Volume 29B, number 1 P H Y S I C S L E T T E R S 31 March 1969

T H E S I G N S O F E 2 , M1 T R A N S I T I O N M A T R I X E L E M E N T S

O F T H E T U N G S T E N , O S M I U M A N D P L A T I N U M N U C L E I

K. KUMAR The Niels Bohr Institute, University of Copenhagen, Denmark

Received 26 February 1969

We extend a previous calculation and give the signs of E2, M1 transition matrix elements, which are of interest in connection with the Coulomb excitation measurement of quadrupole moments, and B(E2) val- ues, and the angular correlation measurement of E2/M1 mixing ratios.

Recen t advances [1-5] in expe r imen ta l t e ch - n iques make it poss ib le to de tec t the inf luence of the relat ive sign * of ce r t a in e l ec t romagne t i c transition m a t r i x e lements . The sign of an in- t e r f e r e n c e t e r m affects the quadrupole moments and B(E2) va lues obtained f r o m Coulomb exc i t a - t ion data [1]. The sign of the E2/M1 mixing r a - t io has been de t e rmined in s e v e r a l angular c o r - re la t ion expe r imen t s for the o smium-p l a t i num reg ion [2-5].

A prev ious ca lcula t ion [6, 7] gave the wave functions and B(E2), B(M1) va lues for a number of s ta tes of the even nuclei 182-186W, 186-192Os and 192-196pt. Since the B(k) value is p r o p o r - t ional to the square of the t rans i t ion m a t r i x e l e - ment , the sign of such a m a t r i x e lement i s not obtainable f r o m ref . 6. T h e r e f o r e , the ca lcu la - tion of e l ec t romagne t i c m a t r i x e l emen t s has been p e r f o r m e d using the wave functions of ref . 6 and the method $ of ref . 7.

The calcula ted va lues of reduced E2, M1 m a - t r i x e l emen t s a r e given in tables 1 and 2. With the Bohr -Mot te l son [8] defini t ions adopted here , these ma t r ix e l emen t s obey the s y m m e t r y r e l a - t ion

(fll~/~(x)l li> = (_ ) / i - I f <i I Pt ( )l If>.

* The absolute sign of an electromagnetic transition matrix element has, of course, no physical meaning since it depends on the arbitrary choice of the rela- tive phase of the initial and final states. However, the sign of a product (ratio) of matrix elements in which each nuclear state appears an even number of times is independent of phase conventions and can be of physical significance.

:~ Although the expressions (136-138) of ref. 7 are for B (E2) values, no sign has been dropped at an inter- mediate stage. The same remark applies to the ex- pressions (151-152) for B(M1) values.

Saladin et al. [1] have m e a s u r e d and analyzed the c r o s s sec t ions for Coulomb exci ta t ion of the f i r s t 2 + s ta te of a number of nuclei . They find that the final va lues of Q2 + depend by as much as 40~o on the sign of the i n t e r f e r ence t e r m (which is independent of phase conventions *)

P3 = Mo2 Mo2, M2, 2 (1)

where Mif = (il~'fl~(E2)l I f>. This t e r m can be unders tood to a r i s e f rom in t e r f e r ence between the d i r ec t exci ta t ion ampli tude propor t iona l to M02 and the ind i rec t one propor t iona l to M 0 2 ' M 2 ' 2" Saladin [1] has concluded that equa l - ly good l eas t squares fit to the expe r imen ta l data is obtained with e i ther sign of P3 and that this lack of knowledge of sign const i tu tes the l a rge s t uncer ta in ty in the expe r imen ta l value of 0 2 ÷ .

