A simplified treatment of octupole vibrations in deformed heavy nuclei

I Nuclear Physics Al l6 (1968) 129--144; (~) North-Holland Publishin9 Co., Amsterdam 1.D.2

I

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

A S I M P L I F I E D T R E A T M E N T O F O C T U P O L E V I B R A T I O N S

I N D E F O R M E D H E A V Y N U C L E I

A. FAESSLER t Physikalisches Institut der Universitiit, Freiburg i. Br., Germany

and

A. PLASTINO Facultad de Ciencias Fisicomatemdticas, Universidad Nacional de la Plata, Argentina t*

Received 25 March 1968

Abstract: Some collective properties of nuclei belonging to the rare-earth and transuranic regions are described within the framework of the quasi-particle random phase approximation utilizing a pairing plus a state-independent octupole force model. Energies corresponding to the K = 0, 1, 2 and 3 octupole vibrational band heads and reduced electric octupole transitions to these bands are calculated. The results thus obtained are compared to the predictions given by the pairing plus octupole force and by the surface delta interaction. It is found that the extremely simple model proposed in this paper gives as good agreement with experiment as the other two models do.

1. Introduction

In recent years, many proper t ies of low-lying nuclear spect ra have been unde r s tood

f rom a microscopic po in t of view. In medium-weigh t and heavy nuclei, the quasi-

par t ic le t r ea tmen t has been successfully app l i ed uti l izing different effective inter-

act ions. As a mat te r o f fact, mos t authors , fol lowing the p ioneer work of Kiss l inger

and Sorensen 1) used a pa i r ing-p lus -quadrupo le (or oc tupole) force, but more in-

volved in terac t ions have also been employed .

In the case of the s t rongly de fo rmed nuclei which can be found in the ra re -ear th

and t ransuran ic regions, one faces, however, a somewha t compl ica ted p rob lem.

F o r these nuclei, the level densi ty is high (one has to take into account a large number

of levels), therefore it is necessary to deal with such a large number o f basic funct ions

tha t a numer ica l so lu t ion is feasible only if the mat r ix elements of the in terac t ion are

separable . This fact p revented the use of more realist ic forces, and all the calcula t ions

o f collective states in heavy deformed nuclei z - l o) were pe r fo rmed uti l izing the

pa i r ing -p lus -quadrupo le ~) on pa i r ing-p lus -oc tupole 7,8) force model . Only recent ly

(refs. t1,12)), a more realist ic force, e.g. the surface del ta in te rac t ion 13), has been successfully app l i ed in this k ind of study.

t Present address: University of Miinster, Germany. tt Work partially done under the auspices of the U.S. Army Research Office (Grant DA-HCI9-67-

G0018) through Laboratorio de Radiaciones, IIAE, DINFIA, Buenos Aires, Argentina.

129

130 A . FAESSLER A N D A . P L A S T I N O

In this work, we report the somewhat surprising result that a simple pairing-plus- state-independent-octupole force (SIF) allows one to obtain a microscopic description of octupole vibrations in deformed heavy nuclei, which is as good as the one given either by the pairing-plus-octupole force (POF) or by the surface delta interaction (SDI).

This constitutes a remarkable result, in view of the fact that the SIF is an even simpler interaction than either the SDI or the POF, and it is also a useful one, considering the difficulties mentioned above.

In sect. 2, the main equations of the random phase approximation (RPA) are specialized for the SIF. The calculation of excitation energies corresponding to the octupole band heads in deformed heavy nuclei is discussed in sect. 3. Sect. 4 deals with the computation of electric octupole transition probabilities and, finally, our results are summarized in sect. 5.

2. Theory

The starting point is the Hamiltonian -'

H EI~o~C~+Ca..] - 1 A + + = E v e, (1) C~ Cp C~ Cr

with the antisymmetrized matrix element

A v=a. ~ = <~(1),/3(2)I EzlT(1), 8 ( 2 ) ) - <~(1),/~(2)i 171218(1), 7(2)).

