A calculation of some ΔK = ±2 band-mixing effects in the odd-mass nucleus W183

1.D.2

[ [

Nuclear Physws 61 (1965) 1--12, @

North-Holland Pubhshmg Co, Amsterdam

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

A CALCULATION O F S O M E AK---- 4 - 2 BAND-MIXING EFFECTS IN T H E ODD-MASS NUCLEUS W tss D. J R O W E

A E R E Harwell, Dtdcot, Berks, England Received 4 June 1964 Abstract: This paper represents an attempt to find out whether the LIK = -4-2 band-mtxmg mteractions play any part in the mLxmg o f rotational bands m odd-mass nuclei, as they are k n o w n to do m even nuclet xl) The a m o u n t o f such an lnteracUon m the W lsa H a m f l t o m a n is f o u n d to be either very small, or small, correspondang to the two solutions for which very good fits to the energy level spectra were obtained Both solutions also fit very well the k n o w n branching ratios for the gamma-ray decay o f the exc~ted states, but only the first solution g~ves good agreement with the few B(E2) values obtained f r o m C o u l o m b excltatlon experiments It thus seems highly probable that only a very small a m o u n t o f the A K = 4-2 mteracttons are present, and whale these are important in giving an excellent fit to the energy levels, they have a neghgtble effect on the wave-functtons and hence on the electromagnetic transition strengths

1. Introduction

It ts well-known that the nuclei m certain regions of the periodic table have deformed eqmhbrmm shapes, and that these nuclei are characterized by their rotational properties In particular their energy levels follow closely the 1(I+ 1) law for rotational bands, and their electromagnetic transition strengths follow generally the simple relationships predicted by the admbatlc model for nuclear rotations. However, the detaded properties of many of these nuclei can often be very much better fitted by the rotaUonal model if the decoupllng effect of the Conohs force is taken into consideration In particular, close-lying bands whose K quantum numbers dafter by umty are often strongly mixed by the Conolls force (or rotaUonal particle coupling, as it is often termed). Such a departure from the adlabaUc model constitutes a weakening of the strong couphng between the particle motion and the nuclear surface deformation This type of band mixing was considered m detail by Kerman 1) for the oddmass nucleus W ls3 For tlus nucleus there is a wealth of accurate experimental data avadable Furthermore, st has a K = ½ ground state band, (see fig. 1) followed by a low-lying K = ½ excited state band, but no other low-lying band that could couple to either of these through the Conohs force One result of Kerman's calculation was to show that for W 183 the mixing of the two lowest bands was so strong that the problem could not be reliably treated using perturbation theory, and an exact dmgonahzat~on procedure was requ|red The perturbation expansion as, however, very enhghtenlng as it shows that to second order lU the energy an lnteracUon of the general 1 January X965

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f o r m o f the C o n o h s force, Le. 111 = h + I _ + h _ I + ,

does n o t h i n g m o r e t h a n r e n o r m a h z e the effectlve m o m e n t s o f inertia. O n e c o u l d express this m o r e p m t o r m l l y b y saying t h a t levels o f the same spin b u t o f different b a n d s t e n d to repel one a n o t h e r . T h e levels o f t h e lower b a n d b e c o m e c o m p r e s s e d a n d effectively increase the m o m e n t o f Inertia, whale the o p p o s i t e occurs for the u p p e r b a n d . T h e thard-order t e r m in the energy expanszon is p r o p o r t i o n a l to 1 2 ( 1 + 1) z, so t h a t this is the first o b s e r v a b l e effect, as far as the energy level sequence is concerned, a%

(ss433)[~ ss ~J

" "/a

,,,2 o6 ~.,2 oe

I" 7~3'

