On the microscopic description of nuclear vibrations when phonons occur at relatively low energy

~

NuclearPhysics A149 (1970)217 --224; (~) North-HollandPublishinq Co., Amsterdam N o t to

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ON THE MICROSCOPIC WHEN PHONONS

DESCRIPTION OF NUCLEAR VIBRATIONS

O C C U R AT R E L A T I V E L Y L O W E N E R G Y

(I1). Octupole states of the even deformed nuclei with A ~ 222 K. N E E R G A R D

The Niels Bohr Institute, University of Copenhayen, Copenhagen and P. VOGEL Department of Theoretical Physics, UniL'ersity of Bergen, Beryen t

Received 20 January 1970 Abstract: The properties of the octupole vibrational states are calculated using the theory of part 1. In this way, it is possible to give a common description of all octupole states in the considered region including the very low-lying K Jr : 0- states in the isotopes of Ra and Th. The Coriolis coupling between the states of the octupole quadruplet is taken into account in the calculations. The resulting theoretical spectra are compared with the experimental data.

1. Introduction The low-lying negative parity states in the doubly even nuclei with A ~_ 228 have been interpreted as octupole vibrational states. Their properties were rather successfully reproduced by R P A calculation i). However, in Ra and lighter Th isotopes, the 1 ~ = 1 - states have very low energies (200-300 keV), and therefore the simple R P A theory is not applicable there. These states could be interpreted either as octupole vibrational states or as rotational states of a permanently oetupole deformed (pearshaped) nucleus. However, theoretical calculations o f the ground state energy indicate a stable equilibrium at zero octupole deformation 2, 3), which supports the first interpretation. Thus, the negative parity states in the actinide region are suitable for application o f the improved treatment of nuclear vibrations developed in part I. In rcf. 4), the properties o f the low-lying octupole states in the rare-earth deformed nuclei were calculated by the same authors. It was found that, by using a modified oetupole-octupole force, and by including the Coriolis interaction between the states o f the intrinsic octupole quadruplet ( K ~ = 0 - , 1-, 2 - , 3 - ) , the agreement with experiment was improved essentially as c o m p a r e d with earlier calculations. In most respects, the present investigation of the actinide nuclei follows the same line as that o f the rare-earths in ref.4). The main difference is that the pure RPA t On leave from the Nuclear Research Institute, Re~, Czechoslovakia. 217

218

K. NEERG,~RD AND P. VOGEL

description of the intrinsic motion has been replaced by that of part I, where fi~ the summation of eqs. (3.8) of part I just the term corresponding to the K = 0 octupole state was included. In this way, we obtain an improved description of those cases where this state comes low, while for higher energies of the K = 0 statc the description is practically cquivalent to the usual RPA. The numerical results are prcsented in sect. 4 of this papcr. In sect. 3, the K = 0 states in the isotopes of Ra, Th and U are discussed. The results obtained from our present theory arc compared with those of the RPA description.

2. Details of the calculations

In all the numerical caiculations, the "standard" Nilsson potential 5) with hr.b =

41/A t McV and ~: = 0.0577 (0.0635), It = 0.65 (0.32) for protons (neutrons) was used. The pairing interaction was included by the usual BCS method with the strength constants

AGz,.~ = 22.15__+ 15.8(m-Z)/A MeV,

(2.1)

which give the best fit to the experimental pairing energies. They are quite close to those used in ref. 4). The modified octupole-octupole force of rcf. 4) was utilized and only states with AN = 1 were included. To diminish the arbitrariness in the choice of the strength constants ~.'~,,a formula already suggested in ref. -*) was used. It involves a dependence on the deformation parameter fl and the main part of the dependence on the projection quantum number u, explicitly, namely K,,(//)=

1+3//2+

~

//3

(1+

~ ] A t .