The basis functions I~), Ifl), [7) and la> are solutions of a deformed potential. As usual, we take them to be the Nilsson functions [see, however, the discussion in ref. **)]. The symbols C + and Cp represent creation and annihilation operators for these states. The ~= are single-particle energies. The approximations employed to solve the Hamiltonian (1) have been described in ref. ~ a) (RPA). In this approximation for the excited states, one has the expression

a > . 8 a > 0

The sum runs over all 12= > f2 a with projections along the intrinsic axes f2= > 0 and K = f2=+f2¢. The ket I~Po) represents the physical ground state. The operator B~ + is the so-called quasi-boson operator. It is a linear combination of two-quasi- particle creation (A +) and annihilation (A) operators

A=a + ( 1 - -~: + +

(3) a ~ + = [2(1+ 6=. a)] - ~ [ ~ + ~ + +

The bar indicates the time-reversed state

1~) = TI~) = ( _ ) i - a I - a ) (4)

OCTUPOLE VIBRATIONS 131

with a negative projection ~2 of the angular momentum along the intrinsic axis. The two-quasi-particle states (3) are symmetrized under time reversal. The quantum number K = f2,+f2 8 is their projection onto the symmetry axis; n is the parity. The connection between quasi-particle and particle operators is given by the Bo- golyubov-Valatin transformation [see for example ref. 1)]

~,+ = u~ C , + - v~ C~ . ( 5 )

We utilize the convention that the coefficients u and r are independent of the sign of f2,, but that

TI~) = T2lct) = -1~) . (6)

The mixing coefficients ~ and 1/ are obtained by diagonalizing a non-symmetric matrix. The superscript i distinguishes solutions belonging to different eigenvalues.

As mentioned in sect. 1, due to the high level density in heavy deformed nuclei, one has a large number of basic functions. This implies that one would have to diagonalize, say in the transuranic nuclei, matrices with a dimension of the order of 500; this is a numerical problem which cannot be dealt with unless our matrix elements are separable. In order to enforce separability one needs to employ additional approximations 11), neglect the exchange terms of the interaction between quasi- particles and take only the leading term in the Slater expansion for the particle-hole matrix element. One obtains thus the secular equation 1~):

1 Z ).r 2 - = {D,8) (EE~8-co]- l+EE~p+to]- t} , (7) F ~>8

~t>0

f is the collective coupling constant of our effective interaction 11) and E~8 the so- called two-quasi-particle energies 11,12) The matrix elements ~ =8=~6 depend, of course, on the effective interaction one wants to utilize. For the SIF model, the Ham- iltonian (1) takes the form

H = Z e~ C + C , - Gp Z C~+C~+C~C8 ¢C>0 P 8 > 0 /(, 7~

I I GN Z + + 1 -- C~t C~ C~Cfl-~X 3 Z S~$8~C+(1)C~(2)C~(3)C~(4) • ( 8 )

ct> O TM aS, 76 f I 8 > 0 K,

The brackets indicate the coupling to the projection K = 0, 1, 2, 3 and the parity rr = - 1 . The indexes P and N refer to protons and neutrons, respectively, and S,r and $8~ the phase factors introduced in order to give octupole coherence to the SIF wave functions. These factors do not affect at all the energy calculations but are very important in considering transition probabilities. Explicitly, we have for the phase factor S,~ the expression

S:r = sign (q:~), where

q,8 = ( ° z l r Z Y 3 o l f l > •

132 A. FAESSLER A N D A. PLASTINO

Note that due to the fact that the whole Hamiltonian (8) is state independent (apart from the single-particle energies e~), it is not necessary to specify the explicit form of the basis functions. One needs only to know the projection ~2~, parity 7r~ and energy e~ for each single-particle state ~.

For the matrix elements D in eq. (7), one has the expressions

D ~ = [-1 - 6~; # ] ( u ~ v p - v~u,)w, D ~ = [1 + a.; ~] - ~-(u. vp + v. u~)w'. (9)

For either the SDI or the POF, we have

w = ~ a~aa~A[(21,,+ 1)(21b+ 1)/(22+ 1)] ~ a l l IA

× c(l. Ib21A.ab K ) ( - 1)~-r~3r.; -r~ F~, w' = ~, a~a aPta[(21a + 1)(21b + 1)/(22 + 1)] ~

a l l lA

x(--)A"c(tolb lA.--Ab/C)c(l.l Xl000)O o;JO . The symbols ata represent normalized Nilsson coefficients 15), I the orbital angular

momentum, A its projection and r the Z-component of the spin along the intrinsic axes.