%

3o694~o87~ ~ %

e ~

~o7oopo~

E%

453 O~

K- :'la

-

ag, ~l ~91 ~3

no0 Ùl Doe 64

K=312c s/2

9Q o~ ~9

• 3/2

~ 6 ~ C~ sO

A I/2--

0 K=I/2-

Fig 1 The level scheme of W z88 The unbracketed energies are those measured by Murray et al s), whale the energies in square brackets are the values calculated by Kerman x) The energy of the level J in round brackets is obtained by assuming that the gamma-ray of 142 25 keV observed by Murray et al ~) belonged to the transition JH

t h a t there is a n y d e p a r t u r e f r o m the p u r e strong c o u p l i n g m o d e l O f course there will be other Indications o f b a n d mlxang f r o m the electromagnetic t r a n s m o n strengths, a n d at least a n inference i f the m o m e n t s o f inertia o f the two b a n d s &ffer to a n y extent. Nevertheless to o b t a i n a q u a n t i t a t i v e estimate o f the C o n o l l s mixing, K e r m a n preferred to w o r k with the energy levels since these are the quantltmS t h a t have been m e a s u r e d to a high degree o f a c c u r a c y 2). F r o m has results he was then a b l e to interrelate the strengths o f the n u m e r o u s electromagnetic t r a n s i t i o n s between the states o f these b a n d s . H i s fits to b o t h the energy levels a n d the electromagnetic t r a n s m o n strengths a r e extremely g o o d a n d are certainly a vast i m p r o v e m e n t o n the a&abat~c model. T h e question n o w arises as to the uniqueness o f K e r m a n ' s solution to the p r o b l e m . It Is evident t h a t the C o r l o h s force Is c a p a b l e o f explaining the nuclear d a t a o f W zs3 to a degree o f accuracy m u c h higher t h a n Is c o m m o n m nuclear physics. Nevertheless small chscrepancles m the energy level s p e c t r u m are still a p p a r e n t , whach are consider-

BAND MIXING-EFFECTS IN W 18a

3

ably larger than the experimental errors. Might it not be possible that some other force is introducing the perturbation into the energy levels, maybe in first or second order, that the Corlohs force introduces m third and tugher order9 One reason for asking th~s question is that the E2 trans~tlon strengths in many even nuclei, indicate that the K = 0 ground state band is considerably mixed with the K = 2 excited state band 3). To explain this mixing it is necessary to consider an interaction of the general form

V2 = h+2IZ_+h_2 I2 Clearly such an interaction could also couple K = ½ and K = ~ bands and would introduce the requlslte 12(1+ 1) 2 term into the energy levels in second order. This then is the kind o f mteractlon we have to consider as an alternatwe to the Corlohs force The possible sources of such an interaction will not for the moment concern us, but we shall return to this later in the discussion Another reason for suspectmg that the Corlolls force might not be responsible for the mlxang it its entirety is the following I f one considers some specific model for the structure of the intrinsic wave-function one is able to estimate the magnitude o f the C o n o h s mixing. In the hmlt of large deformation for example, the asymptotic Ndsson model 4) would predict this to be zero. A more reahstlc deformation 5) of t$ ~, 0.2 would stdl only provide for about half of the value required by Kerman from treating it as a fitting parameter The present calculation, then, represents an attempt to determine the uniqueness of the Kerman solution by finding the extent to which an mteractzon of the general form V2 could be responsible for the perturbations to the W ~s3 energy levels Furthermore better values are obtained for the electromagnetic transition parameters, using the much more accurate and more rehable experimental data, which have recently become avadable 6.7) The method is outlined in sect 2 For various values of a parameter Z, denoting the ratio of the contribution of V2 to that of VI, a leastsquares fit was made to the energy levels, the goodness of each fit being characterized by X2 It was found that an extremely small value of Z could effect a substantmlly better fit than the zero value, but that the total range of Z for a satisfactory fit was not at all large Contained m this range, however, were two particular values of Z yielding especially good fits To dlstlngmsh between these, fits were also made to the electromagnetic transition data Kerman had already done this with the earher less rehable data, for hls Z = 0 solution, using a trml and error procedure Although he could do this with remarkable success, sizable computers are nowadays readdy avadable and there is no excuse for not performing a proper analysis. Results are presented for both of the best solutions and for the Z = 0 solution, together w~th the values of the electromagnetic transition parameters which lead to the best fit in each case Also included are some prechct~ons for some as yet unmeasured electromagnetic transition strengths. A &scusslon o f the results and the conclusions to be drawn are contained m sect. 3.

