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The signs of E2, M1 transition matrix elements of the tungsten, osmium and platinum nuclei
Volume 29B, number 1
THE OF
PHYSICS LETTERS
SIGNS THE
OF
E2,
M1
TUNGSTEN,
TRANSITION OSMIUM
AND
31 March 1969
MATRIX PLATINUM
ELEMENTS NUCLEI
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) values, and the angular correlation measurement of E2/M1 mixing ratios.
R e c e n t a d v a n c e s [1-5] in e x p e r i m e n t a l t e c h n i q u e s make it p o s s i b l e to d e t e c t the influence of the relative sign * of c e r t a i n e l e c t r o m a g n e t i c transition m a t r i x e l e m e n t s . T h e sign of an i n t e r f e r e n c e t e r m a f f e c t s the q u a d r u p o l e m o m e n t s and B(E2) v a l u e s obtained f r o m Coulomb e x c i t a tion data [1]. The sign of the E2/M1 m i x i n g r a tio has been d e t e r m i n e d in s e v e r a l a n g u l a r c o r r e l a t i o n e x p e r i m e n t s f o r the o s m i u m - p l a t i n u m r e g i o n [2-5]. A p r e v i o u s c a l c u l a t i o n [6, 7] gave the wave functions and B(E2), B(M1) v a l u e s f o r a n u m b e r of s t a t e s of the even nuclei 182-186W, 186-192Os and 192-196pt. Since the B(k) v a lu e i s p r o p o r tional to the s q u a r e of the t r a n s i t i o n m a t r i x e l e me nt , the sign of such a m a t r i x e l e m e n t i s not obtainable f r o m ref. 6. T h e r e f o r e , the c a l c u l a tion of e l e c t r o m a g n e t i c m a t r i x e l e m e n t s has b e e n p e r f o r m e d u s i n g the wave functions of ref. 6 and the method $ of r e f . 7. Th e c a l c u l a t e d v a l u e s of r e d u c e d E2, M1 m a t r i x e l e m e n t s a r e given in t a b l e s 1 and 2. With the B o h r - M o t t e l s o n [8] definitions adopted h e r e , t h e s e m a t r i x e l e m e n t s obey the s y m m e t r y r e l a tion
(fll~/~(x)l li>
= (_)/i-If
( )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 r e la 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 intermediate stage. The same remark applies to the expressions (151-152) for B(M1) values.
Saladin et al. [1] have m e a s u r e d and a n a l y z e d the c r o s s s e c t i o n s f o r Coulomb e x c i t a t i o n of the f i r s t 2 + st at e of a n u m b e r of nuclei. They find that the final v a l u e s of Q2 + depend by as much as 40~o on the sign of the i n t e r f e r e n c e t e r m (which is independent of p h ase conventions *)
P3 = Mo2 Mo2, M2, 2
(1)
w h e r e Mif = (il~'fl~(E2)l I f>. This t e r m can be u n d e r s t o o d to a r i s e f r o m i n t e r f e r e n c e b et w een the d i r e c t e x c i t a t i o n am p l i t u d e p r o p o r t i o n a l to M02 and the i n d i r e c t one p r o p o r t i o n a l to M 0 2 ' M 2 ' 2" Saladin [1] has concluded that e q u a l ly good l e a s t s q u a r e s fit to the e x p e r i m e n t a l data is obtained with e i t h e r sign of P3 and that this lack of knowledge of sign c o n s t i t u t e s the l a r g e s t u n c e r t a i n t y in the e x p e r i m e n t a l v a l u e of 02÷. Th e v a r i o u s p o s s i b i l i t i e s a r e b e s t d i s c u s s e d [9] in t e r m s of the sign of the p r o d u c t P 4 = P 3 M 2 2 which is independent of the sign of q u a d r u p o l e m o m e n t (proportional to M22 ) as well as of the iX f a c t o r which is s o m e t i m e s included in the definition of the E2 m a t r i x e l e m e n t . The p r e s e n t cal cu l at i o n