In addition F,~b = 1 for the SDI,

F.~ = <blr~la>(4rc) -~ for the POF.

On the other hand, for the SIF model, the expressions for w and w' are simply

w = w' = 1.

In view of the expressions given above, it is clear that the use of the SIF introduces a great simplification in the calculations.

Our task is now to find the eigenvalues to (and their corresponding eigenvectors) in the secular eq. (7) for the SIF coupling constant Z3.

3. Energy calculation

The level scheme cited first in refs. 12) was used in the rare-earth calculation, while for transuranic nuclei, the level scheme 2 of Soloviev et al. 5) was chosen.

The coupling parameters utilized in the SIF, POF and SDI calculations are listed in table 1, where Gp and GN are the pairing strengths for protons and neutrons, respectively, F the SDI coupling parameter 12), Z3 defined in eq. (8), and the octupole strength in the POF model 12).

k3K = Z3 At~ hoe"

For the oscillator energy, the value hoe = 41 A -~ MeV is used. The coupling constant


for the K ~ = 0 - b a n d has to be chosen in all three forces about twice as large as that

for the K ~ 0 octupole states. It seems that this is due to the highly coherent super-

posi t ion of two quasi-particles in this state. The finite basis introduces a renormaliza-

t ion of the coupl ing constants, which is especially large for the very collective K = 0

states. In figs. 1-8, the energies o~ for the 1- , K = 0, l, 2, 3 and band heads in rare-

earth and t ransuranic nuclei, respectively, are shown. Values obta ined by using the

three models SDI, SIF and P O F are given. Compar i son with experimental results is

made when possible.

TABLE 1

The coupling parameters utilized in this work.

S1F SDI POF

AGp (MeV) 28-29 28-29 28-29 AGn (MeV) 26-27 26-27 26-27

AZ3 AF ka

a) 7.5 34.0 1.04 K = 0 -

b) 6.0 33.5 1.06

a) 3.7 18.5 0.64 K = I -

b) 3.0 18.2 0.60

a) 3.7 18.5 0.64 K = 2 -

b) 3.0 18.2 0.60

a) 3.7 18.5 0.64 K = 3 -

b) 3.0 18.2 0.60

a) Rare-earth nuclei. b) Transuranic nuclei.

For the pairing constants, the same values have been used for both the rare-earth and transuranic nuclei. With respect to the octupole strengths, the first number is the value employed in the rare- earth region, while the second one corresponds to the transuranic case. See ref. 12) for more details about the SDI and POF coupling parameters. (For the differences between the coupling parameters between the states with K = 0 and K @ 0, see the text.)

It is seen that all the interact ions give similar results in most of the cases. When

experimental values are available, one realizes that a reasonably good description of

the octupole v ibra t ional energies is provided by our three forces. Undoubted ly ,

more experimental in format ion is highly desirable if one wishes to make a more

reliable judgement concerning the relative merits of these models.

134 A . F A E S S L E R A N D A . P L A S T I N O

2.2

1.8

1.4

$ h e z 1.0 U J

0.6

I I I I I I I I I I I I I I I I I I I I . . , ~ I I J I I I ]

/ k F \ ' < / - ~ -'X

/ - _ _ Y ~ _ . _ / , " / ~ , , \ ~ . ~ ,

.n_ / ,, ' , , / / \ / ~

/ . " .~ , / I' J'/ "-.,V/ I-~ F,- -O Octupole States

o ..o ~ .,-K" / \J \ ~., =. . i / v ~ j - - = , ~

. f / ~ / ~ _ _ _ , ' - . . . . S O l

%'-" POF o exper imenta l vOlues

152 154 158 182 186 I I I I I I I I I I

154 156 160 180 184 Sm Gd W

158 162 164 168 168 174 174 178 I 1 I I I I I t I I 1 I I I

160 161, 166 170 172 176 176 Dy Er Yb Hf

Mass Number of Rare Ear lh Nuc|ei

186 190- I I I I

184 188 Os

Fig. 1. The energies o f the In = 1-, K = 0 band heads in rare-earth nuclei. They are calculated us ing the three interact ions SIF, SDI and P O F wi th in the f r amework o f the quasi-part icle RPA. The

cor responding coupl ing paramete rs are listed in table 1.