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2. Calculation and Results In the pure adiabatic model the energy levels of the K = ½ and K = ~ bands are given by

E½(I) = E~l)[I(t+l)-l-a(--1)l+½(I-F½)-l-a--¼-],

E~(I) = E~°)-I-E~I)I(I..t-1),

(1)

respectively, where we measure all energies wath respect to the ground state energy, 1 e. E~(½) = 0, by definmon. I f we now admit some perturbing interaction, these energy levels become too&fled and to find their new energies we must dmgonahze the interacting Hamdtonmn. We assume, as Kerman dad, that only m a m x dements between these two low-lying bands are non-vanishing, so that the matrices revolved are only 2 × 2. This is reasonable since, as we have already mentioned, there are no other bands that the Conolis force could couple to eather the K = ½ or K = ½ band, lying low in energy, and tins is also true for the interaction V2 . The energies of the htgher and lower bands then become

E n' r(I) = ½[E~(I)+E~,(I)] +½x/[E~(I)-E~(I)] 2 +41V,0l 2

(2)

I f the mteractmn is the Corlohs force Vx,

V.o = = -- A k x / ( I - K ) ( I + K + 1),

(3)

where K = ½ and where A s is the lntrmsac matrix element defined by h2 As = -
(4)

Eq. (2) may now be written

E"' I'or) = ½[E~(O+ E ~ ( O I + k , / [ E ~ ( Z ) - E d Z ) l % 4 1 A , I ' K ( I ) ,

(5)

K(I) = I ( I + 1 ) - K ( K + 1 I

(6)

where

For the interaction V2,

V.o = < K + l l h + 2 1 2 - I K - - I >

= -Hi(-

1)r+s~/[I(I+ 1 ) - K ( K + 1)1[I(I+ 1 ) - K ( K - 1)l,

(7)

where K = ½ again and where * H s = - < Z s + dh+21Zs-x>

(8)

t We define the phases for the intrinsic wave-functton Z-x according to Bohr and Mottelson'), such that if ZK = ~ j a j ~ then X-~ = ~jaj(--1)z+~Xs_x

BAND-MIXING

EFFECTS

W lsa

IN

5

I f n o w the interactions V 1 and V2 both occur, V.o = - A x [ 1 +

(-- 1)t+XZx/I(I+ 1 ) - - K ( K - 1 ) ] x / I ( I + 1 ) - K ( K + 1),

(9)

where the parameter Z denotes the relative strength o f the interactions and IS defined by Z = Hx/Ar. (10) Inserting eq. (9) into eq. (2) we find that eq (5) still gives the perturbed energy levels provided we generahze K(I) o f eq. (6) to

K(I) = [ I + ( - 1 ) ' + X Z x / I ( I + I ) - K ( K - 1 ) ] 2 [ I ( I + I ) - K ( K + I ) ] .

(11)

F o r a range o f values o f the parameter Z, the parameters o f eqs (5) and (11) were least-squares fitted to the experimental energy levels. The goodness o f each fit, characterized b y the usual g 2 parameter, Is plotted as a function o f Z in fig 2 By

o-\ 20C-IOC 8C 6C

--

4(:

I( 8 -4--

I -O00S

0

000S

1 0 010 Z"--~

I

I

0 015

0.020

0.025

F]g 2 A plot of the goodness of fit parameter X~ against the parameter Z The mm]ma at 0018 and Z = 0 0 1 1 2 correspond to the best fits of the calculated energy levels to experiment

Z = 0

far the best agreement with experiment is obtained when Z = 0 0018, solution 1, or when Z = 0.0112, solution 2, for which X2 has very steep-s]ded minima. The actual energy levels for these Z values are c o m p a r e d with those for Z = 0 in table 1, while the associated parameters are listed in table 2 Solutions 1 and 2 clearly represent a m u c h better fit to the experimental levels than the Z = 0 or K e r m a n solution. To a certain extent this is to be expected since we have introduced an extra parameter into the calculation However, xf we m a k e the null hypothesis that the K e r m a n five-parameter theory is correct, we would expect that for the two degrees o f freedom X2 > 10 6 would occur only Jr % o f the time. In fact for Z = 0, X2 is just ten times this value. F o r our six parameter theory, on the

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TAeL~ 1 A comparison of the theoreUcal energy levels wRh the expertmental values Expermaental Level

B

C D G E F H j a)

TheoreUcal energy (keV)

energy (keV)