p r e d i c t s the sign of P 4 to be n e g a t i v e f o r all nuclei included in table I except f o r 192pt. The sign in the l a t t e r c a s e i s p r o b a bly not significant si n ce the c a l c u l a t e d 0 2 + of 192Pt is quite s m a l l . In this r e g i o n , the c a l c u l at ed O 2 + changes sign f r o m n e g a t i v e (prolafe) f o r tungsten and o s m i u m to p o s i t i v e (oblate) f o r platinum nuclei at N ~ 116. The p r e d i c t e d (see a l s o ref. 6) change of sign has been c o n f i r m e d by the P i t t s b u r g h g r o u p ' s m e a s u r e m e n t s [1] f o r 190-192Os and 194-198pt. It is i n t e r e s t i n g to c o n s i d e r the sign of P4 in the two l i m i t s of c o l l e c t i v e motion. In o r d e r to d e t e r m i n e the sign in the v i b r a t i o n a l l i m i t , we 25
Volume 29B, n u m b e r 1
PHYSICS
LETTERS
31 M a r c h 1969
Table 1 The calculated e l e c t r i c quadrupole m a t r i x e l e m e n t s in e × b. The Bohr-Mottelson definition used here does not include an i~ factor which is s o m e t i m e s used. The diagonal m a t r i x e l e m e n t for I = 3 v a n i s h e s because of the corresponding v e c t o r coupling coefficient [7]. i
f
0+
0'+
2+
182W
184W
2+
2.00
1.94
2 '+
0.17
0.29
2 ''+
0.42
0.37
0.29
2+
186W
186Os
188Os
190Os
192Os
1.87
1.72
0.39
0.44
192pt
1941~
196pt
1.65
1.61
1.60
1.35
1.31
1.20
0.43
0.38
0.19
0.07
-0.07
0.15
-0.21
0.15
0.11
0.08
-0.03
-0.03
0.04
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
-2.37
-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'+
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~+
- 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
-0.72
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
2'+
2"+
2+
0.96
Table 2 The calculated magnetic dipole m a t r i x e l e m e n t s in 10 -2 n.m. The Bohr-Mottelson definition has been used. Values of the diagonal m a t r i x e l e m e n t , which equals a positive constant t i m e s the magnetic m o m e n t , can be obtained from 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"+
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
2'+
26
Volume
29B, number
PHYSICS
1
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 utilizing 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 rather general results that the ratio (J422/M02f)2 = 18 and
P4 = negative in the vibrational limit.
(2)
31 March 1969
LETTERS
Table 3 The E2/Ml mixing ratio 6. The Rose-Brink definition (111, which leads to eq. (5) of the text, has been used.
Nucleus
Transition
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 interference between the direct excitation amplitude proportional to it402~ and the indirect one proportional to MO2M221. Since the matrix element MO2, is small compared to MO2 or M22,, the interference 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, theoretical values are given by the relation 6if = bfi
=
= -0.835 Er (in MeV)
i\\%‘(E2)]lf) (in e x b) (5) [il]m(Ml)//f) (in n.m.) ’
In order not to be exposed to errors in the calculated transition energy, the experimental value 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.
theorv
0.371
+(ll
2+ _) 2+
0.283‘
+(4.7
+ 0.6 a - 0.7)
+5.2
3+ - 2+
0.485
+2.1 b +(10.9_ le5)
+4.8
2’++
2+
0.296
-9.oto-ll.OC
3+ -t 21+
0.308
-6.5 to -8.3 ’
-9.2
3+ - 2+
0.604
+1.9to+2.4
+2.2
2-i _ 2+
0.293
+4 d -(12 _ 2)
2’+ _ 2+
0.333
+(3.8 f 0.1)
p4 = negative if 2’ belongs to a y-band (K=2) (4) 1positive if 2’ belongs to a p-band (K=O)I .
exDt.