I I I [ I I / I I I I I I I I I I I I I I I i I I I I I I I I

1 - ; K=I Octupole State, RPA

2.0 ---- - S [ F t~ / x - - ~

/ It - ' ~ ~ - ~ - ~ : / ~ - I . . . . SDI ill~ f'E -~

1.8L . . . . . P0F /r. " t A //If ' ^ i !

! / ' \ / ~ ,Pi, ,i~l, A /, / ~ ",, / ~,,,'/it k I ~ - - J

7,,_ , %¢ ,jil/\,j .~ • ~ :~

1.2 - \

152 J I r

150 154 Nd Sm

154 158 158 162 164 168 168 174 174 178 182 186 186 190 J I I I I I I I I I J I { I I I I I L r d j i I I I I

156 160 160 164 166 170 172 176 176 180 184 184 188 Gd Dy Er Yb Hf W Os

Mass Number of Rare Earth Nuclei

Fig. 2. Energies cor responding to the I= = 1-, K = 1 band heads. Fur ther details are the same as in fig. 1. No exper imental da ta were available (152 --< A <-- 190).

O C T U P O L E V I B R A T I O N S 135

I I I I I I I 1 I I I I I I I I i [ I I I I I 1 I ~ I I I I |

]

22f, AA 1 "= " !f A. \ f ' 1

e o.s , P A ' - -" A -J ,~ 1.6 2 - ; N = 2 O c t u p o l

m ,,, - - -- S I F ~ X q

z . . . . S D I /

uJ 1.4 . . . . POF J /

1 152 154 158 158 162 164 168 168 17/, 17/* 178 182 186 186 1901

I I I I I I I I I I I I I I [ i I I I I ] 1 I I r r I I I I / 150 154 156 160 160 164 166 170 172 176 176 180 184 184 188

Nd Sm Gd Dy Er Yb Hf W Os

Mass Number of Rare E a r t h Nuclei

Fig. 3. Energies corresponding to the 2-, K = 2 octupole vibrations. Further details are the same as in fig. 2 (152 -<- A ~< 190).

2.4

2.2

2.0 :£

1.8 a-"

LIJ Z , laJ

1.6

I I I I I I i l l l i l t I I I I l l l l I I I I I I I I

I, i \

I \

i I \\

- - ~ \ / , I ^ \\

o¢,u o,, tote \' //\;'

_ _2 I~ - - S D I , POF

152 154 158 158 162 164 168 168 174 17l, 178 182 186 186 19C I I I I I I J I I i I i 1 I I I I I I [ I I I I I I I I I

150 154 156 160 160 164 166 170 172 176 176 180 184 184 188


M o s s N u m b e r of R o r e E o r t h Nuc le i

\ \ \ \

\\\\

Fig. 4. The K = 3 octupole vibrational energies. Further details are the same as in fig. 2 (152 --< A --< 190).

136 A. FAESSLER AND A. PLASTINO

i I I I I I i i I i i I I I I I I I

1.4 i

1.2

o / " / v - - v - ~ ~- o

. P / - . . ' ~ . ~ f ."

. . . . . . . .

0.8 I " I-- K=O Oc tupo le S t a t e s

,,=, ! ,/, \ ~:v/,' o Zo., I t' ',7/ o o SD!

\~ POF

o exper im, va lues

0.4 - 232 232 236 236 240 242 246 252 254 J I I P I [ i i I I 1 J F J i I I I

230 234 234 238 238 242 244 250 252 Th U Pu Cm Cf Fm

Moss Number of T r a n s u r o n i c Nucle i

Fig . 5. T h e 1-, K = 0 o c t u p o l e v i b r a t i o n a l ene rg ies in the t r a n s u r a n i c r eg ion . F u r t h e r de ta i l s are the s a m e as in fig. l .