The p, values should then be rather close to each other. For the arguments leading to this formula, we refer to the cited paper. The values p~, = 18.4 M e V - f m - 6 were used for/~ = 0, 2 a n d p , = 18.2 McV • fin -6 for ~ = 1, 3. Thus, the octupole-octupole interaction contains practically just one free parameter. Let us add that the values of the expression (2.2) extrapolated to the rare-earth region lit nicely with those from ref. 4). For the polarization charge involved in the calculation of the B(E3) values, the value epot = 0 was used. The approximations involved in the theory of part I will have the tendency to give slightly too small values of both the unperturbed phonon propagator G (°) (E) and its derivative Gc°)'(E). The effect for the derivative will be relatively stronger, which may be accounted for by taking too small a value for %ol. However, the main results concerning transition probabilities, i.e. the distribution of the B(E3) strength among different rotational bands in the same nucleus and the relative change from one isotope to another, do not depend sensitively on this value.

NUCLEAR VIBRATIONS

219

The Coriolis intrinsic matrix elements ( K + I I J + I K ) were calculated using the equations given in ref. 4). Note that they do not contain any additional parameter. In order to have perfect consistency, one should really include in the equations for the ( K + I I J + I K ) matrix element corrections analogous to those introduced in the p h o n o n propagator. However, the influence of such corrections in this particular case is expected to be small. Furthermore, we learned from our earlier investigations that the resulting spectra are not too sensitive to the exact value of the matrix elements. In the region considered, almost all octupole states are quite collective, therefore the calculated Coriolis matrix elements are usually close to the spherical limit, i.e., ( K + I l J + I K ) ~ , . / ( 3 - - K ) ( 3 + K + 1).

(2.3)

For the inertial p a r a m e t e r A = h2/2J, we have used the ground state value calculated from the positions of the 2 + and 4 + levels. 3. The K ~

=

0 - intrinsic states in the isotopes of Ra, Th and U

The energies of the K ~ = 0 - states of the nuclei in the beginning of the investigated region are of special interest, because here the equations of part I will give results essentially different from those of the RPA. In fig. 1, we have c o m p a r e d the calculated

j

MeV 0.75-

t/ sIs

////

/

,'

///

/

oJ_

I ~

,

-It/

[

,.. /

/

.,'~/ ~

I /

~ / / )~

// t/

0.25r t Ro O0

220

//-

/ f/" s

I

~

jJ/

/'

//

0.50-

~

Th

,/

/

230

/

/

..... ~ p e r i m e m present t h e o r y , p-18/, ~o-----o l~eser _ . . . . . RPA. p=163 (higher) p=183 (lower)

f

240

A

Fig. 1. Energies of the first K ~r = 0- states calculated by the method of part l, by pure RPA, and the experimental energies. The paramcter Po is defined in eq. (2.2). For clarity isotopes of the same elements are connected by lines. energies for the isotopes of Ra, Th and U with the experimental ones and with the best results obtainable in a pure R P A calculations. It is seen that our equations (3.14), (2.9) of part I are able to explain the general trend of the experimental results, while the R P A fails completely.

220

K. N E E R G A R D A N D P. V O G E L

The occupation numbers p= (cf. eq. (3.8) of part I) are < 0.2 for the single-particle statcs nearest to the Fermi surface. These numbers are somewhat dependent on the magnitude of those matrix elements (/3[~30(~)1=) which involve the state 17); thcy decrease rapidly with the increase of the quasiparticle energy E,. Thus, thc main condition of the validity of our approach is fulfilled. For somewhat higher energies of the K s = 0 - state, like in Z38U, the p~ are ~ 0.01, so there the differences between our results and the results of the pure RPA are negligible. The B(E3) values calculated from the formulas of part 1 are somcwhat larger than those given by the RPA (with interaction strength chosen so as to obtain the same energy). For the K ~ = 0 - states we have obtained monotonically decreasing B(E3) values, f r o m B(E3) ~ 25 s.p.u, in R a to B(E3) ~ 12 s.p.tt, in 238U. T h i s is in a g r e e m e n t with the expected order of magnitude and general behaviour. Note that these B(E3) values calculated without the Coriolis coupling cannot be compared directly with the experimental ones.