Kerman solution Z = 0 46 30 99 35 207 14 308 86 208 68 291 86 412 05 555 53

46 48

99 07 207 00 308 94 208 81 291 71 412 08 554 33

solution 1 Z = 00018 46 43 99 07 207 02 308 94 208 83 291 68 412 09 554 06

soluUon 2 Z = 00112 46 44 99 07 207 02 308 94 208 83 291 68 412 09 554 05

a) Since the existence of the level J is not absolutely certain we dad not fit the theory to Its energy Thus the agreement for all Z, between the energy predicted for level J, and the energy measured gives us confidence that the gamma-ray observed by Murray et al ~) dad m fact originate from such a level J of the K = {-band

TABLE 2 Parameters assocmted wRh the energy levels of table 1 Z

A~

0 0 0018 0 0112

21 716 21 004 15 325

Hi 0 000 0 038 0 172

a

E~1)

E~(0)

0 1680 0 1770 0 1936

15 852 15 607 14 273

146 740 146 703 145 705

E (1) 14-055 14 254 15 587

All quantities are tabulated m keV except Z which is demenslonless

o t h e r hand, there is only one degree o f f r e e d o m an d the value o f Zz = 2.3 hes wlttun the 10 °//o slgmficance level. T h e n u m b e r s d e r w e d f r o m statistical analysis s h o u l d n o t be t a k e n t o o seriously t h o u g h , since n o o n e w o u l d expect the r o t a t i o n a l m o d e l , even with all its refinements, to be c o m p l e t e l y exact It does not, for example, give due conslderaUon to the s h o r t r a n g e c o m p o n e n t s o f the nuclear forces, neither does it separate o u t the centre-of-mass c o o r d i n a t e s It should also be e m p h a s i z e d t h a t the K e r m a n t h e o r y does give the energy levels extremely well, the large value o f X2 really reflecting the h i g h degree or a c c u r a c y ( ~ 0.04 keV) to whach these levels h a v e been m e a s u r e d R a t h e r the p o i n t to be d e d u c e d is that o u r six-parameter t h e o r y does go a long wa y t o w a r d s m a k i n g a t h e o r y which could, f r o m a staUst~cal p o i n t o f wew, be exact, b ut stdl m o r e ~mportant, it shows that the p u r e Corlolls force is n o t u n i q u e m being able to give the c o r r e c t energy levels accurately. T h e sahent q u e s t i o n is then H o w u m q u e is it? F r o m fig. 2 it m a y be seen that a fit just as g o o d as K e r m a n ' s Is o b t a i n e d for all Z m t h e range 0 < Z < 0 02, c o r r e s p o n d i n g to 21 72 > A t > 13.07

BAND-MIXING EFFECTS IN W185

7

keV, and 0 < H½ < 0 26 keV. It was also checked that outside of this range, no fit was obtainable as good as that at Z = 0. For the largest value of Z in th~s range, nearly half of the Corlohs Interaction has been replaced by a very much smaller amount of the V2 mteracUon This is for the reason already mentioned, that the /I2 Interaction Introduces the reqmred perturbations in second order whereas the C o n o h s force does not do so until thard order, in a perturbation expansion for the energy Both interactions wdl, however, cause mlxang of the bands in first order. Thus we should expect to find the mixing strongest for Z = 0 and becoming weaker as Z increases. In unnormahzed form the new wave-functions become V.o

L(I)

+ g.. L(O-e (O

7tt_(i) '

(12)

from which the parameters a H' L(i) and b n' L(I) of Kerman's notation

~H, L(/) -----a H,L(/-) kV~(/) + bH'L(/) ~P~(I)

(13)

are readily obtained, and are given m table 3 for Z = 0 and for solutions 1 and 2. As anticipated the mixing is conslderably less for large Z than It is for Z = 0 O f TABLE 3 The mtxxng parameters Z

I

]

~

½

all(1) = --bL(I)

0 238

0 339

0 463

0 497

aL(I) = bH(I)

0 971

0 941

0 886

0 868

all(I)

0 231

0 324

0 446

0 472

aL(I)

0 973

0 946

0 894

0 882

an(l)

0 170

0 223

0 320

0 304

aL(I)

0 985

0 975

0 948

0 953

0

00018

00112

course the actual mixing is something that can be checked against experiment by looking at the electromagnetic transmon strengths, for which there is a wealth of experimental data 6 - a) For pure K bands the reduced transmon probabllmes are gwen by B(E2; I,K, ~ Idf) =

5 e2{(i~2K,, K f - K , [ I f K f )