1.5

:IE ~--J1,0

0.5

J i i ] i d i i J J i i i J I i I J

o / / , ~

j ~ l > /,~-. v "

/.,G-~ . ~ ///

I- i K=I Octupole V ib ra t ions

S I F

. . . . . S D I

POF

o Exp . va lues

232 232 236 236 240 242 246 252 254 I I ] I I I I I I I l l I I I I I I

230 234 234 238 238 242 244 250 252 Th U Pu Cm Cf Fm

Moss Number of Tronsurcmic biuc[ei

Fig . 6. E n e r g i e s c o r r e s p o n d i n g to the I,~ = l - , K = l b a n d h e a d s in t r a n s u r a n i c nuc le i . F u r t h e r de ta i l s a re the s a m e as in fig. 2.


I I I I I I I I i I i I i 1 I I I !

-2-~ K=2 Octupole S ta tes ~ S I F

~ 1 . 4 ,

$ 1.2 , ' " , / ---, ' V ,,,\\ ' \ ..f_ " ' ~ / \ ~ / . / t - Z V \ " ' . . t - " / IL l ,,...

1,0 o ~'-,. / / o \ /

246 232 232 236 236 240 242 o 252 254

I I I I 1 I I I I I I I I ~ I I I I 2:30 234 234 238 238 2/,2 244 250 252

Th U Pu Cm Cf Fm

Mass Number of Transuranic Nuctei

Fig. 7. Energies of the 2-, K = 2 octupole vibrations. Further details are the same as in fig. 6 (230 ~< A --< 254).

2 . 0 -

1 . 8

1.6

~E

1.4

L d Z

uJ 1.2

1.0

I I I [ I I J I J I I

t", ~

\ I , ,l/ " ~ \ \

I \ 1 7 \ ' \ \ ' , , 'A', ,'/

V V V 3-, ,.+

I I I I I I ~ I

Oc tupole Sta tes

S I F

S D I , (POF)

232 232 236 236 240 242 246 252 254 I I r I r 1 I P I I I I I I i I L I

230 234 234 238 238 242 244 250 252 Th U Pu Cm Cf Fm

Mess Number of Trensuran ie Nucle i

Fig. 8. The 3-, K = 3 octupole vibrational energies. Further details are the same as in fig. 6 (230 < A -<- 254).

138 A. FAESSLER AND A. PLASTINO

4. Transition probabilities

Reduced electric multipole transition probabilities from the ground state rotational band to a vibrational band can be calculated utilizing the expression

B(E2, I i g ---* I f Kf) = cZ(Ii 2Ifl0Kf Kf)M 2, (10) with

M = ~. (l~vvv+u, vv)([(ep+eeff)Q~(1-8,~)(¢,~+rluv) ] t t__v>O

+ (ee + e.ef)Qu, [(1 + 8x; o)/(1 + 5,; ~)] (Cur + r/,~)}. (11)

where Q~ are the multipole matrix elements defined in the second cited ref. ~z). (eq. (4)).

In eq. (11), ep takes the value 1 for protons and 0 for neutrons, while eeff is the effective charge. The mixing coefficients in the SIF model are

= +

~1~., = NS~.,(u# vv + u, v~.)/(E~., + w). (12)

The normalization constant N can be calculated with the help of the condition

2 2 (¢u~-~/,,) = 1. (13) #>--v / t > 0

With respect to the mixing coefficients in the POF and SDI cases, the reader is referred to ref. ~2).

Figs. 9-15 and table 2 display the E3 transitions B(E3; 0 + g -~ 3- , K) corresponding to the K = 0, 1, 2 and 3 octupole vibrational bands in rare-earth and transuranic nuclei. All values are given in single-particle units

B(E2;0 -~ 2) - 22+ 1 3 R [e2cm2a]. (14) 4re

In eq. (14), the nuclear radius R o = 1.2 A ~ (fm). In all cases, values corresponding to the three models SIF, SDI and POF are computed.

Fig. 9 shows the more interesting results corresponding to B(E3, 0 + g ~ 3- , K = 0) values in rare-earth nuclei. These transitions are particularly important because a significant quantity of experimental information is available thanks to the work of Elbek et al. 15) and Hansen and Nathan 16). A very satisfactory agreement between theory and experiment is reached for the three interactions. The only additional case in which a comparison with experiment can be made corresponds to the same kind of transitions in transuranic nuclei (fig. 13), although only two experimental values are at our disposal ~ s), therefore that no definite conclusions can be drawn.

