B(M1)-transition probabilities in odd-A nuclei

Volume 160B, number 1,2,3

PHYSICS LETTERS

3 October 1985

B(M1)-TRANSITION PROBABILITIES IN ODD-A NUCLEI ~ Eva M. M O L L E R and U. MOSEL Instttut fiir Theorettsche Physik, UmversltlJt Gtessen, D-6300 Gtessen, We~t Germany Received 1 April 1985; rewsed manuscript received 5 June 1985

B(M1) values of 157Ho are calculated wlthm a quaslpartlcle-rotor model with a rotation-dependent mteractlon between core and quasiparUcle. The dominant peak in the transmon probabflmes in the backbendmg region as well as the decrease of the signature splitting beyond the band crossmg is reproduced In addition, calculated B(M1)-transmon rates in the low spin regton for several Yb-lsotopes are in accordance with the empmcal data

The effect of signature splitting in yrast energies as well as in B(M1) values is known from theory as well as from experiment. Hamamoto [1] has studied the general structure of electromagnetic transition probabilities in odd-N nuclei in the low spin region within a particle-rotor model. She found that the staggering behaviour between the transitions from the favoured to the unfavoured band and the inverse decay can be described in such a theory. This staggering has, for example, been seen in experiments of Kownacki et al. [2], who have studied the structure of the high spin states of Yb isotopes going from 161yb to 167yb. With increasing neutron number the difference in energy between the Fermi level ~ and the K = 1 / 2 state grows and therefore the signature splitting of the yrast energies decreases. This change in the occupation probability of the K = 1 / 2 state is reflected in the monotonically-decreasing signature dependence in the B(M1) values going from 161yb to 167yb. New and completely unexpected-and not explained so f a r - h a v e been the results of the Copenhagen group [3,4] on 157Ho that showed a striking peak of B(M1) values in the backbending region and a subsequent disappearance of the signature splitting of the B(M1) values at even higher spins. * Supported by BMFT and GSI

In this letter we study this behaviour in an extended particle-rotor model. For this purpose we first discuss the Yb nuclei and then address the question of the striking peak and the following disappearance of staggering of the B(M1) values in ISTHo. For a theoretical description of the data we use a quasiparticle-rotor model with a rotation-dependent interaction between core and quasiparticle incorporated in the hamiltonian

n = H ° + Ec(lRl)+[(1-a)/O]Rjqp.

(1)

H ° is the standard axially-symmetric Nilsson hamiltonian with the parameters ~p,, = 0.0637, /z n = 0.420 and/Zp = 0.600 plus a pairing force. The deformation parameters are chosen to yield the ground-state equilibrium deformation obtained for the neighbouring even-even nuclei by a corresponding Strutinsky calculation [5]. The pairing gaps A are calculated from the empirical masses. R denotes the core angular momentum, which is coupled with the quasiparticle spin jqp to the total spin ! of the nucleus R +i~p = I.

(2)

The term Ec(IR]) represents the collective part of the hamiltonian. In the case where an odd-N nucleus is considered and only one quasiparticle is treated explicitly, we have used for E c the empirical energies of the ground-state bands of the

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21

Volume 160B, number 1,2,3

PHYSICS LETTERS

electromagnetic multipole operator defined in the intrinsic system. The magnetic dipole operator is given by

neighbouring nuclei in order to simulate the blocking mechanism (see ref. [6]). The interaction term [(1 - a ) / O ] R j ~ represents a rotation dependence of the single-particle potential and causes a reduction of the Coriolis force by a factor of roughly a, as well as of the recoil term by approximatively 2a - 1. A more detailed description of the hamiltonian and the determination of the parameters is given in refs. [6-81. The reduced transition probability between an initial state li> and a final state If> is defined by

M ' ( 1 ) -- (3/4~r)1/2/~,

l~=gRl+E(g~--gR)j,+(g~--g,)s,,

where the sum runs over the valence particles only and gR represents the g-factor of the collective core. The spin factor gs has been multiplied in all calculations by 0.6 in order to include polarization effects, whereas for g~ the free values have been used (for gR see the discussion below). We have calculated magnetic transition probabilities for the isotopes 161yb, 163Yb, 165Yb and 167yb. As no experimental B(M1)values are known for spins larger than 27/2, and as the proton alignment only starts at very high spins, a restriction to one valence particle is completely justified. (Values of a = 0.930, 0.815, 0.800 and 0.765 were used in these calculations for the isotopes 161yb, 163yb, 165yb and 167yb, respectively.)