16rr "l"bKtKrt" E2 k - - ]"~If+Kt(I I, , 2K , , -- K,-- KrlI f - Kf)}z(Qs'rf) 2, B(M1; IlK , ~/fKf)

3(eh) ~

= ~

{(I,1K,, Kf-KIIIfKr)

+ b~'K~(- 1)"+K~(I,1K,, - K , - KflI f - Kf)} 2 (GK'r') 2,

(14)

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where bxK,x, = < -- KfIM'(2, -- K f - K,)IK.)
(15)

and M'(2, #) is the approprmte electromagnetm multlpole operator expressed in mtrmsm coordinates. In practice these relatmnships slmphfy considerably For example, we may neglect the off-dmgonal matrix dements for the electric quadrupole transltmns since these are of single-particle magnitude and hence an order of magmrude smaller than the strongly enhanced dmgonal elements. For this reason we also neglect the b ~ x~ since these can only be diagonal when K, = Kf = 1 which does not occur in our problem. The b~'~ ~ m any case vanish unless K, = Kf = ½ due to the trmngle condition. Thus for the present problem, eqs. (14) may be slmphfied to B(E2; I, K - , i , r )

=

5 e2(I. 2 r 0 ] I f K)2(QoK)2, 16rt

3 (\-2-~c,] eh "~2{(I,1K.,Kf_K,[IfKf ) B(M1; I.K, ~ If Kf) = -~ + b i n ( - 1)"+ r~(I,1 K,, - K1- Kfllf-

(16)

Kf))=(Gr'g') 2,

where we have written

Q~

=

QKK,

bm = b ~ for K , =

Kf =½.

= 0 otherwise. For mixed bands, one has just to add the transition amplitudes for the components in the usual manner. The measurements o f Edwards et al. ~) and of Schult and Graber 7) provide us with reformation on twenty of the M1 and E2 transmons between states of these bands, to which we have fitted the parameters o f the above theory m the following comblnatmns.

ct = ~Mc 1--

-~ bm

(G~/Q~o),

g 3 = -~C

9 1 3 --1

h (G~/Q~)=-a~,,

q = Q~o/Q~o.

BAND-MIXING EFFECTS IN W lsa

U n f o r t u n a t e l y only the relative s t r e n g t h s o f t h e t r a n s R l o n s f r o m a given level c o u l d be m e a s u r e d . N e v e r t h e l e s s we are left with rune degrees o f f r e e d o m , w i t h w h i c h o n e m ~ g h t h o p e t o o b t a i n a g o o d m d l c a t t o n o f w h a t ts t h e c o r r e c t a m o u n t o f b a u d m t x m g . TABLE 4

A comparison of the theorettcal wRh the experimental transRlon probabthttes Z Transition

0

00018

00112

texp

ttheor

E2 (%)

texp

ttlaeor

E2 (%)

texp

E2 ttlaeor (%)

BA

(4 5 ~ )

171

171

0

170

170

0

175

175

0

CA CB

(4 1%) (4 2 %)

29 5 23 1

28 2 24 1

100 2

29 3 22 9

28 5 23 5

100 2

31 9 24 9

30 2 26 1

100 2

DB DC

(2 9 %) (3 8 %)

451 1 681

422 1 840

100 1

457 1 701

423 1 883

100 1

53l 1 978

433 2 389

100 0

EA EB EC

(9 2 %) (3 1%) (3 8 %)

182 1 447 177

173 1 429 182

43 7 7

169 1 341 164

172 1 332 165

41 7 7

97 773 95

127 661 69

30 6 4

FA FB FC FD FE

(3 4 ~o) (4 0 ~o) (3 5 %) (4 6 %) (5 0 %)

1 077 102 98 377 121

809 123 102 424 79

100 82 48 1 38

999 95 91 349 113

739 113 95 390 80

100 80 49 1 36

469 44 6 42 7 164 53

350 43 3 43 8 188 62

100 66 62 1 34

GC GD

(3 4 % )

1 747

1 923

90

1749 133

1 911 94

100 8

1 889

133

100 9

1 983

(48%)

151

163

100 3

HB HC HD HE HF HG

(3 7%) (3 5%) (3 4 % ) (3 6%) (5 7%) (8 2%)