With respect to E3 transitions from the ground state to the K = 1 and K = 2 octupole vibrational bands, we find that they seem to possess as collective a character as the transitions to the K = 0 band. Certainly, values corresponding to gamma-ray

30

o i i

- 2C i

o l

. 1 ~ .

"" 1C L ~

I

150 Nd

l l l l l l l l l l l l l l l l l I l l i l l l l l l l

B ( E 3 , o*g - - 3 - , , : o ) [S.P.u.]

SIF e., : 1.0

SDI } POF e., = ZIA

', / ~ --,\ ~ V Etbeck et ol. T T 1 "'-- / o Haos,o u N a,hoo

s 154 158 158 162 164 168 168 174 174 178 182 186 1~6 119( 1 , 2 1 ~ L ~ q I ~ ~ L ~ ~ L I ~ ~ k h ~ ~ ~ ~ ~ 154 156 160 160 164 166 170 172 176 176 180 184 184 188 Sm Gd Dy Er Yb Hf W as

Mess Number of Rare Earth Nuclei

Fig. 9. Reduced electric octupole transit ion probabilities f rom the ground state into the 3-, K = 0 octupole vibrat ional states o f rare-earth nuclei in single-particle units. Values have been calculated for the SIF, SDI and P O F models. Experimental data are f rom refs. ls,1~). The effective charge is equal to 1 in the SIF calculation and to Z/A for both the SDI and POF models. (The difference be-

tween these effective charges is discussed in the text.)

1 I I I I I I I I I I I I I [ I i I 1 I I I I I I I I I I I

"" ', " ' ' o* Is. P. u.] / / v / \~ B ( E 3 1 g ~ 3 - , K = l )

6 \ . \ ~ /~ S I F - " i

u i \ / ~ . . . . . SDI : ~ i I ./~ \~ \ ' \ ; / \ \ . . . . . POF

~'.~ ~ ~ e., = ZIA

,4" 2

, c

150 154 156 160 160 164 166 170 172 176 176 180 184 184 !"}8 Nd Sm Gd Dy Er Yb Hf W as

Mass Number of Rare, Ear th Nuclei

Fig. 10. The B(E3, 0 + g 3-, K = l ) t ransit ion rates in single-particle units. The effective charge is equal to Z/A for the three interactions SIF, SDI and POF. No experimental data were available.


/ ] ~ I i , i i i I ~ I I i , I I [ I I ~ I I i i ~ i I "l'

" ; ' - . . . . so~ e . , , : , , , \,_ L

~52 1¢4 ~ 8 - 1 ( ? ~ 6 2 ~64 ~6~ ~6~ ~74 ~4 ~7~ ~s2 ~86 x'~86--~90- ~ , , , : ~ ; r ~ r , , ~ , , , , , ~ ~ , ~

150 154 156 160 160 164 166 170 172 176 176 180 184 184 188


Mass Number of Rare Ear th Nucle i

Fig . 11. T h e B(E3 , 0 + g 3- , K = l ) t r a n s i t i o n ra tes c a l c u l a t e d for r a r e - e a r t h nuc le i . F u r t h e r de t a i l s a re the s a m e as in fig. 10.

r v

tn" L ~

A 1.0 u

i

t m 0.5

h i v E l

0.0

0,-:

0 2

0.2 n

. , . . .

i

f 0.1

,-6" LLI

m 0.0

~(E3, o ÷ g - 3 - , K.3) [s.Ru,] / /

S I F ( I

. . . . . SDI /

POF p I

eeff = ZIA M / F~ I I

\ / '4 b = ~ - - ~ - j ,,~ ,~

152 154 158 158 162 164 168 J I I L I I I f I I I t I I t

150 154 156 160 160 164 166 170

Nd Sm Gd Dy Er

168 174 I I I

172 176

Yb

t

I & I

\

174 178 1 8 2 186 186 190 I I I I I i I [ I I

176 180 184 184 188

Hf W Os

M a s s N u m b e r o! R a r e E a r t h N u c t e i

Fig . 12. B (E3) , 0 - g ~ 3- , K = 2) t r a n s i t i o n ra tes . F u r t h e r de ta i l s are the s a m e as in fig. 10 (152 G A G 190).