2’+ + 2+
(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. However, away from the transitional region, the deformed nuclei are strongly influenced by higher order anharmonicities and the near closed shell nuclei by deviations from the adiabatic approximation. In the rotational limit, one can use the Alaga rules [8] and find that
6 (E2/Ml)
Er in MeV (expt.)
+ 3) a
+7.6
-14.8
c
-19.9 d
+101.4
a See ref. 5. The Biedenharn-Rose definition of 6 adopted in ref. 5 differs in its sign from the RoseBrink 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 because of a number of approximations and limitations [‘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 nearlyforbidden transition matrix elements is comparatively poor. The calculated Ml transition matrix elements (which vanish in the simple collective model where the gyromagnetic ratio is independent of deformation and the Ml operator is just a constant times the nuclear angular momentum operator) are reduced by one to two orders of magnitude compared to the diagonal matrix element. Attempts are being made to improve 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 deformation. 27
Volume 29B, number 1
PHYSICS
T h e a u t h o r i s g r a t e f u l to J . X. S a l a d i n , W . D . H a m i l t o n , a n d Z. Y¢. G r a b o w s k i f o 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 f o r s e n d i n g t h e i r d a t a b e f o r e publication. He is indebted 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 t h e w a r m h o s p i t a l i t y of t h e N i e l s B o h r I n s t i t u t e , a n d t h e D a n i s h R a s k O r s t e d F o u n d a t i o n f o r t h e a w a r d of a f e l lowship.
References 1. J . X . Saladin, invited talk given at the Miami m e e t ing of the Am. Phys. Society, November 1968 (unpublished) ; J. E. Glenn and J. X. Saladin, Phys. Rev. L e t t e r s 20 (1968) 1298. 2. W.D. Hamilton, private communication. 3. W.D. Hamilton and K. E. Davies, Nucl. Phys. A122 (1968) 165.
28
LETTERS
31 March 1969
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. L e t t e r s 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. B a r a n g e r and K. Kumar, in: P e r s p e c t i v e s in modern physics, ed. R. E. Marshek (Wiley - I n t e r Science, N. Y. 1966) p. 35; K. Kumar, in: Nuclear s t r u c t u r e : Dubna Symposium {International Atomic Energy Agency, Vienna, 1968) p. 419.
THE OF
PHYSICS LETTERS
SIGNS THE
OF
E2,
M1
TUNGSTEN,
TRANSITION OSMIUM
AND
31 March 1969
MATRIX PLATINUM
ELEMENTS NUCLEI
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) values, and the angular correlation measurement of E2/M1 mixing ratios.
R e c e n t a d v a n c e s [1-5] in e x p e r i m e n t a l t e c h n i q u e s make it p o s s i b l e to d e t e c t the influence of the relative sign * of c e r t a i n e l e c t r o m a g n e t i c transition m a t r i x e l e m e n t s . T h e sign of an i n t e r f e r e n c e t e r m a f f e c t s the q u a d r u p o l e m o m e n t s and B(E2) v a l u e s obtained f r o m Coulomb e x c i t a tion data [1]. The sign of the E2/M1 m i x i n g r a tio has been d e t e r m i n e d in s e v e r a l a n g u l a r c o r r e l a t i o n e x p e r i m e n t s f o r the o s m i u m - p l a t i n u m r e g i o n [2-5]. A p r e v i o u s c a l c u l a t i o n [6, 7] gave the wave functions and B(E2), B(M1) v a l u e s f o r a n u m b e r of s t a t e s of the even nuclei 182-186W, 186-192Os and 192-196pt. Since the B(k) v a lu e i s p r o p o r tional to the s q u a r e of the t r a n s i t i o n m a t r i x e l e me nt , the sign of such a m a t r i x e l e m e n t i s not obtainable f r o m ref. 6. T h e r e f o r e , the c a l c u l a tion of e l e c t r o m a g n e t i c m a t r i x e l e m e n t s has b e e n p e r f o r m e d u s i n g the wave functions of ref. 6 and the method $ of r e f . 7. Th e c a l c u l a t e d v a l u e s of r e d u c e d E2, M1 m a t r i x e l e m e n t s a r e given in t a b l e s 1 and 2. With the B o h r - M o t t e l s o n [8] definitions adopted h e r e , t h e s e m a t r i x e l e m e n t s obey the s y m m e t r y r e l a tion
(fll~/~(x)l li>
= (_)/i-If
( )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 r e la 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 intermediate stage. The same remark applies to the expressions (151-152) for B(M1) values.