I ( I , g , h ( g f - g , ) l I , Kf)

× (KflM'(k, ( K , - K,))IK,)

+ (1, - K i k ( K f + K,)IIfKf) X ( K f l M ' ( A , (Kf + K , ) ) I K , ) ( - ) z ' + K, 12 (3)

where the a r,f t are the expansion coefficients in terms of the basic states, M'(A) stands for an r - -

r

T--

I

'

I

'

I

'

-

I

o

(5)

1

K t , Kf

×a~,a~,,,

(4)

and

n ( A , i --* 0 = [ ( f l l n ' ( k ) l l i > 12/(2/, + 1)

= E

3 October 1985

165yb

~

163yb

J

--

•

J

•

I

_

•

167yb

o

o. o

8

10

12 Spin I

d 14

8

10

12

d 14

SpTn I

Fig. 1 Experimental B(M1) values for transitions betweenstates of opposite signature in the 113/2band of odd-A Yb nuclei marked by centered symbols m comparisonto the theoreticalones which are connectedby sohd hnes 22

Volume 160B, number 1,2,3

PHYSICS LETTERS

Fig. 1 shows the theoretical B(M1)-transitlon rates for these four nuclei. In 161yb empirical values exist only for the spins 19/2, 21/2, 2 3 / 2 and 25/2. We can reproduce the empirical values of all four data points very well. In addition, we reproduce the slight increase of the transition probabilities from the favoured to the unfavoured states and the constancy in the rates for the inverse decay. Our results for 163yb are very similar to those in 161yb, but experimentally there is an anomaly: the transition probability for the decay from I = 2 1 / 2 to I = 1 9 / 2 is smaller than the one from I = 1 7 / 2 to I = 15/2. This cannot be explained within our model, because the Clebsch-Gordan coefficients in eq.(3), which determine the general behaviour of the B(M1) values, never decrease. Therefore, our calculation does not reproduce B(M1; 2 1 / 2 ---, 19/2) well. However, for B(M1; 1 7 / 2 ---, 1 5 / 2 ) the agreement with the experiment is again satisfactory. As in 161yb the unfavoured transitions are underestimated a little. On the other hand, the complete vanishing of these transitions in 165yb is reproduced exactly within our model. The favoured transitions are not known empirically, but calculations of Hamamoto [1] show similar results. Finally, the B(M1) values in 167yb starting from states with I = 1 3 / 2 and I = 1 7 / 2 are in good agreement with the data. The unfavoured transitions are relatively large in experiment and do not follow the general trend in this chain of isotopes. They are slightly overestimated in our calculations. Whereas the experimental data mentioned above only deal with transitions below the band crossing, new experiments performed by the Copenhagen group [3,4] have provided B(M1) values within and beyond the backbending region. In the low spin region the transition rates as well as the level energies of the yrast band in 157Ho bases on the ~r7/2-[523] Nilsson level exhibit a distinct signature dependence. However, when the alignment of the ix3/2-neutrons begins, the strong signature splitting of the favoured and unfavoured bands almost entirely disappears [4]. Furthermore, the magnetic transition rates show a strong peak within the alignment region. Beyond the backbend

