Electric monopole transitions and beta bands in even nuclei

I 1.D.2 I

Nuclear Physics 86 (1966) 561--573; ( ~ North-Holland Publishing Co., Amsterdam

Not to be reproduced by photoprint or microfilmwithout written permissionfrom the publisher

ELECTRIC MONOPOLE

TRANSITIONS

AND BETA BANDS IN EVEN NUCLEI J. P. DAVIDSON t U. S. Naval Radiological Defense Laboratory, San Francisco, California

Received 10 December 1965 Abstract: Reduced transition probabilities for electric monopole transitions p~ have been calculated using a collective moO.el with deformation vibrations but no asymmetry vibrations. Both p2 and X -- o2e2R4/B(E2) have been evaluated within the context of the model by exact numerical methods for transitions within positive as well negative parity rotational bands of even nuclei. Comparison of experiment with theory shows quite good agreement in the actinide deformed region; however, for the rare earths l heoretical predictions are generally an order of magnitude greater than the measured values.

1. Introduction R e c e n t investigations o f the low-lying r o t a t i o n a l levels in even nuclei have shown t h a t the g r o u n d state r o t a t i o n a l b a n d structure can be accurately r e p r o d u c e d ~' 2) b y using a very simple collective m o d e l first i n t r o d u c e d b y D a v y d o v a n d C h a b a n 3). I t was also shown t h a t if the stiffness p a r a m e t e r p was d e t e r m i n e d from the energy level spacings in the g r o u n d state r o t a t i o n a l b a n d then the b e t a ( K = 0) b a n d h e a d c o u l d be p r e d i c t e d to within 20 %, a n d often c o n s i d e r a b l y less 1). T h e consistency o f this m o d e l has been checked by calculating exactly the effect b e t a v i b r a t i o n s have on the electric q u a d r u p o l e transitions in such nuclei 4). The results o f this calculation showed by p r o p e r l y including these beta v i b r a t i o n s the E2 b r a n c h i n g ratios c o m p a r e well with experiment. I n particular, in the C o u l o m b excitation o f the b e t a b a n d I = 2 level it was f o u n d t h a t the t h e o r y is within e x p e r i m e n t a l e r r o r of the m e a s u r e d values (recent p u b l i s h e d d a t a on lS¢Sm give a value o f a b o u t half t h a t p r e d i c t e d by t h e o r y s)). A g r e e m e n t between t h e o r y a n d e x p e r i m e n t has also been f o u n d to be quite satisfactory for the C o u l o m b excitation o f the so-called g a m m a b a n d I = 2 level 6). It w o u l d seem t h a t this simple collective m o d e l is c a p a b l e o f quite accurate p r e d i c t i o n s n o t only o f energy levels b u t also o f B(E2) b r a n c h i n g ratios. W i t h increasingly m o r e e x p e r i m e n t a l d a t a available on electric m o n o p o l e transitions it appears useful to c o m p a r e this t y p e o f d a t a with the theory. I n d e e d , D a v y d o v a n d cow o r k e r s have d o n e just that 7, 8); however, their w o r k includes the effects o f g a m m a or a s y m m e t r y v i b r a t i o n s a n d is therefore necessarily a p p r o x i m a t e in nature. It is the * Visiting Scientist-in-Residence, on leave of absence from Rensselaer Polytechnic I~,.stitute, Troy, New York, academic year 1964-65. 561

562

s.P. DAVIDSON

purpose of this paper to treat the effect of deformation vibrations on E0 transitions exactly while at the same time not requiring axial symmetry. The results of refs.l' 2,4) show that asymmetry vibrations can be treated as a small perturbation. In general the g a m m a band mixing into the ground state and beta bands is small and a perturbation treatment of g a m m a vibrations is probably adequate except where rather strong monopole transitions occur between gamma and ground state bands. At present there seems only one nucleide where this occurs and that is in 232Th. Here the I = 2 levels of the beta and g a m m a bands are only 11 keV apart and a relatively strong E0 transition from the g a m m a band I = 2 level to the ground state band has been observed 9). In the remaining part of this section an outline of the vibrational problem will be given in order to fix the notation. In sect. 2 the electric monopole transition probabilities will be derived. Finally, in sect. 3 we compare experiment with theory. In what follows we shall use a generalized treatment of deformation vibrations 1o) equally applicable to quadrupole vibrations and octupole vibrations. While no E0 transitions have been definitely observed between negative parity levels they should in principle be no different from such transitions observed between appropriate positive parity levels. This treatment is essentially the usual one for quadrupole surfaces 3) (it should be noted that a diagonal term omitted in the vibrational Hamiltonian of ref. 3) is included here and has a significant effect on the eigenvalues of the I = 0 and 2 levels 15)). For octupole surfaces the condition which diagonalizes the momental ellipsoid is used lo), so that no terms with K = 1 or 3 appear in the state functions. This is consistent with a recent microscopic calculation of Soloviev and co-workers 11) in which they find that the ~. = 3, [/~1 = 1 and 3 degrees of freedom do not possess very collective properties. As usual one begins by expanding the nuclear surface in the laboratory coordinate system