4. Results and discussion T h c eqs. ( 3 . 1 4 ) , ( 2 . 9 - 2 . 1 2 ) o f part [ w e r e s o l v e d for the first K ~ = 0 -

state in all

nuclei considered. The quantities p~, /:'~*- (see eqs. (3.8), part I) thus obtained were u s e d in the e q u a t i o n s for the first K ~ = 1 - , 2 - , 3 - states. F i n a l l y , the C o r i o l i s inter-

action was included. All four K ~ = 0 - , I -, 2 - , 3- rotational bands of the octupole q u a d r u p l e t w e r e t a k e n i n t o a c c o u n t in the d i a g o n a l i z a t i o n o f the r o t o r H a m i l t o n i a n . 20

.2.0 5

t,

--5" --t,"

-~-3"

1.5,

--5-

4----- - 5 " 763" T--

5

--5-

2. I -

--s-

703--5"

91 3--2-

/- ~-0~3" 1

I

/-2B~_ 3" 2---1"

2

i -

~Rct

-X-3"

-x-l-

1. ~...~33" -X-3"

~4Ro

-x-~-

1.5

.,SJ. 3.

5 6g_s%-

--5" -×-I"

0.0

923 -

/973-

__1" 2--

0.5

--5-

--2"

1 ~ 3L'--

1.0

__/. -

_~.y

0.5

--1" 22eRa

~BRa

~.0

Fig. 2. Calculated and experimental spectra of low-lying octupole states in Z22Ra, 224Ra, 2 2 6 R a and 22SRa. The figs. 2-8 contain for each state the energy in MeV, quantum numbers 1 = and, for I n = 3 - states, the B(E3; 0 + --* 3 - ) value in 1 0 - 7 4 c m • e 2. The states are tentatively ordered into "rotational bands". The experimental energies are shifted to the right and denoted by -x-. A general reference for the experimcntal data is 7); the most important specific references are given in the caption o f each figure.

:J

2.0 5

---5" --6-

--L"

--5"

--5

77 3"

~..~%~-

1.51

__s -y-~ --5"

c"

s--- r__-x- 2-

C20~_3 ~3" 1"

---5"

---5"

5"---X-

81, B ~. -

--I-

-765

31-~

-

---1-

~Z,.Th

- x - ~"

,k

~ - - r - - : t : ~: 1.~2"

~STh

-x-l"

I

1.0

2-~1-

~.~.-~-~-

r---x-,-

5-

-X-

1.5

963"

--5"

y3i-x-

--2"

---2"

0.5.

--~"

-~.

--z.-

7 ~ 3-

1.0-

2.0

"05

3"

-x-i-

~OTh

2~Th

20j

O0 - - -

~00

Fig. 3. Spectra o f 224Th, 226Th, 228Th and 23°Th. Exp. ref. a) (cf. caption to fig. 2).

2.0 --5--4"

1.5

--b"

.9~_3--5"

-9--23"

--5"

--5"

7 .t.-

3--W- ; 3 3"

~'---x-]-

1.0¸

2.

-~-q;"

--5"

-~L3-

rc~,:

--T

--5"

--S"

---5" ~ 9 3---I-

5_8~3-i-

--5"

1.0

5 - - - X - 33" 60~3--Y:-- V F--.

-.--5"

N_6y

232Th

I~_g.:f ~-

2"~;.

~2-

0.5

~4._ 3--2"

5"--

Js._.2r

.1.5

.191. 3-

230U

2~Th

0.5 232 u

0.0 0.0 Fig. 4. Spectra o f 232Th, 234Th, 23°U and 2azu. Exp. refs. 9.6) (cf. caption to fig. 2). 20

.

.

.

.

2.0

1.5 -=C

1.0-

--5-

1063-

----5"

--5"-

"

-

-

-

_

-

~ --T --t-

__.

.

