' f i fe PO,qlTIWE-I'ARITY STATE8 IN :70 AND JTb~ 35

prox ima te ly twice the observed value, 6.5e~-fm% while the predicted B(E2), 13.5 e-~ fro% for the same t ransi t ion in '7F is approxima.tely five t imes smaller t han the ( .xperimental value, 65 c-'fm*.

The calculated lifetimes and mixing ratios for ~"O and '7F are compared with av%ilable exper imenta l da ta (') in table IV. Unfor tunate ly , there is not enough exper imenta l da ta avai lable on mixing ratios and lifetimes of ~70 and 'VF to be compared with the predicted values. I lowevcr , it is also noted, in table IV, theft the observed l ifetime, (258.6 :t: 2.6) ps, for the �89 s tate in '70 is app rox ima te ly twice the predicted value, 126 ps, while the predicted life- t ime, 2030 ps, for the same s ta te in 'TF is approx imate ly five t imes larger than the exper imenta l value, (412 _-:' 9)ps.

TAm,~- V. - ,Vuclear ,re,amen, is in ~70 and ~71,~.

Nuclei Magnetic-dipole moment/~g.,. (n.,n.)

Electric-quadrupole moment Qg.,. (b)

Experimental (~) Calculated Experimental Calculated

170 -- 1 .89379 ~ 0 . 0 0 0 0 9 - - 1.95 - - 0 . 0 2 5 7 8 ( l e ) - - 0 . 0 4 4

I~F +4.7223 -]=0.0012 +4.50 --- 0.10 + 0.02 ( l a ) --0.045

The calculated and exper imenta l nuclear moment s for ~K) and ~TiF are com- pared with each other in table V. I t is seen fl 'om the table t h a t the calculated magnet ic-dipole m om en t s and electr ic-quadrupole momen t s for the ground s tates of ~vO and ~ F are in good agreement with the available exper- imenta l da ta (~.~,~3).

4 . - C o n c l u s i o n s a n d d i s c u s s i o n s .

We m a y conclude f rom the preceding section t h a t the in termediate-coupl ing v ibra t ional model provides a reasonably good description of the proper t ies of the low-lying states in ~70 and ~TF.

As can be seen in the energy level d iagrams (fig. 3 and 4), agreement exists in general between theory and exper iment . For example , the predicted �89 anti ~ levels in both nuehq are in good agreement with the exper imenta l data. In ~TO, the exper imenta l ly observed levels 7.202 MeV a (~), 7.383 MeV (~), 7.956 MeV (�89 and 8.070 MeV (~) wouhl be predicted theoretically a t proper energy locations. However , there are certain discrepancies between theory and exper iment . For example , in the middle region near 6 MeV of exci tat ion three higher spins,

(12) It. F. SOnAEH.m I I I and R. A. ]~LEMN: Phys..Rev., 1111, 137 (1969). (13) I ' . ~IINAMISONO, Y. NAJIRI, A. ~IIZO:BUCItl a n d K . SUGIMOTO: Nucl . Phys. A, 236, 416 (1974).

36 w . H . CHUNG and v. )~. StIIN

TABLE VI. - Expansion coefficients corresponding to states [E (MeV); I~> of 170.

Basi~ [o;6> IO.871;~> 15.086;~> [7.202;6> 17.383;6> [7.956;�89 [8.070;~) states [NR; j> 100; 6> 0.9319 --0.2222