The va r ious poss ib i l i t i e s a r e bes t d i s - cussed [9] in t e r m s of the sign of the product P4 = P3M22 which is independent of the sign of quadrupole moment (proport ional to M22 ) as well as of the iX fac tor which is s o m e t i m e s included in the definit ion of the E2 ma t r ix e lement . The p r e sen t calculat ion p red ic t s the sign of P4 to be negat ive for all nuclei included in table I except for 192pt. The sign in the l a t t e r case i s p r o b a - bly not s ignif icant s ince the calcula ted 02+ of 192Pt is quite smal l . In this region, the ca l cu - la ted O 2 + changes sign f rom negat ive (prolafe) for tungsten and osmium to pos i t ive (oblate) for plat inum nuclei at N ~ 116. The p red ic ted (see a lso ref . 6) change of sign has been conf i rmed by the Pi t t sburgh g roup ' s m e a s u r e m e n t s [1] for 190-192Os and 194-198pt.

It is in te res t ing to cons ider the sign of P4 in the two l imi t s of co l lec t ive motion. In o rde r to d e t e r m i n e the sign in the v ibra t iona l l imi t , we

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Volume 29B, number 1 P H Y S I C S L E T T E R S 31 March 1969

Table 1 The calculated e lec t r i c quadrupole ma t r ix e lements <il ~fL(E2)I ] f > in e × b. The Bohr-Mot te lson definition used here does not include an i~ fac tor which is somet imes used. The diagonal ma t r ix e lement for I = 3 van ishes because of

the corresponding vec tor coupling coefficient [7].

i f 182W 184W 186W 186Os 188Os 190Os 192Os 192pt 1941~ 196pt

0 + 2 + 2.00 1.94 1.87 1.72 1.65 1.61 1.60 1.35 1.31 1.20

2 '+ 0.17 0.29 0.39 0.44 0.43 0.38 0.19 0.07 -0.07 0.15

2 ''+ 0.42 0.37 0.29 -0.21 0.15 0.11 0.08 -0.03 -0.03 0.04

0 '+ 2 + 0.41 0.48 0.51 0.40 -0.35 0.28 0.20 0.26 0.38 0.46

2 '+ -1.73 -1.58 -1.36 -0.88 0.79 -0.73 -0.85 0.65 0.56 -0.49

2 "+ 0.92 1.32 1.77 -1.57 -1.60 1.59 1.53 -1.18 -1.12 1.08

2 + 2 + -2.37 -2.24 -2.05 -1.86 -1.53 -1.18 -0.47 0.12 0.65 0.92

2 '+ 0.72 0.94 1.23 1.13 1.42 1.64 1.93 1.67 1.50 -1.24

2 "+ 0.29 0.19 0.10 0.10 -0.12 -0.12 -0.03 -0.04 -0.05 0.06

3 + 0.65 0.70 0.72 0.72 0.68 0.59 0.30 0.11 -0.10 -0.21

4 + 3.29 3.22 3.14 2.86 2.75 2.67 2.65 2.24 2.19 2.07

2 '+ 2 '+ 0.18 0.97 1.38 1.72 1.47 1.16 0.45 -0.06 -0.56 -0.79

2 "+ 2.22 1.92 1.41 -0.42 0.21 0.13 0.22 -0.38 -0.35 -0.46

3 + -1.33 -1.67 -2.00 -2.21 -2.29 -2.32 -2.27 -1.97 -1.86 1.65

4 + -0.46 -0.40 -0.32 -0.05 0.10 0.15 0.02 0.09 -0.08 -0.14

2 "+ 2 ~+ - 0 .41 -1.45 -2.27 -1.84 -1.57 -0.92 -0.75 -0.59 -0.51 0.35

3 + -2.28 -1.93 -1.53 1.13 -1.08 -1.12 -1.19 -1.05 -1.14 0.83

4 + 0.58 0.77 0 .96 -0.72 0.62 0.48 0.47 -0.46 -0.51 0.69

3 + 4 + -0.62 -0.74 -0.92 -0.91 -1.17 -1.35 -1.44 -1.26 -1.13 -0.90

4 + 4 + -3.06 -2.91 -2.70 -2.35 -1.82 -1.24 -0.54 0.25 0.88 1.28

Table 2 The calculated magnet ic dipole ma t r ix e lements <illC~(M1)ll f> in 10 -2 n.m. The Bohr-Motte lson definition has been used. Values of the diagonal ma t r ix e lement , which equals a positive constant t imes the magnet ic moment , can be

obtained f rom ref. 6.