R(0,

= Ro[ o+ Z#

q )l,

*

(1)

here 2 = 2 for positive parity and ~. = 3 for negative parity states, while ~o is unity to first order and the second-order differences are usually neglected. It is known, however, that these volume conserving second-order terms are important in electric monopole calculations 12) and they will be retained here. Small oscillation theory then yields the classical Hamiltonian Ha = -~Ba • Ioi~.,,[2+~-C~2 [ tt

~

~.u[2 ,

(2)

//

H = ~Ha. 2

For vibrational nuclei this can be quantized straight away and, using the number representation IN), one can easily show that E0 transitions are allowed between any two states of the same spin for which A N = 2. This selection rule then prohibits such transitions between the one and two phonon I = 2 + states. Making use of the trans-

E0 TRANSITIONS

563

~z. = Z DZu*(O~)az~,

(3)

formation v

where the DuZvare the (2I+ 1) dimensional representations of the rotation group 13) and the 0 i the Euler angles relating lab and body-fixed axis systems. It is useful if the body expansion coefficients in (3) are parametrized as ~o) aa. = fl;oo'~.

(4)

where ~ are the 2th order deformation parameters while the asymmetry parameters 0-zu may be subjected to the additional condition 2 2

= 1.

(5)

#

The expansion of the nuclear radius in the body-fixed system becomes

[

R(0',•') = Ro 1 - ~ 1t 3 / ~ 2+,3Tz)_l_flaEa.~,~yau(O,,d/) ? ,

(6)

#

The quantity Ta, which arises from the condition of volume conservation, is V22 + 1 Tz = - - 47r C(222; 000)"~ 0-;.,cr~, 0-~.u-~' C(222; #-/z',/1', #)

(7a)

/t/~'

which for deformed nuclei will give rise to E0 transitions between gamma-like and ground bands. From the properties of the Clebsch-Gordan coefficients C(L 1, Lz, L3; m~, m z, m3) it is seen that Tz vanishes for 2 odd. Thus, in principle this model permits no monopole transitions between negative parity states except between zeta and ground bands. For positive parity states on the other hand transitions between gamma and ground state bands are possible. By expressing the asymmetry parameters in the familiar form 0"20 =

COS ])~

0"30

COS t],

=

0-2±2 = x/{- sin 7, rr3 _+2 = .]'½ sin q,

Tz can be expressed as 7) Tz = 7

~zcos37'

2=2 (7b)

=0,

2=3.

The Hamiltonian of the system is obtained from eq. (2) by using relation (3) and recalling that since the deformation vibrations are to be treated exactly and the asymmetry vibrations only in perturbation theory the space with respect to which the quantization is carried out contains but four dimensions: /3z and the three Euler angles 0 i. The 0-~, are taken as fixed. The Schr6dinger equation separates into rota-

564

S. P. DAVIDSON

tional and vibrational parts: [½Y

2.3. 3. (Ik/tk)-e,U]~,N(O,)

= O,

(8)

k

2B3. fl~ d[3 fl~

+ 4B~fl 2 elN+]C;~(flz--fizo)

--EIN n

~lnS(fl3. ) = O.

(9)

The I k are the body-fixed angular momentum operators while the reduced moments of inertia of eq. (8) are defined by 2-).