12;6> --0.2714 0.4731 0.4436 --0.7879 0.0046 0.7594 --0.3412

20;6> 0.0473 0.1859

22; ~> 0.0211 --0.1055 --0.1103 --0.0257 0.3421 --0.3427 0.2451

24;~> 0.0523 --0.2602 --0.2196 0.2885 0,3331

30;6> --0.0052 --0.0447

32;~> --0.0100 0.0251 0.0197 --0.0076 --0.0410 0.1021 --0.0249

33; ~> --0.0002 --0.0035 --0.0150 --0.0549 0.0750 --0.0287 0.0662

34; 6> --0.0082 0.0594 0.0101 --0.0549 --0.0766

36; ~>

00;6> 0.6484 0.0548 0.6950

12;~> --0.0693 0.1898 --0.1786 0.0028 0.0203 --0.0822 --0.1985

20; ~> 0.0806 0.0058 --0.0595

22; 6> 0.0131 --0.0415 0.0472 0.1591 0.0839 --0.0443 --0.0639

24; ~> 0.0396 0.1950

30; ~} --0.0143 --0.0021 0.0188

32;6> --0.0028 0.0140 --0.0168 0.0028 --0.0101 0.0452 0.0107

33;~> --0.0010 0.0009 0.0387 0.0290 --0.0109

34; ~> --0.0088 --0.0370

36;~>

I00;�89 0.8474 --0.4793

]12;�89 0.2080 --0.4946 --0.5237 0.8040 0.4062

]20;�89 0.0904 0.2241

[22;~> --0.0470 0.1057 --0.1466 --0.0039 --0.0215

124; ~>

]30;�89 --0.0156 --0.0618

[32;�89 0.0110 --0.0441 --0.0403 0.0965 0.0710

133; �89 0.0017 0.0000

134; �89

[36; �89

T I I E P O S I T I V E - P A R I T Y S T A T E S I N 170 A N D 17F 37

TABLE VII . - Expansio~ coej/icie~ts corres'pondi~g to states ]E (MeV); I.n> o/ 17F.

Basic ]0 ;~) ]0.495;�89 ]5.103;-~> ]6.556;�89 16.774;~> 17.356;~> 18.416;~> states I~v~;j>

[00; ~-> 0.9334

[12; ~> - -0 .2614 0.4347 0.3620 0.7861 0.7644 - -0 .4648 0.1065

[20; -}> 0.0452

122; ~} 0.0220 - -0 .0915 - -0 .0868 - -0 .3300 0.0377 0.2244 0.2610]

]24; ~> 0.0499 - -0 .2216 0.2180 0.3176 0.2018

]30; -~> - -o .oo5o

132; ~> - -0 .0092 o.o214 o.0156 0.0987 0.0074 - -0 .0256 - -o .o146

I33; ~) o.ooo0 - -0 .0026 - -o .0111 - -0 .0255 0.0554 0.0489 0.0369

134; ~> - -0 .0078 0.0475 - -0 .0088 - -0 .0751 - -0 .0934

]36; ..~> - - 0.2208

]00; ~> 0.7270 0.0386 0.6251

112; ~> - -0 .0726 0.1953 - -0 .1920 --0 .0711 - -0 .0360 - -0 .1643 0.7161

]20; 23-> 0.0742 - -0 .0070 - -0 .0734

122; .23-> 0.0131 - -0 .0403 0.0482 - -0 .0496 - -0 .1675 - -0 .0564 - -0 .0589

]24; ~-) 0.0411 - -0 .2063

130; ~> - -0 .0123 0.0028 0.0198

]32; ~> - -0 .0028 0.0127 - -0 .0148 0.0473 0.0034 0.0146 0.0676

]33; ~) - -0 .0008 0.0015 --0 .0401 - -0 .0060 - -0 .0221

134; ~> --o.oo87 0.0292

[36; ~>

00; -~> 0.8690 - - 0.4408

~12; ~-) 0.2141 - -0 .4757 0.5529 0.4349

20; �89 0.0829 0.2349

22;�89 - -0 .0456 0.0917 0.1536 - -0 .0588

24; �89 - -0 .4980

30; �89 - -0 .0135 - -0 .0597

32; ~> 0.0107 - -0 .0371 0.0410 0.0667

33; �89 0.0015 - -0 .0474

134; �89 0.0756

[36; �89

~ ~V. ] [ . CIIU~NG a n d Y. ~I. SHI:N

~ ~ a n d -~-, a p p e a r in t h e t h e o r e t i c a l s p e c t r u m which a r e no t o b s e r v e d expe r -

i m e n t a l l y .

The e l e c t r o m a g n e t i c p r o p e r t i e s of t h e l ow- ly ing s t a t e s in 370 a n d ~TF whose

e x c i t a t i o n energ ies a re wel l r e p r o d u c e d b y t h e m o d e l h a v e been c a l c u l a t e d

in t h i s w o r k a n d c o m p a r e d w i t h e x p e r i m e n t s in t a b l e s I I I - V . T h e r e is n o t

e n o u g h e x p e r i m e n t a l i n f o r m a t i o n a v a i l a b l e , a t p r e s e n t , on m i x i n g rnt ios , life-

t i m e s a n d r e d u c e d t r a n s i t i o n p r o b a b i l i t i e s for t h e l o w - l y i n g s t a t e s in ~:O a n d 'TF

for c o m p a r i s o n w i th t h e t h e o r e t i c a l p r e d i c t i ons . H o w e v e r , t h e c a l c u l a t e d

m a g n e t i c - d i p o l e m o m e n t s a n d e l e c t r i c - q u a d r u p o l e m o m e n t s for t h e g r o u n d

s t a t e s of ~70 a n d 17F ~re in good a g x e e m e n t w i t h t h e e x p e r i m e n t a l va lue s

in t a b l e V.

