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E2 transitions within the ground-state band of deformed even-even nuclei
NUCLEAR
AND
INSTRUMENTS
METHODS
146 (1977) 189-192 ; 0
E2 TRANSITIONS WITHIN THE GROUND-STATE OF DEFORMED EVEN-EVEN NUCLEI A. COVELLO lstituto
di Fisica
lstituto
Nazionale
NORTH-HOLLAND
PUBLISHING
CO.
BAND
and G. GIBERTI Teorica
dell’llniversitci
di Fisica
Nucleare.
and Naples.
Italy
1. Introduction The E2 transitions within the ground-state band of deformed even-even nuclei have been the object of many experimental investigations. In the last few years B(E2) values for transitions between high-spin YRAST states in several rare-earth nuclei have been reported’-r6). In particular, evidence has been presentedi3) for significant retardations in some even Dy, Er and Yb isotopes compared to the rotational predictions. Knowledge of E2 transition rates is of crucial importance for the understanding of the nuclear rotational motion. As is well known, several rotational nuclei undergo a sudden increase in their moments of inertia around I= 14. If this increase is attributed to the destruction of the pair correlations by the Coriolis force (the Coriolis antipairing effect), then the reduced E2 transition probabilities between states in the “phase transition” region should be severely hindered. On the other hand, the model of Stephens and Simoni7), which involves the decoupling of two i,3,z neutrons from the core, indicates that the B(E2) values for the decay of high-spin states are changed only slightly from the rotational values. In the present report we shall discuss from the phenomenological point of view the problem of the deviations from the rotational behaviour of the ground-state band transition probabilities. In particular, we shall examine, on the basis of the present experimental information, the possibility of explaining the spin dependence of the B(E2) values in terms of a simple band-mixing theory.
B(E2; I-+1-2)
B(E2; 2+0); x
l+ [
‘df;-“;
x
1’ (1)
LX(Z2--Z-2) 2 1+3c(
which gives the Z-dependence of the E2 transition probabilities within the ground-state band in terms of the parameter a. This relation, which has been first suggested in a slightly different form by Bohr and Mottelson (see ref. 18) can be easily derived by a band-mixing calculation in lowest-order perturbation theory. The coupling Hamiltonian ZZ, may be written”.*O) in the usual notation as Z-Z,= h,(z:+z;)
+ k,(z:-Zjt),
(2)
where the ho and h2 terms represent the interactions between the ground-state band and excited bands with K = 0 and K = 2, respectively. Making use of the perturbed wave functions and neglecting second-order terms in the coupling strengths one obtains the following expression for E2 transition probabilities within the ground-state band: B(E2; Z-+Z-2)
= B,(E2; 240)
; w
x
X [I + t%(Z2-Z+ 1) + 3cC212, (3) with a= ~,+a,. Here a0 and a, are constants which are proportional to the strengths of the couplings, and B. (E2; 2 --*0) is the pure rotational value. This expression can be written in the alternative form B(E2; Z-+1-2)
2. The effect of band mixing on the E2 transitions within the ground-state band The phenomenological analysis of B(E2) values, which will be discussed in the next section, is based on the following formula:
=
= B(E2; 2-O) x
FB
x
I + a(Z2--Z-2) 1+3(Cr+cl,)
1’ 2
(4)
which reduces to eq. (1) by neglecting a,. If one neglects both the small quantities a and a, in the II.