s01

"I ~ B(E3~ O'g--3-,K:O) [S.P.U.] 40

S I F , e e l f = 1,0 r ~ /-%\

fL \ I t / \

u~ I\. / ~ . . . . . S D I , eeff= Z I A

o 30 \ k . - - . . . . P O F , e,ff = Z / A

~ \u /"~

m 1(3 \-Z ~xx. •

232 232 236 236 240 242 246 252 254 I i i i I t I i I ', I ] I ~ I I I I

230 234 234 238 238 242 244 250 252

Th U Pu Cm CJ Fm

Moss Number of T r o n s u r o n i c Nuc le i

Fig . 13. The B(E3 , 0 + g --~ 3- , K = 0) t r a n s i t i o n ra tes in the t r a n s u r a n i c r eg ion . F u r t h e r de ta i l s a re the s a m e as in fig. 9.

14

12

oz 1(3

- - 8

II

0

w 2 m

I l l J i l l I J I I I l l i ] I I

'\ \ B(E3,.O+g --->3-, K : I ) ~S.P.U.~ \ \

~ \ S I F

\ \ ~ . . . . SDI " \ \ \ / / \

/ \ , ~ \ . . . . . P O F

' " " \ = Z / A

0 232 232 236 236 240 242 246 252 254 I i I I I I I I I I 1 I I L I L I I

230 234 234 238 238 242 244 250 252 Th U Pu Cm Cf Fm

MOSS Number of T ronsurcmic Nucte i

Fig . 14. T h e B(E3 , 0 - g ~ 3% K = 1) t r a n s i t i o n ra tes i n the t r a n s u r a n i c reg ion . F u r t h e r de ta i l s a re the s a m e as in fig. 10.


t12

~e

f ÷ o

~7 W

en

l I 1 l I } I 8

\ 6i "

I [ I I I I I l I l I

B(E3, O+g--> 3-,K:2) [ S . P . U . ~

SIF . . . . SDI e., , = Z/A

POF 0 232 232 236 236 240 242 246 252 254

I I I I I r I I I r I I I i q I I I 230 234 234 238 238 242 244 250 252

Th U Pu Cm Cf Fm

Mass Number of Transuranic Nuclei

Fig. 15. The B(E3, 0 + g ~ 3-, K = 2) transition rates. Further details are the same as in fig. 10 (230 ~ A ~ 254).

TABLE 2

B(E3; 0 + g --~ 3-, K = 3) transitions between the ground state and the K = 3 octupole vibrational bands in single-particle units for the SIF, SDI and POF models

Z A SIF SDI POF

230 0.66 0.022 0.011 9oTh 232 0.21 0.010 0.004

234 0.12 0.016 0.004

232 0.59 0.110 0.057 234 0.23 0 .0l l 0.004

92U 236 0.12 0.011 0.004 238 0.12 0.011 0.004

236 0.18 0.011 0.004 238 0.10 0.012 0.004

~4Pu 240 0.10 0.012 0.004 242 0.33 0.347 0.107

242 0.07 0.013 0.004 96Cm 244 0.48 1.981 0.903

246 0.46 2.050 1.104

250 0.23 0.924 0.036 9sCf 252 0.23 0.397 0.026

252 0.48 2.136 1.244 looFn 254 0.47 2.110 1.206

These values are calculated with in the RPA framework. The effective charge was taken to be eert = Z/A. N o experimental data were available.


transitions to this last band are generally larger than those corresponding to the K = 1 and 2 bands, a result which is due to the greater number of two-quasi-particle com- binations with K = 0 relative to those available when K = 1 or 2. However, the values obtained for these last two bands are large enough to justify the assertion that they display a collective character. This contradicts a statement by Vogel and Solo- viev 7, 8,17) that only gamma ray transitions to the K = 0 octupole band show such

a collective behaviour. For transitions to the K = 3 octupole band, on the other hand, remarkably smalller

values are obtained in all three models. It is not possible to ascribe a collective character to this band (here we have in most of the cases almost pure two-quasi-particle

states). Note, however, that the SIF yields in this case values which are on the aver- age ten times larger than those given by the other two models (for this reason, two

energy scales are needed in fig. 12). Finally, it should be pointed out, that no theoretical argument is known which

could be used as a guide with respect to the value of the effective charge. We have

employed in all but two cases the value eef f = Z/A. This choice can be justified only for E2 transitions utilizing a self-consistent argument given by Mottelson 18). The two exceptions above correspond to the SIF calculation of E3 transitions to the