Ytk3.)(S3., ~r3.u) -- 4Bail3. lk(a2#), and the functional forms are known for the cases )0 = 2 (ref. 1~)) and 2 = 3 (ref. is)). The solutions to eq. (8) have also been given for these two cases 16, lo). The particular vibrational potential used in eq. (9) has been justified from binding energy data tT) and need not be discussed further here. By expanding the sum of this term and the previous one about the new equilibrium position f13.(I, N), keeping only the quadratic term for the new equivalent potential and defining new independent and dependent variables by Y = ZI [ f i x - flz(lN)]/fl3.(IN), NvDv(\/~y )

~ m

=

Nv being a normalization constant, eq. (9) can be placed in the form I~

+(2v+l-y2)]

D~(v"SY) = 0'

which is Weber's equation for parabolic cylinder functions 18). The quantity v is determined by the boundary conditions ~,,u(fi3. = O) # O, lira (~n/N(/~3.)= 0 , f l z ~ co

while the Z are known ~o) functions of v. Calling I~ the integral Iv =

D~(~/2x)dx

the nolmalization constant Nv can be expressed in the form N 2 = Zl/ItflzoZI,,,

where # is the stiffness parameter and Z is the positive real root of' ,

3.

3;

2

-- :(F.IN.~ 3) .~_ 0

and is related to Z~ by ~

3~

Z41 = 2 4 ~- 2~elS ~-2-]"

565

E0 TRANSITIONS

2. Electric Monopole Transitions The absolute transition probability for electric monopole conversion has been defined by Church and Weneser 19) as T(EO) = f2p z, where the electronic factor f] is available in graphic form 19) while the nuclear strength parameter is

the ~b being the nuclear wave functions, rp the radius vector of the pth proton, R the nuclear radius and ~ a numerical coefficient of the order of O. 1. In collective model calculations it is customary to neglect all but the first term of eq. (10). Using the assumption of a uniform charge distribution one can define an electric monopole operator O(EO) such that p = (¢~,N'(fiz)lO(EO)14/,Y'(fi~,)).

(11)

O(EO) =

(12)

Then 4re

+ x, p ,3 r 3 ;

is the atomic number. The first term in (12) makes no contribution to p, the second term induces transitions between "'beta" bands and ground state bands while the third term induces transitions between the gamma and ground state bands for 2 = 2. For 2 = 3 this term is identically zero so that no E0 transitions can occur between the octupole analogue of the gamma band (sometimes called the g-band 2o)) and the octupole ground state band, e.g., in the notation of ref. 15) the model permits no E0 transitions between the 3 l 1 - and 321 - levels, even i f a s y m m e t r y vibrations are included. Since we are going to consider monopole transitions between "beta" and ground state bands only the term proportional to f12 in eq. (12) will be retained. Thus eq. (11) can be written /

=

\4re/ \ Zl /

Iv~Iv~

where the overlap integral/~fv,(0) is defined by lyre(O) =

D ~ ( x / 5 [ x - Z i ] ) D v , ( x / 2 [ x - Zi])x 2 dx,

(14)

which is identical in form with the overlap integrals arising from the vibrational contributions to the electric quadrupole reduced matrix elements between states of positive parity 4). In fig. 1, p2 is plotted for transitions between the positive parity beta band and the positive parity ground state band for I = 0, 2, 4, 6 and 8 as a function of the stiffness

566

J.P. DAVIDSON

parameter/1. T h e a s y m m e t r y p a r a m e t e r has been t a k e n as 7 = 0°, i.e., axial s y m m e t r y ; however, the curves for other values o f 7 are only slightly different f r o m those given here. I n fig. 2 we have p l o t t e d pZ as a function o f g for m o n o p o l e transitions between the ~ b a n d a n d the g r o u n d state octupole b a n d for I = 1, 2 a n d 3. T h e o c t u p o l e a s y m m e t r y p a r a m e t e r has been t a k e n as t/ = 5 ° since r/ = 0" the 2 - level lies at infinitely high energy. These curves d o show m o r e v a r i a t i o n with a s y m m e t r y p a r a m e t e I t h a n d o those for transitions between positive p a r i t y states. ~C

9

BB ~'8 gnd _

--

6/3 ~6gnd

8

/

4t3 ~4 gr'd ] ~ 7

-

2,3 -~2 gr~d / / .

"o

% -,-ognd. / / /

6

%

0

01

08

03

,u

1 0,4

I 05

i 0.6

0.7

Fig, 1. The monopole nuclear strength parameter p~/Z2flo4 of eq. (13) plotted as a function of the stiffness parameter # for an axially symmetric system (7 = 0°) • The curves are labelled with the spin of initial and final state and are transitions from the beta band to the ground state band for quadrupole deformations. Here Z is the atomic number and. /~0 the equilibrium deformation.