NUCLEAR
SPECTROSCOPY
190
A. COVELLO
AND G. G I B E R T I
TABLE 1
TABLE 2
values obtained from eq. (7) for various values of I m. Im ~z× 103
4
6
8
10
-5.0
- 1.4
-0.55
-0.27
Values of ~z as obtained by means of eq. (1). The experimental B(E2) values used in the least-squares fits are taken from the references listed in the last column. If denotes the largest value of 1. Nucleus
denominator, eq. (4) becomes B(E2; 1--,1-2) = B(E2; 2 ~ 0 ) 15 I ( I - 1 ) 2 412-1 x
x [1 + ~(I 2 - I - 2 ) ] 2,
(5)
which is just the same as that given by Symons and Douglas in ref. 18. This latter expression is the one usually reported in the literature. Let us now examine the behaviour of the E2 transition rates predicted by the band-mixing model21). The rotational B0 (E2; I--, 1 - 2) increases with I according to the well-known relation
15 I ( I - 1) Bo(E2; 1--.1-2) = 3-~n Q~ 4 1 2 - 1 ,
(6)
and for large values of I tends to constant value ~ B 0 (2 --, 0). For ~z>0 the B(E2; 1---,I - 2) predicted by the band-mixing model increases with 1 more rapidly than the rotational one. Actually, the slope of the B(E2) versus I curve may be taken as a measure of the amount of mixing. For ~z< 0, it is easily seen that B(E2;I---,I-2) first increases reaching a maximum for some value of I = Ira, and then decreases as long as ( F - l + l ) < l / l ~ z l . For ( / 2 - 1 + 1 ) = 1/[~z], B ( E 2 ; I + I - 2 ) has a minimum and then keeps increasing for all values of I. The parameter ~z is related to Im (of course, I m need not be an even integer) through = -
412m- 2I,n + 1 (412m--2I~+ 1) (I2--Im+ 1) + + 2Ira(Ira--1) (2Ira--1) (412--1)
(7)
By means of eq. (7) one may calculate the o~-values corresponding to various values of Ira. It turns out that cr decreases by an order of magnitude when going from Im= 4 to 1~ = 10 (see table 1). Thus, according to eq. (1), for o~<0 the position of the maximum in the B(E2) versus I curve is directly related to the amount of mixing, namely the higher the value of 1.1 the smaller the mixing. 3. A n a l y s i s o f e x p e r i m e n t a l data
Eqs. (1) and (5)have been used by several authors 9,22) to analyze the experimental B(E2) values
152Sm /54Sm 154Gd J56Gd 16°Dy 162Dy ~64Dy ~58Er 16°Er ¿6°yb 162yb 164yb 166yb
lj
~z× 103
Ref.
10 8 10 10 12 12 12 16 10 8 8 18 10
2.0 _+0.4 0.67_+0.66 1.7 _+0.5 0.30_+0.33 -0.88_+0.30 --0.82_+0.30 -0.33_+0.34 -0.23-+0.46 -0.82-+ 1.I 2.4 _+ 1.5 - 1 . 4 _+2.7 -0.58_+0.27 -0.57_+0.79
9 5 9 9 10, 13 10, 13 10, 13 8 1 14 7 15 16
for (1--,1-2) transitions within the ground-state band of deformed even-even nuclei. In view of some additional recent experiments 1°-~6) and bearing in mind the possibility of systematic measurements of the lifetimes of high-spin rotational states opened up by Coulomb excitation with heavy ions, it is of interest to further inspect the possible range of validity of these expressions. We are presenting here the results obtained by using eq. (1) with B ( E 2 ; 2 ~ 0 ) and ~z as parameters to fit the B(E2) values for transitions wihin the ground-state band in a number of deformed even-even nuclei, for which enough experimental information is available. The values of the parameter ~z are given in table 2*. The dependence of the B(E2) values on I is shown in figs. la and lb for 152'1545m, 154't56Gd, 16°,162Dy, 16°Er and 162yb. The normalized Z 2 is less than 0.6 for all cases, with the exceptions of r64Dy(z2= 0.9) and 158Effz2 = 1.1). * The s-values for 16°,162,164Dy have been obtained by using for the (12~10), (10~8) and (8~6) transitions the B(E2) values derived from the lifetimes measured by Kearns et al) 3) by Doppler-broadened lineshape analysis. It should be mentioned that Sayer et al) l) have obtained much larger B(E2) values for the (12~10) and (10~8) transitions by comparing their experimental Coulomb excitation probabilities with theoretical values calculated with the Winther - de Boer computer code. However, these latter data are affected by very large uncertainties and do not take into account all sources of error. t For ~58Er the fit to the data does not include the (12~10) transition, for which only an upper limit of the lifetime has been set in ref. 8.
