Nuclear Physics A160 (1971) 665--672; @ North-Holland Publishing Co., Amsterdam

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

POSITIVE PARITY STATES IN 16’Er

I. KANESTR0M and G. L0VM0IDEN

Institute of Physics, University of Oslo, Norway

Received 2 1 September 1970

Abstract: The positive parity states in i6’Er are described in terms of the rotational model, where

the particle-rotation coupling is taken into account. The moment of inertia, the Fermi energy and the coupling strength are determined by fitting the energies. The wave functions thus obtain- ed are used to calculate the (d, p) stripping and (d, t) pick-up cross sections and branching ratios. The results obtained are in agreement with experimental data.

1. Introduction

Recently, strongly perturbed positive parity bands were populated in the nucleides

161,163*165Er by means of the (c(, xn) reaction ‘). The observed band in each

nucleus is nicely explained as the lowest energy band which results from strong

coupling through the Coriolis force between the 4’ [660], 4’ [651], 3’ [642], 3’ [633]

and 8’ [624] Nilsson orbitals. However, since only one positive parity band has been

identified in each of the nuclei mentioned, the calculations performed may not yield

a decisive test of the model used. It is therefore interesting to use the same model on

the neighbouring nucleus 16’Er where members of different positive parity bands

have been identified experimenta;lyZ-4). The ground state band in 1 67Er is generally

believed to be built on the 3’[633] orbital 2Y3P “). However, the experimental (d, t)

cross sections for the 8’ and y’ members of this band are larger than the predicted

values by factors of about 3 and 2 respectively. Furthermore the rotational param-

eter (A % 8.6 keV) for the ground state band is lower than that found for the negative

parity bands in the same nucleus “). These facts suggest a strong coupling to other

N = 6 orbitals through the Coriolis interaction.

Except for the ground state band, the assignments of the positive parity states are

somewhat uncertain. Based on their (d, t) experimental cross sections, Tjsm and

Elbek ‘) suggest that the levels at 812, 933 and 1109 keV are the 3+, 9’ and y’

states in the 3_’ [642] band. On the other hand, Michaelis et d3) placed the t3’ [642]

state at 592 keV in order to explain a strong Ml transition in their spectrum. They

also investigated the effect of the Coriolis force, but claimed that the +‘[660]

orbital had to be excluded from the calculation in order to reproduce the experi-

mental data. However, this orbital has a large positive decoupling parameter, and

previous works i* ‘,‘) have shown that the influence of this orbital on other bands is

strong. The 3’ [660] orbital should therefore be taken into account in the calculation.

665

666 I. KANESTRBM AND G. L0VH0IDEN

The aim of the present calculation is to give a consistent description of available

experimental data for the pcsitive parity states in 16’Er, such as energies, transfer

reaction cross section and branching ratios.

2. Calculation

The only positive parity shell-model states in the erbium region are the degen-

erate i+ states. In a deformed nucleus, these states are split to form the +‘[660],

3+[651], . . ., y+ [606] Nilsson orbitals. The Coriolis matrix elements between these

states are large, but the probability of interaction with other N = 6 orbitals is small

since the energy differences are large and the connecting matrix elements relatively

small. The only other positive parity states that might interact with the states consider-

ed are the 4’ [400] and $’ [402] orbitals. However, such dN = 2 couplings are not

observed ‘) in 1 6 ‘Er and are excluded in the present calculation. We assume therefore

that the observed positive parity states are based on the i, states. In order to take

the Coriolis interaction into account, an energy matrix is constructed and diagonalized.

The diagonal terms are the usual unperturbed rotational energy,

E(Z) = &,+A [Z(Z+ l)-K(K+ l)+s,,,(-)‘+9a(z+3)], (1)’

where A is the rotational parameter and Z&, the quasiparticle energy given by

J?& = [(EK-A)2+d2]Gl. (2).

In eq. (2), sK is the single-particle energy, L the chemical potential and A half the

energy gap. The off-diagonal matrix elements consist of the Coriolis matrix elements

M ILK+1 = -aA(Klj-liY+1)[(z-K)(z+K+l)lf((i,U,+,+ VKVK+r). P),

The parameter a is introduced to adjust the coupling strength. The occupation

amplitudes UK and V, are determined from the equations:

2cv; = n,

A = G c UiVi, Uf + V; = 1,

(4)

(5)

where n is the number of neutrons and G the strength of the pairing force.

The rotational parameter A is assumed to be a constant for all the bands considered.

The deviations from the Z(Z+ 1) rule which are found in even nuclei, an effect caused

by the decrease of the pairing gap with the frequency of rotation, are expected to be

small in odd nuclei “) and are neglected.

The reduction factor tl may depend on Z and K, but has been considered to be a

constant. This approximation implies that the interactions not included in the calcu-

lation affect all the states in the same manner.

Coupling to /?- and y-vibrational states may also occur in 167Er [refs. “S3)l-

There is some uncertainty in the position of these vibrational states. However,

r6’Er POSITIVE PARITY STATES 667

the effects of the rotation-vibration interaction are in general small, and the wave

functions for the N = 6 positive parity states are expected to have only minute

components of the vibrational states “). This assumption is confirmed by the present

results.