K = 0 octupole band, where the value eeff = 1 was taken. The large effective charge for the SIF and the AK = 0 transitions is connected with the fact, that the K = 0 octupole states are strongly collective and that the SIF distributes the strength over

more two-quasi-particle states than the SDI or POF. A truncation of the basis en- larges therefore mainly the effective charge of these transitions.

5. Conclusions

The simplest Hamiltonian which is able to produce an energy gap and a coherent octupole vibrational function has been assumed. Such a Hamiltonian is composed of a shell-model term and a state-independent pairing plus octupole force.

Although a state-independent pairing force is very widely used, we believe this is the first time that an octupole force of such characteristics is employed.

Octupole vibrational energies and E3 transition rates have been calculated and found in reasonably good agreement with experiment, whenever such experimental information was available. Moreover, comparison with results obtained using both a surface delta interaction and the (conventional) pairing-plus-octupole force shows that the three models give similar results in most of the cases. More experimental information is needed in order to gain a deeper insight into the relative usefulness of these models. It is our belief, however, that due to its extreme simplicity, the SIF model may prove to be useful at least in obtaining a first orientation.

We are grateful to Dr. B. Elbek for sending us a list of his experimental results prior to publication and are indebted to Dr. S. A. Moszkowski for very valuable dis- cussions. We want to thank the computer centers of UCLA, Institute of Physics of

144 A. FAESSLER A N D A. PLASTINO

t h e U n i v e r s i t y o f F r e i b u r g a n d U n i v e r s i d a d de L a P l a t a fo r m a k i n g c o m p u t i n g t i m e

a v a i l a b l e to us.

References

1) L. S. Kisslinger and R. A. Sorensen, Mat. Fys. Medd. Dan. Vid. Selsk. 32, No. 9 (1960) 2) Z. Szymafiski, Nucl. Phys. 28 (1961) 63 3) D. R. B6s and Z. Szymafisky, Nucl. Phys. 28 (1961) 42 4) D. R. B6s, Nucl. Phys. 49 (1963) 544 5) V. G. Soloviev, Atom. Energy Rev. 3, No. 2 (1965) 6) V. G. Soloviev, Nucl. Phys. 69 (1965) 1 7) V. G. Soloviev, P. Vogel and A. A. Korneichuk, Bull. Acad. Sci. USSR (phys. ser.) 28 (1964) 1512 8) P. Vogel, Dubna Report No. E 1703 (1964) unpublished 9) E. R. Marshalek, Phys. Rev. 139 (1965) 770

10) T. Udagawa and R. K. Sheline, Phys. Rev. 147 (1966) 671 11) A. Faessler, A. Plastino and S. A. Moszkowski, Phys. Rev. 156 (1967) 1066 12) A. Faessler and A. Plastino, Nucl. Phys. A94 (1967) 580;

Phys. Rev. 156 (1967) 1072; Z. Phys. 203 (1967) 333 13) I. M. Green and S. A. Moszkowski, Phys. Rev. 139 (1965) 790 14) S. G. Nilsson, Mat. Fys. Medd. Dan. Vid. Selsk. 29, No. 16 (1965) 15) B. Elbek, Determination of nuclear transition probabilities by Coulomb excitation (Ejnar

Munksgaards Forlag, Copenhagen, 1963); B. Elbek et al., private communication

16) O. Hansen and O. Nathan, Nucl. Phys. 42 (1963) 197 17) V. G. Soloviev and P. Vogel, Phys. Lett. 6 (1963) 126 18) B. R. Mottelson, The many body problem (John Wiley, New York, 1958) p. 283

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A simplified treatment of octupole vibrations in deformed heavy nuclei