191
E2 T R A N S I T I O N S
152 Sm
(a)
~60 Dy
1546d
(b)
162 Dy
I 1.8
1
//
1.6 1.4
J ,!
//
1.2 1.0 0.8 f I
I
1'54S m
T
I'56G'd
T
'160E r
'
,
8
10
~62y' b
1--4
"* 1.8 cq LU c'n 16
14 1.2 1.0
b; off
j
0.8
2
4
6
8
10
12
2
4
6
8
10
12
2
4
6
12 I 2
4
6
8
10
12
Fig. 1. Dependence of the B(E2) values on 1. The solid lines are least-squares fits of eq. (1) to the experimental data using B(E2; 2 ~ 0 ) and cz as parameters. The dashed lines show the
dependence predicted by the rotational model and result from the fitting of eq. (6) with Q~ as free parameter.
It appears that, to the present accuracy of the data, eq. (1) gives a good fit of the experimental B(E2;I--,I-2) values for transitions within the ground-state band up to relatively high angular momenta. It should be noted, however, that seven of the thirtheen cases considered yield values of cx consistent with zero and are of little use in testing the band-mixing model. An interesting feature observed in figs. la and lb is the remarkable similarity of the behaviour of the E2 transition rates for the three isotonic pairs of nuclei 152Sm-lS4Gd, 154Sm-]S6Gd and 1 6 ° E r J 6 2 y b . Concerning the six nuclei, for which there are significant deviations from the rotational model behaviour, ac is positive for 152Sm, 154Gd and 16°yb, whereas it is negative for 16°'162Dyand 164yb 21). AS discussed in the preceding section, the appearance of negative at-values is related to hindrances in the B (E2) values.
4. Concluding remarks We have seen that the experimental B(E2;I--,I-2) values for transitions within the ground-state band in several rare-earth nuclei are well fitted by the expression (1). In particular, this expression, for at<0, can explain the retardations of E2 transition rates observed in some nucleiC3:5). This is certainly a point of interest, as retardations of E2 transition rates are generally expected as a consequence of a phase transition from a superconducting to a normal state. In this connection, it is worth noting that, at the present time, there is no definite evidence that the backbending effect is associated with hindrances in the E2 transition rates. For instance, in 164yb, which is a strongly backbending nucleus, some hindrances seem to have been observed~5). On the other hand, the B(E2) values measured by Ward et al. 8) in 158Er, which also exhibits a large amount of backbendI1. N U C L E A R
SPECTROSCOPY
192
A.
COVELLO
AND
ing, seem to be consistent with rotational values. Unfortunately, the errors on most lifetimes of high-spin rotational states, which have been recently measured using Doppler shift techniques, are too large to permit any definite conclusion on this point. As is well known, there have been many attempts to explain in a consistent way the deviations from the rotational values of both the level spacings and the E2 transition rates in terms of a simple band-mixing model, but they have not been generally successful (see for instance ref. 5). In these attempts, the K = 0 and K = 2 excited bands which mix into the ground-state band have been usually identified with the /3- and y-vibrational bands, respectively. This might indicate 5) that there are excited bands which contribute to the deviations in the energy level spacings, but have little effect on the E2 transition rates. It should be stressed, however, that formula (1) holds irrispective of the specific nature of the excited bands. Clearly, more accurate experimental data are needed in order to make a stringent test of the expression (1). However, the present evidence seems to suggest that the deviations from the rotational values of the E2 transition rates within the ground-state band of deformed even-even nuclei may be explained in terms of a simple band-mixing mechanism. References l) R. M. Diamond, F. S. Stephens, W. H. Kelly and D. Ward, Phys. Rev. Lett. 22 (1969) 546. 2) I. A. Fraser, .J.S. Greenberg, S. H. Sie, R. G. Stokstad and D. A. Bromley, Phys. Rev. Lett. 23 (1969) 1051. 3) R. O. Sayer, P. H. Stelson, F. K. McGowan, W, T. Milner and R. L. Robinson, Phys. Rev. 4C (1970) 1525.