3 October 1985

a decrease of the absolute value as well as of the signature splitting is observed. Describing nuclei with odd proton number and again treating only one quasiparticle explicitly, one has to use the yrast energies of 158Er for the collective energies E c, which contain the alignment of the same two neutrons as 157Ho. In addition, for a calculation of B(M1) values one has to use spin-dependent g a factors. For even-even nuclei these factors are known from microscopic cranking calculations [9] as well as from experimental data [10]. They show a strong decrease due to the alignment of two neutrons in the i13/2 shell in the crossing region. An analysis of the effect of such spin-dependent gR factors is displayed in fig. 2a. The upper part shows the results for a calculation w~th a constant g a factor of Z/A = 0.43. Only a small increase of the B(M1) values is obtained, but a decrease of the signature splitting at the backbend can be seen. In order to get a feeling for the influence of the spin dependence of g a we have used values extracted from a cranking calculation with number projection and monopole pair field, [9], because these have not been measured yet for the core nucleus ~58Er. The lower part of this figure shows results obtained with this prescription. Now a stronger increase in the transition rates is found, which reaches nearly the correct maximum value, whereas the sharpness of the peak seen in the experiment cannot be reproduced at all. Furthermore, the signature splitting at the backbend goes down again, but does not vanish entirely. Nevertheless, these results show that the B(M1) values are sensitive to a spin dependence of the core g-factors and, therefore, to alignment processes among the core nucleons. We now extend our model to an explicit treatment of three quasiparticles in the valence shell, namely an h11/2 proton and two i13/2 neutrons. Therefore, the necessary degrees of freedom for a neutron i13/2 pair alignment are included and thus the gR(I) dependence can be obtained with this three-quasiparticle model. For such a configuration, rotor energies can be obtained by a proper extrapolation of the reference band used in the low spin region. We have found [8] that the best fit of the backbending 23

Volume 160B, number 1,2,3 I

0

PHYSICS LETTERS '

F

r

3 October 1985

]

t

lSTHo

-~

157H0

favoured f I

•

"

J

I

i

I

-,

I

I

157Ho #i~' .S

~

I

I

" 5

I

I

10

I

I

t5

unfovoured

o 0.0

I

20

I

]

'

,

=

ra

I

10

i

1

15

I

I

.

I

20

Fig. 2 (a) Theoretxcal values of B ( M 1 ) / Q 2 (Qo2 = 50e2b 2) calculated within a one-quaslparucle rotor model using a

0.4

At I = 3 1 / 2 , where the alignment of the pair begins, as reflected in the usual alignment plots (see fig. 3), the structure of the wavefunction is determined by a mixture of a one- and a three-quasiparticle excitation. This corresponds to a partial alignment of the i13/2 neutrons with intermediate values of the projection quantum

constant gp. factor (upper part) in comparison to the use of spin dependent gR factors (lower part) (b) The expenmental values

,

5 behaviour is given by using a quadratic extrapolation formula so that the energies of the quasiparticle band are well reproduced. The rotational energies yielded by this calculation are used to construct fig. 3 giving' the spin I versus the frequency a~ = d E/dI. In fig. 4 we show the corresponding transition probabilities obtained without any new parameters. In the low spin region the results exhibit the same transition probabilities as seen in the upper part of fig. 2a. This is because we now use a constant gR-factor of Z/A by treating the alignment microscopically and because the rotational band possesses a pure one-quasiparticle structure up to spin I _< 29/2. 24

J

Fag 3. l ( a 0 for 157Ho for the favoured and unfavoured band (Dashed hne experiment, sohd line. calculation with three quaslparticles and the same set of parameters as used for fig. 4.)