TABLE 1

Positive parity states in r6’Er

Spin Level energy Amplitude of wave functions

(keV) -

talc exp g+ [660] $+[651] _ % + [642] $+ [6331 %+[624] +++ [615]

1819

1490 2088

769 1514 1812

0 847

1626

80 584 945

1595 1977

174 711

1068 1863

296 859

1205 1735

434 1028 1379

592 1217 1543 1972

770 1427 1783

( 812) (1525)

0

79 592 933

(1558)

178 711

296 826

(1205)

434 962

592

772

1 .ooo 0.117 0.993

0.993

-0.117

0.013 0.113 0.993 0.413 0.904 -0.109 0.910 -0.412 0.035

0.001 0.010 0.116 0.993 0.018 0.173 0.978 -0.116 0.202 0.964 -0.173 0.011

0.002 0.019 0.170 0.977 0.127 -0.001 -0.008 -0.051 -0.119 0.99 1

0.053 0.23 1 0.955 -0.174 0.03 1 0.689 0.695 -0.203 0.021 -0.001 0.723 -0.680 0.122 -0.010 0.000

0.003 0.029 0.213 0.960 0.178 0.008

-0.002 -0.019 -0.094 -0.160 0.980 0.061

0.039 0.273 0.932 -0.225 0.058 0.005

0.250 0.928 -0.274 0.033 -0.003 -0.001

0.007 0.040 0.249 0.944 0.214 0.014

- 0.009 -0.034 -0.140 -0.183 0.968 0.089

0.131 0.337 0.888 -0.270 0.089 0.011

0.799 0.520 -0.299 0.053 -0.007 -0.002

0.006 0.050 0.281 0.927 0.242 0.021 -0.007 -0.051 -0.184 -0.193 0.956 0.112

0.062 0.356 0.869 -0.315 0.121 0.018

0.015 0.063 0.309 -0.031 -0.079 -0.232

0.266 0.436 0.771 0.819 0.360 -0.430

0.263 0.026 0.941 0.132 0.161 0.029

-0.024 -0.007

0.010 0.073 0.334 -0.016 -0.095 -0.269

0.084 0.42 1 0.791

0.911 -0.191 -0.344

0.115

0.896 -0.188 -0.391

0.281 0.032

0.928 0.151 0.188 0.040

The Coriolis coupling calculations are performed with deformation parameters ~2 = 0.267, E.+ = 0.015, the inertial parameter A = 11.35 keV, the reduction factor CL = 0.55 and the decoupling constant (I = 6.59 for the K = $ band. Calculated states are listed for spin Z ( 4 up to an energy of about 2 MeV. The contributions from the $$ + [606] states are less than 5 x 10m5 and are omitted in the table.

668 I. KANESTRBM AND G. LBVHOIDEN

The single-particle energies sK in eq. (2) are calculated from the Nilsson model

with both quadrupole and hexadecapole deformations. The parameters used lo) are

p = 0.420, K = 0.0637, s2 = 0.267 and .sq = 0.015. Thus the only free parameters

needed to calculate the absolute excitation energies of the members of the positive

parity bands are A, GI and i, and they are fixed by a search routine to fit the known

energies.

A calculation including the single-particle energy of the ;‘[633] orbital &-I as a

fourth parameter, was also performed. This improved the fit to the ground state band,

as shown in fig. 1. The stripping and pick-up cross sections are evaluated from the

expressions

do 2( F C’S, taiUi)‘+,(d, P),

dn = I i 2( C Cj, iaiI’i)2@,(d, t)*

(6)

Here C,,, are the expansion coefficients for the Nilsson wave functions, ai the mixing

amplitudes due to the Coriolis coupling, Ui and Vi the occupation probabilities. The

cross sections 41 are obtained from a DWBA calculation using the optical-model

parameters given in ref. ‘I).

The transition probabilities are calculated using the energies and wave functions

given in table 1. The formulae needed and the computational procedure are outlined

in ref. ‘). The matrix elements bhll Ght and GE:’ are evaluated from the Nilsson

model 12) for a deformation 6 = 0.3.

3.1. ENERGIES OF 16’Er 3. Results and discussion

The energies obtained from the calculation are shown in fig. 1. In spite of the simple

model used the fit to the experimental data is good. Case b corresponds to the

calculation with three free parameters. As seen from the figure, the ground state band

is a little compressed.This may indicate that the calculated particle energies are slightly

incorrect. Therefore, as mentioned above a fit was made including the particle energy

for the $‘[633] orbital treated as a free parameter. The results are indicated by a in

fig. 1. The change in the particle energy sI from case a to case b was only 0.25 % or

140 keV. This improved the energy fit for the ground state band, whereas the wave

functions were insignificantly changed. Therefore the following discussion will be

based on the last calculation (a) with the wave functions given in table 1.