--El&-18,1 -'X-'a"

~ . ~ # i - ~- ~------~"

-

so,_~s53-

05

_

- X - 2"

--5-

-~"

'~-

2-- ',:~3 I

-15

__%-

--5

--"S"

--2" --i-

--/.U2y

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

-1.0

I" - - - X - 3"

-X- 1"

--I-

.05 23,~u

236 u

~3aU

236pu

00 Fig. 5. Spectra o f 234U, 236U, 23sU and 236pu. Exp. refs. 6.1o) (cl'. caption to fig. 2).

2.0'

-2.0

1.5--~"

--5"

--5"

___=-7-4"

.

--2"

tO

~]:

-15

--5"

".-o-2:

--X-S"

- - '},--:re q-

--r

---5"

--5"

--5"

--l"

---2--I"

--5"

"~:

--5"

,---~-

$ --4" --

-1.0

I"'--

2.~3" -X- 3" -X- F

-X-l"

-O5

05 :~epu

2~2p u

~4op u

~4~pu

-00

O0

Fig. 6. Spectra o f 238Pu, z+°Pu, 242pu and z++Pu. Exp. ref. it) (cf. caption to fig. 2). 2.0

2.0

1.5.

--5"

--ss

__~-

1.0.

5

--5"

--5" --4"

24 3--4--,'-.~++:

_2-

3--

s.tm~ - - v

.1.5

--5-

5

3_o_~--I"

r

--5" --4"Z.O3-

1.0

4---

-

--I"

--5-

--2-

05

0.5. 246Cm

2~Cm

2¢2Cm

2~.sCm

0.0

(10

Fig. 7. Spectra o f 242Cm, 244Cm, 2a6Cm and 2+SCm. Exp. ref. 12) (cf. caption to fig. 2). -2 +0

2.0

1.5

--5"

--5" 5

63 3"

--5"

-'

1.0

"-+-":

03

--

"-- t," r - - r 3"

-

~ 1S

--1"

--5" 4 ~__s~

co=hr --S" .~_~-

-m'1

='-- 1-

-1.5

• --5"

473 -

--5-

~66 3-

--r

-1.0

5

0.5

-05

--T

zsocf

252Cf

252Fm

254F m

00

0,0 Fig. 8. Spectra o f 25°Cf,

252Cf9

252Fm and 254Fm (cf. caption to fig. 2).

NUCLEAR

223

VIBRATIONS

Let us m e n t i o n t h a t the o r d e r i n g o f the states w i t h different a p p r o x i m a t e K - v a l u e s is r a t h e r similar to t h a t in the r a r e - e a r t h nuclei - n a m e l y for the lightest nuclei in the a c t i n i d e r e g i o n the K " = 0 - state c o m e s lowest, while for h e a v i e r o n e s the K " = 2 a n d K " = 1 - states share the l o w e s t p o s i t i o n . T h i s is e x p l a i n a b l e in a w a y similar to the g e n c r a l d i s c u s s i o n given in ref. 4). "FABLE I

Encrgics and the corresponding B(E3; 0 + -~- 3-) valucs for somc I rr -- 3- states Nuclcus

2261~:. t

232Th 23s U

Enclgy (keV)

B(E3) exp

exp.

theory

253 770 ll00 724 998 1170

397 546 945 778 922 1236

10 -2 e 2 • b 3 theory

Ref.

77 59 26 44 18 It

16) ts) la) 16) a~) 16)

~ 70 ~ 50 30 ~ 10 50~ 6 22~ 5 20~10

2.0-

-2.0 . . . . 4" --3"

--5-

1.5-

7~ 3-

--5-

--5"

-1.5

2._--3" --I-

--5" "t- - - 5 -

1.0-

--5-

r~:

3"5-

,.Z2;.,~,-

---5"

--S" ¢.6~3"

--7"

-1.0

--3--2"

--3"

-0.5

0.5-

a) 0.0

b)

c) -0.0

Fig. 9. Spectra of 234U. a) calculated with the usual value of the interaction strength pg = 18.4 (K -- 0, 2) and Px = 18.2 (K ~ 1,3). b) with p~ -- 18 (K -" 0,2) andp~ = 17 (K = 1,3). c) Experimcnt according to refs. t3, 14).