,--~

Experiment

•

[

0.3

[MEV]

o

~" 5

i

0.2

Spin I

~]

I

I

0.1

a~.10

m7 Ha

~o

5

f

,

:

~

10

b

15

i

20

Spin I Fig. 4 Values of B(M1)/Q02 extracted from the experimental branc~ng raUos for 1sYria are marked by the centered symbols. The theoretical values calculated in a model which treats three quasipartmles explicitly are represented by the sohd and dotted hnes

Volume 160B, number 1,2,3

PHYSICS LETTERS

numbers K on the symmetry axis. At higher spins these nucleons are aligned maxamaUy with very small K-values. The peak in the transition probabilities m this region can be explained by the action of the spherical components of the/~-operator and the specific form of the intrinsic structure of the wavefunction. .1 If the initial and the final states differ in the occupation strength of the various K states, as it is the case in the region where the pair aligns, the/~ s c o m p o n e n t s become very strong, because they connect states with IAKI = 1. Beyond the backbend, when the ahgnment is completed, the structure of the wavefunction is again fixed and these components lose their importance. Then the B(M1) values fall off immediately. Using the same model with three quaslparticles included we have also calculated the B(M1) values for 159H0; the results are shown in fig. 5. Again the B(M1) values show a signature splitting and a peak in the band-crossing region although the peak is weaker and less pronounced than in 157H0. This difference corresponds to the fact that 159H0 exhibits only an upbend compared with a backbend in X57Ho. It would be interesting to obtain data for 159H0 to check our prediction. Altogether we found a satisfactory description of the B(M1) values of odd Yb nuclei in the low spin region using a constant ga-factor. Due to the blocking mechanism, it seems reasonable to use the same gR-factor also for intermediate frequencies, because the it3/2 neutron alignment is hindered by the additional neutron. For odd proton nuclei like t57Ho this argument does not hold. Due to the alignment of two it3/2 neutrons the g a factors of the nelghbouring even-even nuclei go down drastically. Inclusion of this spin dependence in the calculated B(M1) values of odd proton nuclei leads to a strong increase during the backbending region and also a decrease of the signature splitting. The sharp peak and the total vanishing of the signature dependence beyond the backbend, however, can only be reproduced by a ,1 The calculated peak ts shifted down by two umts of angular momentum w~th respect to experiment Tins ~s due to a shght inaccuracy m the detmled structure of the wavefunctlon m the backbendmg region

3 October 1985

o

IS9H0 p

"s: i

f

I

o~

\

\

~ f

\

o

15

10 Spin I

Fig 5 Prediction of B(M1)/Q 2 for tS9Ho

microscopic treatment of the alignment process. This causes a mixed structure of the wavefunction between a one- and a three-quasiparticle excitation yielding a large increase of the B(M1) values during the reordering of the occupation strength for different basis states and also a sharp fall when the structure is again stable. An introduction of triaxiality, therefore, seems to be unnecessary for a description of all the high spin data in 157~"Io. Stimulating discussions with Toni Reitz and Dr. E. Wiast are warmly acknowledged.

References [1] I Hamamoto, Phys Lett 106B (1981) 281 [2] J. Kownacki, J D Garrett, J J Gaardhoje, G B Hagemann B Herskmd, S Jonsson, N. Roy, H Ryde and W Walus, Nucl Plays A394 (1983) 269, and references thereto [3] G B Hagemann, J D. Garrett, B Herskmd, J Kownackt, B M Nyako, P.L Nolan, J F Sharpey-Schafer and P.O Tjom, Nucl Phys A424 (1984) 365. [4] J D Garrett, G B Hagemann, B Hersland, J Kownaclo and P O Tjom, Nordic Meeting on Nuclear physics (Fuglso, 1982). [5] S Aberg, Phys Scr 25 (1982)23 (6] E M Mtiller and U Mosel, J Phys G10 (1984) 1523 [7] E M Mi~ller and K Neergard, Phys Lett 120B (1983)

28O [8] E M Miiller,Thesis Justus-Lxeblg-Umverslt~itGlessen (1985) [9] A Ansan, E Wiist and K Miihlhans,Nucl Phys A415 (1984) 215. [10] G Seller-Clark,D. Pelte, H Emhng. A Balando, H Greta, E. Grosse, R Kulessa, D Schwalm, H-J. Wollersheam,[3 Hass, G J Kumbartzkxand K -H Spe~del Nucl Phys A399 (1983) 211 25