i f 182W 184W 186W 186Os 188Os 190Os 192Os 192pt 194pt 196pt

2 + 2 '+ 4.02 1.37 -0.29 -4.05 -5.94 -6.70 -8.79 2.78 1.84 0.34

2 "+ -5.58 -5.48 -4.66 2.26 1.99 4.45 3.49 4.31 4.03 -4.85

3 + -1.49 -1.99 -1.91 -3.45 -3.44 -2.84 -2.52 -2.55 -2.96 -2.21

2 '+ 2 "+ 0.87 1.13 1.34 -2.93 1.96 0.75 0.94 0.48 1.02 0.74

3 + 0.65 2.32 3.45 3.31 4.86 5.37 6.87 -5.51 -3.71 0.99

2" 3 + 3.51 3.94 3.41 -3.30 2.91 2.84 3.75 -1.61 -1.14 -0.30

3 + 4 + 1.50 2.29 2.80 3.80 4.80 5.12 6.41 -4.90 -3.91 -1.93

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Volume 29B, number 1 PHYSICS LETTERS 31 March 1969

expand the wave functions for the O+, 2+ and 2’+ states up to three phonon states [6,7, lo]. The phonon mixing amplitudes are obtained by utiliz- ing the fact that in our calculation [6] most of the anharmonicity comes from the prolate-oblate difference term in the potential energy function which is proportional to 63 cos 3~. We get rath- er general results that the ratio

(J422/M02f)2 = 18 (2)

and P4 = negative in the vibrational limit. (3)

To leading order, these results are independent of the magnitude and sign of the cubic term in p and are not affected by a possible /34 term. How- ever, away from the transitional region, the de- formed nuclei are strongly influenced by higher order anharmonicities and the near closed shell nuclei by deviations from the adiabatic approxi- mation. In the rotational limit, one can use the Alaga rules [8] and find that

p4 = negative if 2’ belongs to a y-band (K=2) (4)

1 I positive if 2’ belongs to a p-band (K=O) .

The available experimental data [l] do not discriminate between the two signs of P4 and P-J. It would be interesting to extend this kind of analysis to the excitation of the second 2+ state. There, the same term P3 would cause interfer- ence between the direct excitation amplitude proportional to it402~ and the indirect one pro- portional to MO2 M221. Since the matrix element MO2, is small compared to MO2 or M22,, the in- terference term P3 can be expected to have a significant influence on the final value of the cross-over transition probability B (E2; 0+ - 2I+). A combined analysis of excitation data for the first and second 2+ states could lead to a unique experimental determination of the sign of P3.

The sign and magnitude of the E2/Ml mixing ratio 6 have recently been determined [2-51. With the Rose-Brink [ll] definition of 6, theo- retical values are given by the relation

6if = bfi =

= -0.835 Er (in MeV) i\\%‘(E2)]lf) (in e x b)

[il]m(Ml)//f) (in n.m.) ’ (5)

In order not to be exposed to errors in the cal- culated transition energy, the experimental val- ue of E,, is used. Comparison with experiment is given in table 3. In view of the remarkably good agreement in 1gO-1S20s and lg2-124Pt, the large discrepancy in 186pt is somewhat puzzling but not completely unexpected.

Table 3 The E2/Ml mixing ratio 6. The Rose-Brink defini- tion (111, which leads to eq. (5) of the text, has been

used.