G.
4)
GIBERTI
R. M. Diamond, F. S. Stephens, K. Nakai and R. Nordhagen, Phys. Rev. 3C (1971) 344. 5) R. M. Diamond, G. D. Symons, J. L. Quebert, K. H. Maier, J. R. Leigh and F. S. Stephens, Nucl. Phys. AI84 (1972) 481. 6) N . Rud, G. T. Ewan, A. Christy, D. Ward, R. L. Graham and J. S. Geiger, Nucl. Phys. AI91 (1972) 545. 7) B. Bochev, S. A. Karamian, T. Kutsarava, E. Nadjakov, Ts. Venkova and R. Kalpakchieva, Physica Scripta 6 (1972) 243. 8) D. Ward, H. R. Andrews, J. S. Geiger, R. L. Graham and J. F. Sharpey-Schafer, Phys. Rev. Lett. 30 (1973) 493. 9) D, Ward, H. R. Andrews, R. L. Graham, J. S. Geiger and S. H. Sie, 24th Nat. Conf. on Nuclear spectroscopy and structure oi' the atomic nucleus. Kharkov (1974). 10) R. N. Oehlberg, L. L. Riedinger, A. E. Rainis, A. G. Schmidt, E. G. Funk and J. W. Mihelich, Nucl. Phys. A219 (1974) 543. ~l) R. O. Sayer, E. Eichler, N. R. Johnson, D. C. Hensley and L. L. Riedinger, Phys. Rev. 9C (1974) 1103. 12) D. Ward, Proc. Int. Conf. on Reactions between complex nuclei (eds. R. L. Robinson, F. K. Mc Gowan, J. B. Ball and J. H. Hamilton; North-Holland Publ. Co., Amsterdam, 1974) vol. 2, p. 417. 73) F. Kearns, G. D. Dracoulis, T. lnamura, J. C. Lisle and J. C. Willmott, J. Phys. 7A (1974) 11. 14) B. Bochev, S, A. Karamian, T. Kutsarova and V. G. Subbotin, Preprint of the Joint Institute for Nuclear Research, Dubna (1974). 15) B. Bochev, S. lliev, R. Kalpakchieva, S. A. Karamian and T. Kutsarova, Preprint of the Joint Institute for Nuclear Research, Dubna (1975). 16) B. Bochev, R. Kalpakchieva, S. A. Karamian, T. Kutsarova, E. Nadjakov abd V. G. Subbotin, Preprint of the Joint Institute for Nuclear Research, Dubna (1975). 17) F. S. Stephens and R. S. Simon, Nucl. Phys. A183 (1972) 257. 18) G. D. Symons and A. C. Douglas, Phys. Lett. 24B (1967) 11. 19) p. O. Lipas, Nucl. Phys. 39 (1962) 468. 20) A. Bohr and B. Mottelson, N u d e a r structure, vol 2 (W. A. Benjamin, Inc., Reading, Massachusetts, 1975), ch. 4. 21) A. Covello and G. Giberti, to be published in Lett. Nuovo Cimento (1976). 22) D. Ward, R. L. Graham, J. S. Geiger, N. Rud and A. Christy, Nucl. Phys. A196 (1972) 9, and references therein.
AND
INSTRUMENTS
METHODS
146 (1977) 189-192 ; 0
E2 TRANSITIONS WITHIN THE GROUND-STATE OF DEFORMED EVEN-EVEN NUCLEI A. COVELLO lstituto
di Fisica
lstituto
Nazionale
NORTH-HOLLAND
PUBLISHING
CO.