1 568 23 470 2 800 1 232 211 590

1 836 11 270 3 306 1203 127 328

100 4 1 100 83 2

1494 22370 2 668 1 174 201 562

1 739 10 590 3 151 1 154 134 300

100 4 1 100 77 2

986 14 760 1 761 775 133 371

1 074 5 830 1 866 827 153 132

100 2 1 100 53 1

JC JD JF JG JH

2209 3 175 6 421 115 309

100 43 100 3 46

2 082 2 962 6 188 93 334

100 42 100 7 42

1202 1 281 4471 42 4 352

100 38 100 46 34

g2

559

530

626

The quantity t is related to the transRlon probabdRy T according to the relationship t =

lO°T](eQo~)

2

and has the damenslon (MeV) 5 m natural umts, l e h = c = 1, 1 m u = 931 MeV The experimental quantmes measured were relative mtensmes and were taken as the mean of the values of refs. s, ~) The accuracy o f each result is gtven m brackets after the transition label The absolute values of texp have been normahzed to g~ve the best agreement wRh expertment for each group of transRlons from a given level

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In table 4 are g w e n the best fits to e x p e r i m e n t fo r each o f the three solutions u n d er consideration. A t the same t i m e a few o th e r t r a n s i t i o n probabdlt~es are predicted, w ~ c h h a v e n o t so far been observed. T a b l e 5 contains the values o f the p a r a m e t e r s leading to the best fits TABLE 5 The parameters leadang to the best fits to the electromagnetic trans~tlon data Z

~.

fl

g3

0 0 0018 0 0112

q:0 0366 q:0 0362 q:0 0374

q:0 122 :~0 122 q:0 125

4-0 0362 4-0 0363 q-0 0303

ff13

q

4-0 0162 4-0 0152 4-0 0098

+ 1 231 + 1 200 + 0 993

It m a y be seen f r o m the values o f Z 2 that e q u a ll y g o o d fits are o b t a i n a b l e with either solution, so that u n f o r t u n a t e l y we are u n a b l e to dlstlngmsh between t h e m by this means. A c o m p a r i s o n o f the predicted B ( E 2 ) values with those o b t a i n e d f r o m C o u l o m b excitation ex p er i m e n t s s), is h o w e v e r m o r e r e w a r d i n g (table 6). In each case the n u TABLE 6

A comparison of some predicted B(E2) values with those obtamed from Coulomb excitation

Experiment B(E2, AB) B(E2, AC) B(E2, AE) B(E2, AF) Qo½

1 524-0 07 2 044-0 08 0 084-0 02 0 304-0 05

Kerman solution Z = 0

Solution 1 Z = 00018

Solution 2 Z = 00112

1 52 2 14 0 091 0 28

1 52 2 15 0 085 0 26

1 52 2 23 0 044 0 12

4 03

4 01

3 91

The B(E2) values are quoted m umts of ez × 10-4s cm4, whde Qot is m umts e × 10-2` cm2 The B(E2) value for the transition AB ~s used to determine Q0½,from which the other B(E2) values are calculated clear q u a d r u p o l e m o m e n t is d e t e r m i n e d by n o r m a h z m g the B( E2 ) v al u e f o r the t r an sition A ~ B. T h e a g r e e m e n t between the o th e r predicted and e x p e r i m e n t a l B ( E 2 ) values is then seen to be excellent for Z = 0 a n d Z = 0 0018, b u t to be rather p o o r for Z = 0 0 1 1 2 . 3. D i s c u s s i o n T h e c o n c l u s i o n to be d r a w n f r o m the a b o v e analysis seems to be that there co u l d very well be c o n t r i b u t i o n s f r o m an i n t e r a c t io n o f the type Vz I n particular there are tw o values o f its off d i a g o n a l m a t r i x e le m e n t HK for w h i ch the energy level s p e c t r u m is fitted very well T h e first value is v e r y small and leads to a w a v e - f u n c t i o n that is n o t