Er in 6 (E2/Ml) Nucleus Transition MeV

(expt.) exDt. theorv

2’+ + 2+

2+ _) 2+

3+ - 2+

2’++ 2+

3+ -t 21+

3+ - 2+

2-i _ 2+

2’+ _ 2+

0.371 +(ll + 3) a +7.6

0.283‘ +(4.7 + 0.6 a - 0.7)

+5.2

0.485 +2.1 b

+(10.9_ le5) +4.8

0.296 -9.oto-ll.OC -14.8

0.308 -6.5 to -8.3 ’ -9.2

0.604 +1.9to+2.4 c +2.2

0.293 +4 d -(12 _ 2) -19.9

0.333 +(3.8 f 0.1) d

+101.4

a See ref. 5. The Biedenharn-Rose definition of 6 adopted in ref. 5 differs in its sign from the Rose- Brink definition.

b See ref. 4. c See ref. 3. Similar values have been communicated

in ref. 4. d Preliminary results communicated in ref. 2.

The E2/Ml mixing ratio provides a rather sensitive test of nuclear wave functions. We do not expect the present theory to fare too well be- cause of a number of approximations and limita- tions [‘I, 121. Two of these are most relevant to the mixing ratio. (i) Since the matrix elements are evaluated by summing over a large number of terms corresponding to different points of a 67 mesh [?‘I, the numerical accuracy of nearly- forbidden transition matrix elements is com- paratively poor. The calculated Ml transition matrix elements (which vanish in the simple col- lective model where the gyromagnetic ratio is independent of deformation and the Ml operator is just a constant times the nuclear angular mo- mentum operator) are reduced by one to two or- ders of magnitude compared to the diagonal ma- trix element. Attempts are being made to im- prove the numerical accuracy. (ii) Deviations from the adiabatic assumption of the present, microscopic treatment of collective states may be particularly serious for Ml matrix elements, which in contrast to E2 matrix elements are only mildly affected by nuclear collectivity or de- formation.

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Volume 29B, number 1 P H Y S I C S L E T T E R S 31 March 1969

The au tho r i s g r a t e f u l to J . X. Sa lad in , W . D . H a m i l t o n , and Z. Y¢. G r a b o w s k i fo r s t i m u l a t i n g c o r r e s p o n d e n c e and fo r s e n d i n g t h e i r da t a b e - f o r e pub l i ca t i on . He i s i n d e b t e d to P r o f e s s o r s B. M o t t e l s o n and A. W i n t h e r f o r v a l u a b l e d i s c u s - s i o n s , P r o f e s s o r A. B o h r f o r the w a r m h o s p i t a l - i ty of the N i e l s B o h r I n s t i t u t e , and the D a n i s h R a s k O r s t e d Founda t ion f o r t he a w a r d of a f e l - l owsh ip .

References

1. J .X . Saladin, invited talk given at the Miami meet - ing of the Am. Phys. Society, November 1968 (unpub- lished) ; J . E. Glenn and J. X. Saladin, Phys. Rev. Let ters 20 (1968) 1298.

2. W.D. Hamilton, private communication. 3. W.D. Hamilton and K. E. Davies, Nucl. Phys. A122

(1968) 165.

4. z . w . Grabowski, private communication. 5. R.L. Robinson, F .K. McGowan, P.H. Stelson, W.T.

Milner and R. O. Sayer, Nucl. Phys. A123 (1969) 193.

6. K. Kumar and M. Baranger, Phys. Rev. Le t te rs 17 (1966) 1146; Nucl. Phys. A122 (1968) 273.

7. K. Kumar and M. Baranger, Nuel. Phys. A92 (1967) 608.

8. A. Bohr and B. R. Mottelson, Mat. Fys. Medd. Dan. Vid. Selsk. 27, No, 16 (1953); K. Alder, A. Bohr, T. Huus, B. Mottelson and A. Winther, Revs. Mod. Phys. 28 (1956) 432.

9. A.Winther, private communication. 10. T. Tamura and T. Udagawa, Phys. Rev. 150 {1966)

783. 11. H.J . Rose and D. M. Brink, Revs. Mod. Phys. 39

(1967) 306. 12. M. Baranger and K. Kumar, in: Perspec t ives in

modern physics, ed. R. E. Marshek (Wiley - In te r - Science, N. Y. 1966) p. 35; K. Kumar, in: Nuclear s t ructure: Dubna Symposi- um {International Atomic Energy Agency, Vienna, 1968) p. 419.

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