BAND
and G. GIBERTI Teorica
dell’llniversitci
di Fisica
Nucleare.
and Naples.
Italy
1. Introduction The E2 transitions within the ground-state band of deformed even-even nuclei have been the object of many experimental investigations. In the last few years B(E2) values for transitions between high-spin YRAST states in several rare-earth nuclei have been reported’-r6). In particular, evidence has been presentedi3) for significant retardations in some even Dy, Er and Yb isotopes compared to the rotational predictions. Knowledge of E2 transition rates is of crucial importance for the understanding of the nuclear rotational motion. As is well known, several rotational nuclei undergo a sudden increase in their moments of inertia around I= 14. If this increase is attributed to the destruction of the pair correlations by the Coriolis force (the Coriolis antipairing effect), then the reduced E2 transition probabilities between states in the “phase transition” region should be severely hindered. On the other hand, the model of Stephens and Simoni7), which involves the decoupling of two i,3,z neutrons from the core, indicates that the B(E2) values for the decay of high-spin states are changed only slightly from the rotational values. In the present report we shall discuss from the phenomenological point of view the problem of the deviations from the rotational behaviour of the ground-state band transition probabilities. In particular, we shall examine, on the basis of the present experimental information, the possibility of explaining the spin dependence of the B(E2) values in terms of a simple band-mixing theory.
B(E2; I-+1-2)
B(E2; 2+0); x
l+ [
‘df;-“;
x
1’ (1)
LX(Z2--Z-2) 2 1+3c(
which gives the Z-dependence of the E2 transition probabilities within the ground-state band in terms of the parameter a. This relation, which has been first suggested in a slightly different form by Bohr and Mottelson (see ref. 18) can be easily derived by a band-mixing calculation in lowest-order perturbation theory. The coupling Hamiltonian ZZ, may be written”.*O) in the usual notation as Z-Z,= h,(z:+z;)
+ k,(z:-Zjt),
(2)
where the ho and h2 terms represent the interactions between the ground-state band and excited bands with K = 0 and K = 2, respectively. Making use of the perturbed wave functions and neglecting second-order terms in the coupling strengths one obtains the following expression for E2 transition probabilities within the ground-state band: B(E2; Z-+Z-2)
= B,(E2; 240)
; w
x
X [I + t%(Z2-Z+ 1) + 3cC212, (3) with a= ~,+a,. Here a0 and a, are constants which are proportional to the strengths of the couplings, and B. (E2; 2 --*0) is the pure rotational value. This expression can be written in the alternative form B(E2; Z-+1-2)
2. The effect of band mixing on the E2 transitions within the ground-state band The phenomenological analysis of B(E2) values, which will be discussed in the next section, is based on the following formula:
=
= B(E2; 2-O) x
FB
x
I + a(Z2--Z-2) 1+3(Cr+cl,)
1’ 2
(4)
which reduces to eq. (1) by neglecting a,. If one neglects both the small quantities a and a, in the II.