BAND-MIXING EFFECTS IN W 18s

]I

significantly different from that obtained when such an interaction is ignored. The second value IS somewhat larger and effects a considerable reduction in the amount of band m~xing Usually translUon probabilities are a good test of wave-functions, but in the present case, where we only have the branching ratios for the gamma-ray decay of excited states, the data are fitted equally well by either wave-function. To really distinguish between the solutions, one would need to know the absolute hfetimes of some of the levels for which widely &fferent values are predicted. The small number of absolute B(E2) values that have been measured in Coulomb excltatatlon experiments 8) are strongly m favour of the first solution (cf. table 6). Tlus is of course, the solution that one would prefer to be substantiated. One would then be more confident that the V2 interaction could be ignored in future A K = ___1 band mlxmg calculations, without seriously impairing the wave-function. On the other hand, should the second solution prove to be correct, this interaction could certainly not be overlooked without serious error. So far we have made no mention of how an interaction of the type 112 could be present in the nuclear Hamdtonlan Knowing that such an mteractxon does occur In the even nuclei, we have merely gone on to see what would be its effect on an oddmass nucleus There are in fact several ways in which this interaction could arise. It could for example, be simulating a fourth order Coriolls effect, when the coupling would take place through another higher K = ½ band. The V2 term might even be the coherent result of coupling through several tugher K = ½ bands On the other hand a similar Interaction arises if we allow the nucleus to have an axial asymmetry as proposed by Davydov and Pluhppov io) There is then the extra term in the rotational H a m d t o m a n

J,

J2

+I-+J+

which does not however conserve K - ~ It could not therefore &rectly couple the unperturbed K = ½and ~ bands, for both ofwhlch K = ~ I f we now extend our model of the rigid rotor and incorporate wbratlons as well as rotations, then [ ( 1 / J 1 ) ( l / J 2 ) ] becomes an operator which can excite a gamma-wbratlonal mode It is just this term which is beheved 11) to be responsible for the A K = -t-2 band mixing in even nuclei, where the K = 2 intrinsic state is most likely a gamma-vibration based on the ground state It is also reasonable that an odd-mass nucleus should contain some fraction of a gamma-wbratlonal mode in its parentage This would then lead to just such an interaction between the low-lying bands of W 183 as we have discussed m this paper It should be emphasized however that there is no real necessity at present to require the presence of any such interaction at all Certainly an explanation of the W Iss data does not demand it, although it is very usefel m making the fine corrections to the energy level spectrum. Rather the problem has been the reverse Such an lnteraction is known to be ~mportant m even nuclei, and one would like to know the extent

12

I) J ROWE

to whach ~t could also be important m odd-mass nuclei. The answer for W 183 IS that it could be important, but probably only so far as the detads o f the energy level spectrum are concerned. The second solution re&eating a much greater contribution f r o m the interaction ~s however, not entirely ehmmated, and a measurement of some absolute hfetlmes is reqmred to decide unambiguously in favour o f one or the other. One can nevertheless say defimtely that there Is no other reasonable solution mvolwng a larger amount o f Vz. The author wishes to thank Professor B. R. Mottelson for has advice and for suggesting the problem and Professor A Bohr for the hospltahty of the Umverslty Institute for TheoreUcal Physics, Copenhagen where a large part of tbas work was completed. He is also indebted to Dr. J. G u n n for has assistance with the computer programming, to the Ford Foundation for the award o f a research fellowshap at Copenhagen and to the U.K.A.E.A. for a research fellowshap at the A.E.R E. H a r well, where the work was completed.

References 1) 2) 3) 4) 5) 6) 7) 8) 9)

A K Kerman, Mat Fys Medd Dan Vld Selsk 30, No 15 (1956) J J Murray, F Boehm, P M a m e r and J W M Dumond, Phys Rev. 97 (1955) 1007 O B Nielsen, Proc Rutherford Jubilee Int C o n f , Manchester (1961) p 317 S G Nllsson, Mat Fys Medd Dan Vld Selsk 29, No 16 (1955) B R. Mottelson and S G Nllsson, Mat Fys Skr Dan Vld Selsk 1, No 8 (1959) W F Edwards, F Boehm, J Rogers and E J Seppl, (1963) to be pubhshcd O W B Schult and U Graber, preprmt, Laboratory for Tcchmcal Physics, Mumch (1962) O Hansen, M C Olesen, O Skllbreld and B Elbeck, Nuclear Physics 25 (1961) 634 B R Mottelson and A Bohr, Lectures on nuclear structure and energy spectra (1962) to be pubhshed 10) A S Davydov and G F Fdlppov, Nuclear Physics 8 (1958) 237 11) V Radojeclc, A SobleZewskl and Z Szymanskl, Nuclear Physics 38 (1962) 607