T h e q u a n t i t y p2 is n o t the m o s t useful one to c o m p a r e with experiment. F r e q u e n t l y one c o m p a r e s the relative rates o f E0 a n d the c o m p e t i n g E2 transitions defined b y 2~)

/&(I i ~ I f ) = T ( E 0 : I o I )

(15)

T(E2: Ii ~ It) A s o m e w h a t m o r e useful q u a n t i t y is the dimensionless r a t i o defined b y R a s m u s s e n ~2)

E0 TRANSITIONS

567

for transitions between the beta b a n d I = 0 level and the ground state b a n d as p2 e2 R4° X(0~- ~ 0+d) = B(E2: 0~ --* 2g.d)'

(16a)

where Ro = 1.2 A ~ fm is the nuclear radius. F o r transitions other than f r o m the beta band head we m a y generalize this quantity to + = p2e2R4 X(I-~ --* Ig.d) B(E2: Ip --* Ig.d) '

(16b)

lO

i

%

I ~= 5°,r- ] l

l

6

~5 %

2 ~/-"2.,

~4

0 0

' 0.!

02

, 04

0.3

I 05

I 0.6

0.7

~z

Fig. 2. T h e m o n o p o l e nuclear strength p a r a m e t e r p~/Z2'~o4 plotted as a function o f the stiffness p a r a meter # for an almost axially s y m m e t r i c octupole system ( ~ / = 5°). T h e curves are labelled with the initial a n d final state spin a n d are for transitions f r o m the b - b a n d to the octupole g r o u n d state. Here Z is the atomic n u m b e r a n d ~0 the octupole equilibrium deformation.

of course Ia =

Ignd # O. The relation between X and ~uK is just X = 2.53

I~KA~E5 x 109, Q

where the g a m m a transition energy Er is in MeV.

(17)

568

J.P.

DAVIDSON

By making use of the expression for the reduced E2 transition probabilities developed elsewhere 4) these expressions for the dimensionless ratio X can be expressed in terms of overlap integrals (14) and the other parameters of the theory. For transitions from the beta band head eq. (16a) becomes

x(o;

OLd) = (l~fl2o)2(Z,/Z,i)5(Zaf/Zf)3I~,I~,~,(O)Z/I~,I,,~,(2)Zb(E2:01

+ ~ 21+),

(18a)

4.5

,u.= 0 . 7 4.0

~o

_

_ _

\

\ \ \ 25

!

!

!

~

!

5

10

;5 y ( dc.j,'ees )

20

2,5

ZO

Fig. 3. The dimensionless ratio X(0~---~0gna)//302plotted as a function of the quadrupole deformation parameter 7, for several values of the stiffness parameter ~. while for transitions from other beta b and levels eq. (16b) may be written

X ( I ; --, l+d) + = (#Bz0ZI/Z,j)2I .... (0)2/I .... (2) 2b(E2: I~ N~ ~ Ig+dNg.d).

(lSb)

In eqs. (18) the subscripts i and f refer as usual to initial and final states while the notation (0) and (2) on the overlap integrals refer to the integral of eq. (14) for monopole transitions or a similar one for E2 transitions 4). (For the latter the functional dependence is different since Z and Za are different in initial and final states). The quantity b(E2: IN ~ I ' N ' ) is defined from the adiabatic reduced transition ploba-

E0

569

TRANSITIONS

bility of ref. 4) by

B~(E2:IN~I'N')=

(~)2b(E2:IN~I'N'),

3e being the nuclear charge. In fig. 3 the ratio X(Oa ~ Og,d)//320 is plotted as a function of the asymmetry parameter 7 for various values of/t. In figs. 4 and 5 the ratio X(I a ~ lg.d))/fl20 are plotted 34 I

4

32!