Saladin et al. [1] have m e a s u r e d and a n a l y z e d the c r o s s s e c t i o n s f o r Coulomb e x c i t a t i o n of the f i r s t 2 + st at e of a n u m b e r of nuclei. They find that the final v a l u e s of Q2 + depend by as much as 40~o on the sign of the i n t e r f e r e n c e t e r m (which is independent of p h ase conventions *)
P3 = Mo2 Mo2, M2, 2
(1)
w h e r e Mif = (il~'fl~(E2)l I f>. This t e r m can be u n d e r s t o o d to a r i s e f r o m i n t e r f e r e n c e b et w een the d i r e c t e x c i t a t i o n am p l i t u d e p r o p o r t i o n a l to M02 and the i n d i r e c t one p r o p o r t i o n a l to M 0 2 ' M 2 ' 2" Saladin [1] has concluded that e q u a l ly good l e a s t s q u a r e s fit to the e x p e r i m e n t a l data is obtained with e i t h e r sign of P3 and that this lack of knowledge of sign c o n s t i t u t e s the l a r g e s t u n c e r t a i n t y in the e x p e r i m e n t a l v a l u e of 02÷. Th e v a r i o u s p o s s i b i l i t i e s a r e b e s t d i s c u s s e d [9] in t e r m s of the sign of the p r o d u c t P 4 = P 3 M 2 2 which is independent of the sign of q u a d r u p o l e m o m e n t (proportional to M22 ) as well as of the iX f a c t o r which is s o m e t i m e s included in the definition of the E2 m a t r i x e l e m e n t . The p r e s e n t cal cu l at i o n p r e d i c t s the sign of P 4 to be n e g a t i v e f o r all nuclei included in table I except f o r 192pt. The sign in the l a t t e r c a s e i s p r o b a bly not significant si n ce the c a l c u l a t e d 0 2 + of 192Pt is quite s m a l l . In this r e g i o n , the c a l c u l at ed O 2 + changes sign f r o m n e g a t i v e (prolafe) f o r tungsten and o s m i u m to p o s i t i v e (oblate) f o r platinum nuclei at N ~ 116. The p r e d i c t e d (see a l s o ref. 6) change of sign has been c o n f i r m e d by the P i t t s b u r g h g r o u p ' s m e a s u r e m e n t s [1] f o r 190-192Os and 194-198pt. It is i n t e r e s t i n g to c o n s i d e r the sign of P4 in the two l i m i t s of c o l l e c t i v e motion. In o r d e r to d e t e r m i n e the sign in the v i b r a t i o n a l l i m i t , we 25
Volume 29B, n u m b e r 1
PHYSICS
LETTERS
31 M a r c h 1969
Table 1 The calculated e l e c t r i c quadrupole m a t r i x e l e m e n t s
f
0+
0'+
2+
182W
184W
2+
2.00
1.94
2 '+
0.17
0.29
2 ''+
0.42
0.37
0.29
2+
186W
186Os
188Os
190Os
192Os
1.87
1.72
0.39
0.44
192pt
1941~
196pt
1.65
1.61
1.60
1.35
1.31
1.20
0.43
0.38
0.19
0.07
-0.07
0.15
-0.21
0.15
0.11
0.08
-0.03
-0.03
0.04
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
-2.37
-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'+
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~+
- 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
-0.72
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
2'+
2"+
2+
0.96
Table 2 The calculated magnetic dipole m a t r i x e l e m e n t s
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"+
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
2'+
26
Volume
29B, number
PHYSICS
1
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 utilizing 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 rather general results that the ratio (J422/M02f)2 = 18 and
P4 = negative in the vibrational limit.