T h e w a v e f u n c t i o n s for s eve ra l l o w - l y i n g levels in ~70 and 17F a re l i s t ed

in t a b l e s V I a n d V I I . I n t hese t ab l e s , t h e squa re of t h e e x p a n s i o n coeff icients

of t h e b a s i c s t a t e s c o r r e s p o n d i n g to t h e h i g h e s t p h o n o n s t a t e s is n e a r l y negl i -

g ib le as c o m p a r e d to u n i t y , i n d i c a t i n g t h e v a l i d i t y of on ly i n c l u d i n g s t a t e s of

u p to t h r e e p h o n o n s in t h e p r e s e n t ca l cu la t ions .

TABLE VIII . - Values o/ C1, ~, ]or some states in, 170 a n d 17F.

1 X

0 1 2 3

J7 0

~1 0.8684 0.1217 0.0094 0.00041

�89 0.718 1 0.259 8 0.0210 0.001 1

0.420 4 0.473 3 0.099 8 0.006 6 21

0.0030 0.895 1 0.095 7 0.006 3 23

~2 0.049 37 0.646 8 0.279 9 0.023 94

• 0.229 7 0.583 4 0.169 6 0.017 11 23

~4 0.483 0 0.3214 0.1791 0.016 50

17 F

~1 0.8712 0.1194 0.0089 0.00037

�89 0.7552 0.227 1 0.0169 0,00081

0.528 5 0.394 2 0.072 9 0.004 37 21

�89 0.1943 0.6230 0.1665 0.0162

3 0.00145 0.891 3 0.1006 0.00651 ~23

0.390 8 0.432 2 0.163 3 0.013 78 ~4

-'~'1 0.000 0.524 1 0.402 9 0.072 92

THE FOSITIV:E-PA]~ITY STATES IN 170 AND 17F 3~

The structure of the lowest states in 1:O and 17F is schematically shown in

table V I I I with the quantit ies defined as

ar 2 (7) c . = Z (A,,~..), j,R

which are the total weights of the ZT-phonon components in each eigenfunction

of the coupled system. As seen from table V I I I , the ~4 state in both nuclei contains 0-phonon and 1-phonon of almost equal strength. We also note tha t

the ground state I~1) has a very large particle component (87 % for both nuclei) and the first excited state I1~) contains a particle component of about 75 %, i.e. 7 2 ~ (170) and 76% (~TF) and a 1-phonon component of about 25%,

i.e. 26% (~70) and 23% (~TF). I t is also noted from table V I I I tha t beyond 3-phonons vibrat ional components are negligible in all the low-lying states of 170 and 17F.

In the vibrational model, the root-mean-square deformation (~4), fir ..... , is given by

�89 ]~<o

in terms of the intermediate-coupling parameters. The calculated values,

fl ...... = 0.34 and 0.32, for ~70 and 17F, respectively, are in the range of the experimental values (15) in this mass region, suggesting tha t the parameters used in the present calculations fall in the correc$ range.

In conclusion we believe tha t the present calculations hel l) in understanding many of the interesting features of the experimental results pertaining to the

proI)erties of the low-lying levels in ~70 and ~T.

The authors wish to thank Mr. L. CUSTEAD for assistance in using the U:NIX computer system at the Saskatchewan Accelerator Laboratory.

(14) D. M. BRINK: Prog. 5fuel. Phys. , 8, 99 (1960). (15) 1:~. C. RITTER, 1 ). H . STELSON, F . K . McGowA~ " and R. L. Ronlxso~: Phys. .Rev. , 128, 2320 (1962).

�9 R I A S S U N T 0 (*)

I1 inodello unifieato di accoppiamcnto intcrmedio ~ stato applicato ai nuclei (< dello strato chiuso pih uno ~, 170 e 17F. I livelli di energia, i valori di transizione elettromagnetica e gli impulsi statici sono stati calcolati e confrontati con i risultati sperimentali. Si ottiene un accordo soddisfacente con i dati sperimentali.

(*) Traduzio~e a cura della •edazione.

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