I l

28

~o 26

24

×

22

20

76

]4

i

0

5

10

]5

20

25

30

7' (degrees)

Fig. 4. The dimensionless ratio X(2/~ -+ 2gnd)/fl0~ plotted as a function of the quadrupole deformatio1~ p a r a m e t e r 7 for several values of the stiffness parameter /~.

as a function of 7 again for various values of p. Fig. 4 is for the 2 + ~ 2 + transition while fig. 5 is for the 4 + ~ 4 + transition. Examples of both have been measured. Finally, for monopole transitions between negative parity levels the ratio X(I~ ~Ig,a ) is especially simple since the E0 and E2 overlap integrals are identical and thus cancel. Thus, eq. (16b) can be written for such transitions as

X(I~-

~ If, a) = 3rc(p¢o Z/Z,)Z/4b(E2: I¢ N~- ~ [gnd Ngnd)'

(l 9)

The quantity X(I~ ~ l;oa)( 2, where ¢o = f13o, is plotted in fig. 6 as a function of the octupole asymmetry parameter q for several values of/~ for the case I = 3-.

570

S. P. DAVIDSON

3O

g

2O

1

5

~

0

I 5

10

I

I

15 20 ¥ (degrees)

25

30

Fig. 5. The dimensionless ratio X(4p --* 4gnd)/flo z plotted as a function of the quadrupole deformation parameter 7 for several values of the stiffness parameter #. 14 p_=07 12

~

]0

~=0.6

B

,o-05

,u.=O 3

0

]0

20

30

40

50

~,0

70

80

90

(degrees)

Fig. 6. The dimensionless ratio X(3; ~ 3g-nd)/~o2 plotted as a function of the octupole deformation parameter z/ for several values of the stiffness parameter #. This ratio is for transitions between the lowest 3- state in the octupole b band to the lowest 3- state in the octupole ground state band.

E0 TRANSITIONS

571

3. Comparison with Experiment I n t a b l e 1 we c o m p a r e this c a l c u l a t i o n w i t h t h e a v a i l a b l e e x p e r i m e n t a l results f o r E0 t r a n s i t i o n s f r o m v a r i o u s levels o f t h e b e t a b a n d to t h e g r o u n d state b a n d b o t h f o r r a r e e a r t h d e f o r m e d n u c l e i a n d the actinides. I n c o l u m n s 2 a n d 3 are g i v e n t h e v a l u e s o f ~ a n d / t w h i c h r e d u c e t h e r.m.s, d e v i a t i o n b e t w e e n t h e o r y a n d e x p e r i m e n t o f t h e e n e r g y level s t r u c t u r e to a m i n i m u m . I n p a r t i c u l a r it has b e e n f o u n d t h a t the v a l u e s TABLE l Comparison of experimental values of electric monopole transition probabilities with theory for transitions from the beta to the ground state band Nucleus

7

#

flo

Et

I

(keV) 152Sm

10.73

0.3996

0.314

154Gd 156Gd XSSDy 164Er ~ssYb 178Hf

11.62 10.04 11.81 12.39 10.85 9.750

0.4006 0.2735 0.2916 0.2509 0.2561 0.2614

0.303 0.342 0.335 0.329 0.325 0.281

ls4pt

12.43

0.5192

0.198

0.2714 0.2464 0.2482 0.2201 0.2030 0.2068 0.2131

0.267 0.272 0.279 0.285 0.297 0.281 0.283

23°Th 232Th 282U 2~4U 28sU 28Spu 24°pu

9.375 9.049 8.365 8.619 7.914 7.898 8.345

685 689 681 1010 986 1217 1104 1197 1440 677 796 634 730 693 811 993 941 870

X/flo 2 theory

0 2 0 0 2 2 4 0 0 2 4 0 0 0 0 0 0 0

3.40 16.6 3.42 3.30 15.8 15.6 18.0 3.33 3.33 20.45 28.75 3.25 3.40 3.39 3.48 3.54 3.53 3.49

Ref. exp

0.88i0.07 8.7 0.84 6.8 4.61 1.35 0.731 2.3 -t0.5 6.7 ±2.0 :>7.2 3.3 ± 1.0 3.1 ±1.4 4.6 ± 1.1 2.2 ±0.5 6.2 ±1.0 2.4 ±0.6 8.0 ±2.5 0.62±0.12

23) 23) 24) 23) 2~) 2~) 2~) 27) 2D 9) 91 2s) ~s) 2s) 2s) 2s) 28) 2s)