(2)
31 March 1969
LETTERS
Table 3 The E2/Ml mixing ratio 6. The Rose-Brink definition (111, which leads to eq. (5) of the text, has been used.
Nucleus
Transition
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 interference between the direct excitation amplitude proportional to it402~ and the indirect one proportional to MO2M221. Since the matrix element MO2, is small compared to MO2 or M22,, the interference 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, theoretical values are given by the relation 6if = bfi
=
= -0.835 Er (in MeV)
i\\%‘(E2)]lf) (in e x b) (5) [il]m(Ml)//f) (in n.m.) ’
In order not to be exposed to errors in the calculated transition energy, the experimental value 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.
theorv
0.371
+(ll
2+ _) 2+
0.283‘
+(4.7
+ 0.6 a - 0.7)
+5.2
3+ - 2+
0.485
+2.1 b +(10.9_ le5)
+4.8
2’++
2+
0.296
-9.oto-ll.OC
3+ -t 21+
0.308
-6.5 to -8.3 ’
-9.2
3+ - 2+
0.604
+1.9to+2.4
+2.2
2-i _ 2+
0.293
+4 d -(12 _ 2)
2’+ _ 2+
0.333
+(3.8 f 0.1)
p4 = negative if 2’ belongs to a y-band (K=2) (4) 1positive if 2’ belongs to a p-band (K=O)I .
exDt.
2’+ + 2+
(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. However, away from the transitional region, the deformed nuclei are strongly influenced by higher order anharmonicities and the near closed shell nuclei by deviations from the adiabatic approximation. In the rotational limit, one can use the Alaga rules [8] and find that
6 (E2/Ml)
Er in MeV (expt.)
+ 3) a
+7.6
-14.8
c
-19.9 d
+101.4
a See ref. 5. The Biedenharn-Rose definition of 6 adopted in ref. 5 differs in its sign from the RoseBrink 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 because of a number of approximations and limitations [‘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 nearlyforbidden transition matrix elements is comparatively poor. The calculated Ml transition matrix elements (which vanish in the simple collective model where the gyromagnetic ratio is independent of deformation and the Ml operator is just a constant times the nuclear angular momentum operator) are reduced by one to two orders of magnitude compared to the diagonal matrix element. Attempts are being made to improve 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 deformation. 27
Volume 29B, number 1
PHYSICS
T h e a u t h o r i s g r a t e f u l to J . X. S a l a d i n , W . D . H a m i l t o n , a n d Z. Y¢. G r a b o w s k i f o 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 f o r s e n d i n g t h e i r d a t a b e f o r e publication. He is indebted 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 t h e w a r m h o s p i t a l i t y of t h e N i e l s B o h r I n s t i t u t e , a n d t h e D a n i s h R a s k O r s t e d F o u n d a t i o n f o r t h e a w a r d of a f e l lowship.
References 1. J . X . Saladin, invited talk given at the Miami m e e t ing of the Am. Phys. Society, November 1968 (unpublished) ; J. E. Glenn and J. X. Saladin, Phys. Rev. L e t t e r s 20 (1968) 1298. 2. W.D. Hamilton, private communication. 3. W.D. Hamilton and K. E. Davies, Nucl. Phys. A122 (1968) 165.
28
LETTERS
31 March 1969
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. L e t t e r s 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. B a r a n g e r and K. Kumar, in: P e r s p e c t i v e s in modern physics, ed. R. E. Marshek (Wiley - I n t e r Science, N. Y. 1966) p. 35; K. Kumar, in: Nuclear s t r u c t u r e : Dubna Symposium {International Atomic Energy Agency, Vienna, 1968) p. 419.