NUCLEAR
SPECTROSCOPY
190
A. COVELLO
AND G. G I B E R T I
TABLE 1
TABLE 2
values obtained from eq. (7) for various values of I m. Im ~z× 103
4
6
8
10
-5.0
- 1.4
-0.55
-0.27
Values of ~z as obtained by means of eq. (1). The experimental B(E2) values used in the least-squares fits are taken from the references listed in the last column. If denotes the largest value of 1. Nucleus
denominator, eq. (4) becomes B(E2; 1--,1-2) = B(E2; 2 ~ 0 ) 15 I ( I - 1 ) 2 412-1 x
x [1 + ~(I 2 - I - 2 ) ] 2,
(5)
which is just the same as that given by Symons and Douglas in ref. 18. This latter expression is the one usually reported in the literature. Let us now examine the behaviour of the E2 transition rates predicted by the band-mixing model21). The rotational B0 (E2; I--, 1 - 2) increases with I according to the well-known relation
15 I ( I - 1) Bo(E2; 1--.1-2) = 3-~n Q~ 4 1 2 - 1 ,
(6)
and for large values of I tends to constant value ~ B 0 (2 --, 0). For ~z>0 the B(E2; 1---,I - 2) predicted by the band-mixing model increases with 1 more rapidly than the rotational one. Actually, the slope of the B(E2) versus I curve may be taken as a measure of the amount of mixing. For ~z< 0, it is easily seen that B(E2;I---,I-2) first increases reaching a maximum for some value of I = Ira, and then decreases as long as ( F - l + l ) < l / l ~ z l . For ( / 2 - 1 + 1 ) = 1/[~z], B ( E 2 ; I + I - 2 ) has a minimum and then keeps increasing for all values of I. The parameter ~z is related to Im (of course, I m need not be an even integer) through = -
412m- 2I,n + 1 (412m--2I~+ 1) (I2--Im+ 1) + + 2Ira(Ira--1) (2Ira--1) (412--1)
(7)
By means of eq. (7) one may calculate the o~-values corresponding to various values of Ira. It turns out that cr decreases by an order of magnitude when going from Im= 4 to 1~ = 10 (see table 1). Thus, according to eq. (1), for o~<0 the position of the maximum in the B(E2) versus I curve is directly related to the amount of mixing, namely the higher the value of 1.1 the smaller the mixing. 3. A n a l y s i s o f e x p e r i m e n t a l data
Eqs. (1) and (5)have been used by several authors 9,22) to analyze the experimental B(E2) values
152Sm /54Sm 154Gd J56Gd 16°Dy 162Dy ~64Dy ~58Er 16°Er ¿6°yb 162yb 164yb 166yb
lj
~z× 103
Ref.
10 8 10 10 12 12 12 16 10 8 8 18 10
2.0 _+0.4 0.67_+0.66 1.7 _+0.5 0.30_+0.33 -0.88_+0.30 --0.82_+0.30 -0.33_+0.34 -0.23-+0.46 -0.82-+ 1.I 2.4 _+ 1.5 - 1 . 4 _+2.7 -0.58_+0.27 -0.57_+0.79
9 5 9 9 10, 13 10, 13 10, 13 8 1 14 7 15 16
for (1--,1-2) transitions within the ground-state band of deformed even-even nuclei. In view of some additional recent experiments 1°-~6) and bearing in mind the possibility of systematic measurements of the lifetimes of high-spin rotational states opened up by Coulomb excitation with heavy ions, it is of interest to further inspect the possible range of validity of these expressions. We are presenting here the results obtained by using eq. (1) with B ( E 2 ; 2 ~ 0 ) and ~z as parameters to fit the B(E2) values for transitions wihin the ground-state band in a number of deformed even-even nuclei, for which enough experimental information is available. The values of the parameter ~z are given in table 2*. The dependence of the B(E2) values on I is shown in figs. la and lb for 152'1545m, 154't56Gd, 16°,162Dy, 16°Er and 162yb. The normalized Z 2 is less than 0.6 for all cases, with the exceptions of r64Dy(z2= 0.9) and 158Effz2 = 1.1). * The s-values for 16°,162,164Dy have been obtained by using for the (12~10), (10~8) and (8~6) transitions the B(E2) values derived from the lifetimes measured by Kearns et al) 3) by Doppler-broadened lineshape analysis. It should be mentioned that Sayer et al) l) have obtained much larger B(E2) values for the (12~10) and (10~8) transitions by comparing their experimental Coulomb excitation probabilities with theoretical values calculated with the Winther - de Boer computer code. However, these latter data are affected by very large uncertainties and do not take into account all sources of error. t For ~58Er the fit to the data does not include the (12~10) transition, for which only an upper limit of the lifetime has been set in ref. 8.
191
E2 T R A N S I T I O N S
152 Sm
(a)
~60 Dy
1546d
(b)
162 Dy
I 1.8
1
//
1.6 1.4
J ,!