T h e final c a l c u l a t e d s p e c t r a for states with I __< 5 h a v e been collected in figs. 2-8. F o r c o m p a r i s o n , the e x i s t i n g e x p e r i m e n t a l d a t a are s h o w n , t o o . T h e c a l c u l a t e d B ( E 3 ) v a l u e s are given for all the I " = 3 - states. T h e r e are v e r y few e x p e r i m e n t a l d a t a on thc B ( E 3 ) to c o m p a r e with. T h e m e a s u r e m e n t s in 226Ra, 23SU a n d 232Th are c o m p a r e d with o u r t h e o r e t i c a l results in table I. T h e a g r e e m e n t is v e r y g o o d . F o r 234U, the best s t u d i e d n u c l e u s in this r e g i o n , the a g r e e m e n t is u n f o r t u n a t e l y n o t t o o g o o d . H o w e v e r , m u c h b e t t e r results c o u l d be a c h i e v e d by slightly a d j u s t i n g

224

K. NEERG~RD AND P. VOGEL

s o m e o f the parameters. This is d e m o n s t r a t e d in fig. 9, where a n o t h e r s p e c t r u m o f 234U with the octupole interaction strength changed by ~ 7 % for K = 1 and 3 is shown. The agreement with experiment is then much better. This illustrates also the expected accuracy o f the calculations, which is certainly not better than ~ 100-200 keV for the r o t a t i o n a l b a n d heads. In o r d e r not to m a k e the p a p e r too extensive, we have not included tables o f the Coriolis matrix elements, wave functions and the microscopic c o m p o n e n t s o f the p h o n o n s . However, these d a t a are available for those interested in them. Inspection o f all the figures c o m b i n e d with o u r earlier results on the rare-earth nuclei shows clearly that the present model is able to explain satisfactorily the main features o f the o c t u p o l e states in a very b r o a d region o f the d e f o r m e d even nuclei. The a u t h o r s wish to t h a n k Professor A. Bohr, Professor B. M o t t e l s o n and D o c t o r B. S~rensen for valuable discussions. Drs. C. E. Bemis a n d T. Thorsteinsen a n d stud. scicnt. F. Videbaek kindly supplied us with their u n p u b l i s h e d experimental data. Financial help f r o m N O R D I T A m a d e possible our trips between C o p e n h a g e n a n d Bergen. The numerical p a r t o f the work was d o n e at the G I E R c o m p u t e r in the Niels B o h r Institute, which was kindly p u t at o u r disposal.

References 1) V. G. Soloviev, P. Vogel and A. A. Korneichuk, lzv. Akad. Nauk USSR (ser. phys.) 28 (1964) 1599 2) P. Vogel, Nucl. Phys. All2 (1968) 583 3) H. C. Pauli, private communication 4) K. Nccrggtrd and P. Vogel, Nucl. Phys. A145 (1970) 33 5) I. L. Lamm, Nucl. Phys. A125 (1969) 504 6) B. Elbek, Determination of nuclear transition probabilities by Coulomb excitation (Munksgaard, Copenhagcn, 1963) 7) C. M. Ledercr, J. M. Hollandcr and I. Perlmann, Table of isotopes, (John Wiley, N.Y., 1967) 8) E. Arbman, S. Bjornholm and O. B. Nielsen, Nucl. Phys. 21 (1960) 406 9) S. Bj~rnholm, F. Boehm, A. B. Knutscn and O. B. Nielsen, Nucl. Phys. 42 (1963) 469 10) C. M. Lcdcrer et al., Nucl. Phys. A135 (1969) 36 l l ) C. E. Bemis, private communication 12) F. S. Stephens, F. Asaro, S. Fried and 1. Perlmann, plays. Rev. Lett. 15 (1965) 420 13) S. Bjernholm et aL, Nucl. Phys. All8 (1968) 261 14) S. Bj~rnholm, I. Dubois and B. Elbck, Nuel. Phys. All8 (1968) 241 15) F. Videbaek, private communication 16) T. Thorstcinsen, private communication