In column 1 are the nuclei and A values, columns 2-4 give the best fit values to the quadrupole asymmetry parameter, stiffness parameter and equilibrium deformation parameter. Column 5 is the energy of the monopole transition, column 6 the spin of the levels involved while columns 7 and 8 give the theoretical value and experimental value, with errors where quoted, of the dimensionless ratio X/flo2. The quantity X is defined in eq. (16a) for 0 -+ 0 transitions and in eq. (16b) for all other E0 transitions. o f / ~ are h i g h w h e r e a d e f o r m e d r e g i o n o p e n s a n d d e c r e a s e r a p i d l y to a b o u t 0.2. T h i s v a l u e is m a i n t a i n e d i n t o t h e t r a n s i t i o n a l r e g i o n w h e r e / z b e g i n s a s l o w increase. T h e best fit v a l u e s o f # s e e m to b e c o r r e l a t e d w i t h t h e n u m b e r o f n e u t r o n s b e y o n d t h e n e a r e s t c l o s e d shell a n d are i n d e p e n d e n t o f t h e d e f o r m e d r e g i o n in w h i c h t h e n u c l e i d e in q u e s t i o n is f o u n d 6). T h e f o u r t h c o l u m n gives t h e e q u i l i b r i u m d e f o r m a t i o n s fi0 = //2o w h i c h h a v e b e e n fit to 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 y for C o u l o m b excit a t i o n to t h e first e x c i t e d ( H = 2 +) state 4) B ( E 2 : 0 1 1 ~ 211) = ( 3 ~ e / / ° ' u R 2 ] 2 b ( E 2 : 0 1 \ 4n /

~ 2l)(Zli/ZiI~ft,,,)(ZdZ,t)Slviv~(2)

2. (20)

572

J.P. DAVIDSON

Where the Coulomb excitation data are either not available or of insufficient accuracy use was made of Grodzins' analysis of the E2 transition probabilities from which one can obtain the empirical relation zg)

[30 = II08/AT/6E~,

(21),

E~ is the energy of the first excited state in keV. This relation is quite accurate in the deformed regions and the values of [3o given by eqs. (20) and (21) are very close. Table 1 shows that except for ~VSHf, lS6Gd and the 2~ --, 2gnU transition in 152Sm the monopole transitions in the rare earth deformed nuclei are from five to ten times smaller than predicted by theory. (The experimental value for the 0~ --* 0g,a transition in lSZSm is that reported in ref. 22). Rasmussen 12) making use of some unpublished data of the Chalk River group takes half of this value which clearly leads to a no m o r e favourable comparison.) On the other hand for the actinide nuclei agreement between experiment and theory is quite satisfactory except for the nucleus/*°Pu. However, Bjornholm 28) has pointed out that in this case the very low value may be associated with the fact that the beta band head lies very close to the neutron energy gap. One might expect that this would offer an explanation for the failure of the theory in the rare earth region, and certainly in the middle of the region the beta band heads are, rather close to the gap. However, this can hardly account for the lack of agreement for the 0 r --* 0g,d transition in 152Sm where the band head at 685 keV is well below the gap. Furthermore, for the 2~ -~ 2gnd transition the comparison is adequate especially in view of the fact that the B(E2) from the beta band is somewhat smaller than predicted by theory 4). It would seem quite worthwhile for both of these monopole transitions in 152Sm to be measured by the same group to see if this discrepancy persists. Several of the rare earth nuclei listed in the table are known to have more than one 0 + excited state and for 178Hf the X ratios for two such levels have been measured. In this nucleus we have taken the lower 0 + level at 1197 keV as the beta band head for which the /~ value is consistent with the general trend in the region. The X/[32 ratio for the 1400 level has been calculated assuming that it too is a 012 level. It very probably is not the 013 level which the model would predict for these parameters in the neighbourhood of 2 MeV. This level must then have a different character from that of a 0 + beta vibrational level - perhaps it is a two-quasi-particle state. No attempt has been made to calculate the interaction of these two states and it seems hardly worthwhile to construct a theory of higher excited 0 + states until more are known and their characteristics determined. A true beta band should not only have enhanced reduced E2 matrix elements to the ground state band but should as well have large E0 transition probabilities. Thus investigation of the excited 0 + levels must involve not only the determination of E2 strengths but E0 strengths as well. It has been suggested that the failure of the theory in the rare earth region might be due to the influence of a non-uniform charge distribution, which was assumed in order to derive the operator O(E0) of eq. (12) or of the lack of irrotational nuclear