//
1.2 1.0 0.8 f I
I
1'54S m
T
I'56G'd
T
'160E r
'
,
8
10
~62y' b
1--4
"* 1.8 cq LU c'n 16
14 1.2 1.0
b; off
j
0.8
2
4
6
8
10
12
2
4
6
8
10
12
2
4
6
12 I 2
4
6
8
10
12
Fig. 1. Dependence of the B(E2) values on 1. The solid lines are least-squares fits of eq. (1) to the experimental data using B(E2; 2 ~ 0 ) and cz as parameters. The dashed lines show the
dependence predicted by the rotational model and result from the fitting of eq. (6) with Q~ as free parameter.
It appears that, to the present accuracy of the data, eq. (1) gives a good fit of the experimental B(E2;I--,I-2) values for transitions within the ground-state band up to relatively high angular momenta. It should be noted, however, that seven of the thirtheen cases considered yield values of cx consistent with zero and are of little use in testing the band-mixing model. An interesting feature observed in figs. la and lb is the remarkable similarity of the behaviour of the E2 transition rates for the three isotonic pairs of nuclei 152Sm-lS4Gd, 154Sm-]S6Gd and 1 6 ° E r J 6 2 y b . Concerning the six nuclei, for which there are significant deviations from the rotational model behaviour, ac is positive for 152Sm, 154Gd and 16°yb, whereas it is negative for 16°'162Dyand 164yb 21). AS discussed in the preceding section, the appearance of negative at-values is related to hindrances in the B (E2) values.
4. Concluding remarks We have seen that the experimental B(E2;I--,I-2) values for transitions within the ground-state band in several rare-earth nuclei are well fitted by the expression (1). In particular, this expression, for at<0, can explain the retardations of E2 transition rates observed in some nucleiC3:5). This is certainly a point of interest, as retardations of E2 transition rates are generally expected as a consequence of a phase transition from a superconducting to a normal state. In this connection, it is worth noting that, at the present time, there is no definite evidence that the backbending effect is associated with hindrances in the E2 transition rates. For instance, in 164yb, which is a strongly backbending nucleus, some hindrances seem to have been observed~5). On the other hand, the B(E2) values measured by Ward et al. 8) in 158Er, which also exhibits a large amount of backbendI1. N U C L E A R
SPECTROSCOPY
192
A.
COVELLO
AND
ing, seem to be consistent with rotational values. Unfortunately, the errors on most lifetimes of high-spin rotational states, which have been recently measured using Doppler shift techniques, are too large to permit any definite conclusion on this point. As is well known, there have been many attempts to explain in a consistent way the deviations from the rotational values of both the level spacings and the E2 transition rates in terms of a simple band-mixing model, but they have not been generally successful (see for instance ref. 5). In these attempts, the K = 0 and K = 2 excited bands which mix into the ground-state band have been usually identified with the /3- and y-vibrational bands, respectively. This might indicate 5) that there are excited bands which contribute to the deviations in the energy level spacings, but have little effect on the E2 transition rates. It should be stressed, however, that formula (1) holds irrispective of the specific nature of the excited bands. Clearly, more accurate experimental data are needed in order to make a stringent test of the expression (1). However, the present evidence seems to suggest that the deviations from the rotational values of the E2 transition rates within the ground-state band of deformed even-even nuclei may be explained in terms of a simple band-mixing mechanism. References l) R. M. Diamond, F. S. Stephens, W. H. Kelly and D. Ward, Phys. Rev. Lett. 22 (1969) 546. 2) I. A. Fraser, .J.S. Greenberg, S. H. Sie, R. G. Stokstad and D. A. Bromley, Phys. Rev. Lett. 23 (1969) 1051. 3) R. O. Sayer, P. H. Stelson, F. K. McGowan, W, T. Milner and R. L. Robinson, Phys. Rev. 4C (1970) 1525.
G.
4)
GIBERTI
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