E0 TRANSITIONS

573

flow 21). It is d o u b t f u l if any but the m o s t drastic i n n o v a t i o n w o u l d induce the needed o r d e r o f m a g n i t u d e change in X a n d this in turn w o u l d influence greatly, no d o u b t u n f a v o u r a b l y , the a g r e e m e n t already o b t a i n e d for the E2 transitions 4, 6). Also it is difficult to believe that such a change in the theory w o u l d n o t destroy the a g r e e m e n t for the E0 transitions so evident in the actinide region. In any event, we should like to see m a n y m o r e m e a s u r e m e n t s o f the m o n o p o l e transition p r o b ab i l i t y in the rare e a rt h region especially for transitions other t h a n 0" ~ 0g,d. A particularly g o o d candidate is aS°Nd whose level structure seems quite a n a l o g o u s with 152Sm. I should like to a c k n o w l e d g e m a n y stimulating conversations with R. M. D i a m o n d a nd F. S. Stephens as well as permission to use their data on 184pt before publication. I should also like to t h a n k Dr. B. E l b e k for m a k i n g available before p u b l i c a t i o n n o t only the d a t a a p p e a r i n g explicitly in table 1 but a great deal m o r e i n f o r m a t i o n conc e r n i n g beta bands and C o u l o m b excitation which has been used in the fitting procedure. Finally, I should like to a c k n o w l e d g e the s u p p o r t given by the staff o f this L a b o r a t o r y where this investigation was u n d e r t ak en .

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) 25) 26) 27) 28) 29)

F. S. Stephens, N. Lark and R. M. Diamond, Phys. Rev. Lett. 12 (1964) 225 F. S. Stephens, N. Lark and R. M. Diamond, Nuclear Physics 63 (1965) 82 A. S. Davydov and A. A, Chaban, Nuclear Physics 20 (1960) 499 J. P. Davidson and M. G. Davidson, Phys. Rev. 138 (1965) B316 Y. Yoshizawa, B. Elbek, B. Herskind and M. C. Olesen, Nuclear Physics 73 (1965) 273 J. P. Davidson, Bull. Am. Phys. Soc. 10 (1965) 468 A. S. Davydov, V. S. Rostovsky and A. A. Chaban, Nuclear Physics 27 (1961) 134 A. S. Davydov and V. S. Rostovsky, Nuclear Physics 60 (1964) 529 J. Burde, R. M. Diamond and F. S, Stephens, UCRL-16163 (1965), unpublished S. A. Williams and J. P. David.son, Can. J. Phys. 40 (1962) 1423 V. G. Soloviev, P. Vogel and A. A. Korneichuk, Izv. Akad. Nauk SSR (ser. fiz.), to be published; V. G. Soloviev, private communication John O. Rasmussen, Nuclear Physics 19 (1960) 85 M. E. Rose, Elementary theory of angular momentum (John Wiley and Sons, New York, 1957) A. Bohr, Mat. Fys. Med.d. Dan. Vid.. Selsk. 26, N'o. 14 (1952) J. P. David.son, Nuclear Physics 33 (1962) 664 R. B. Moore and W. White, Can. J. Phys. 38 (1960) 1149 R. M. Diamond, F. S. Stephens and J. S. Swiatecki, Phys. Lett. 11 (1964) 315 E. T. Whittaker and. G. N. Watson, Modern analysis 4th ed., (Cambridge University Press, Cambridge, 1950) E. L. Church and. J. Weneser, Phys. Rev. 103 (1956) 1035; L. A. Borisoglebskii, Soviet Phys. Usp. 6 (1964) 715 P. O. Lipas and. J. P. David.son, Nuclear Physics 26 (1961) 80 A. S. Reiner, Nuclear Physics 27 (1961) 115 I. Marklund, O. Nathan and. O. B. Nielsen, Nuclear Physics 15 (1960) 199 O. Nathan and S. Hultberg, N'uclear Physics 10 (1959) 118 B. Harmatz, T. H. Handley and J. W. Mihelich, Phys. Rev. 123 (1961) 1758 L. V. Groshev et aL, Bull. Acad.. Sci. USSR, Phys. Set. 26 (1962) 1127 R. Graetzer, G. Hagemann, K. A. Hagemann and B. Elbek, private communication from B. Elbek C. J. Gallagher, Jr., H. L. Nielsen and O. B. Nielsen, Phys. Rev. 122 (1961) 1590 S. Bjornholm, Nuclear excitations in even isotopes of the heaviest elements (Munksgaard, Copenhagen, 1965) L. Grodzins, Phys. Lett. 2 (1962) 88