Some of the states deserve special comments. The calculation indicates that the

$3’[642] state has higher energy than proposed by Michaelis et al. “). Instead, the

strong Ml transition (592 keV) observed, corresponds to a transition from the

$4’ [624] state to the ground state. This transition is calculated to be 3.4 s.p.u. using

the wave functions given in table 1. Moreover, this offers a natural explanation of the

positive parity states observed “) at 711, 826 and 962 keV as the -i?_, y and y mem-

bers of the $+ [624] band.

16”Er POSITIVE PARITY STATES 669

keV

EXP CALC

b a 13

--------- i

16’Er

EXP CALC

____~ $ a b 15

1000 - CALC **,----- -9 EXP -’ 1 __-- ____

7 ----- ? a

b 13 -_-_--- -_ 1

___.- 5 --__ 19

$‘[642] -_-_- z ___..-_..~ y

500

~

------------- f ___.~___ -3

++ [624] -.-. -____

____-c- _- 7

----___.--- !$

____-. <-- $ O-

**- ___.-- 4 ;” [633]

i

Fig. 1. Level scheme for positive parity states in 167Er, Cases a and b are calculated with four and three parameters, respectively (see text).

z 2 9 0 EVEN NUCLEI

b = 10 I I I I I I I I I

160 161 162 163 164 165 166 167 168

MASS NUMBER

Fig. 2. Rotational parameters for erbium isotopes.

The energies obtained for the states of the 4’ 16421 band agree reasonably well with the assignments given by Tjom and Elbek “). The calculated cross section for the 3 state of this band is small, and the experimental yield is probably hidden by the strong $$- [521] peak. There are two possible candidates for the 9 state in the (d, t) spectrum, and the calculated state coincide with the level observed at 1205 keV.

670

Spin

ZK"[Nn,A]

I. KANESTRIZIM AND G. L0VHBIDEN

TABLE 2 Energies and cross sections in 16’Er

do/dQ(d, p), 90”&b/sr) do/d.f&d, t), 90”@b/sr) Energy (keV) Q = 3 MeV Q= -2MeV

talc. exp. pure mix. exp. pure mix. exp.

0

80 178 296 434 592 170 584 711 859

1028 169 847 945

1068 1205 1490 1514 1626 1595 1863 2284 1819 2086 1812 2396 1977 2802 1735 2261 2412 4263

0

79 178 296 434 592 772 592 711 826 962

933

1205

(1525)

(1558)

5 4 3 1 0 0 0 0 0

32 32 w30 5 0

0 0 3 5 0 0 1 1 1 1 35 50 76 0 0 0 0 1 1 30 37 33 0 0 1 1 0 0 16 36 (% 19) 0 0 1 1 0 0 53 91 (50) 0 0 0 0 0 0 29 0 0 0 5 5 0 0 1 1 0 0 38 16 0 0 0 0 0 0 66 3 0 0 0 0 0 0 27 31 0 0 0 0

38 35 0 0 38 35 0 0

0 w 0 0 0 w 1 2 11 17 28 47 0 % 1 1 1 w2 9 13 31 54 63

TABLE 3 Calculated and experimental branching ratios Z,,(Z + I-2)/ZY(Z --f I- 1) for the ground state band

in le7Er

Z Calculated Experiment

mixed wave function pure K = $ wave function

ss = gs.rree gs = 0.6 gs.rrec ss = &rrec 98 = 0.6 g..rrcc

-ti- 0.38 0.62 0.15 0.25 0.60f0.34 “) 0.34kO.03 b)

D 0.84 1.37 %

0.35 0.58 0.81 kO.05 ‘) 1.37 2.25 0.62 1 .oo 1.32hO.09 “)

“) Taken from ref. 14). ‘) Taken from ref. Is).

16’Er POSITIVE PARITY STATES 671

The calculated band head energies of the unperturbed +‘[6603 and $‘[651]

orbitals are 1819 and 1498 keV, respectively, while the ++” [400] and 4%’ [402] states

are found at 1135 and 1086 keV. The relatively large energy differences between these

N = 4 and N = 6 orbitals explain why the AN = 2 coupling is unobserved “) in

1 6 7Er.

The value of the rotational parameter A obtained by the search routine is 11.35 keV,

which is slightly lower than the value (12.08 keV) found for the lowest negative

parity band in the same nucleus “). The situation is summarized in fig. 2, where the

rotational parameters found for the positive parity states in the nucleides

161S163,165,167Er are displayed and compared with the parameters for the negative

parity bands. The results seem to indicate that the quantity A is characteristic for the

nucleus and depends only weakly on the parity and K quantum numbers. However,

there remains to be understood why the rotational parameters in the odd nuclei are

systematically smaller than the corresponding values found in the neighbouring even

nuclei, and why this difference increases so rapidly towards lower mass numbers.

The reduction factor ff in the Coriolis matrix elements is equal to 0.55, which is

somewhat lower than found for other nuclei ‘*13).

It might be mentioned that a calculation has been performed with single-particle

energies evaIuated with Ed = 0. In this case we were not able to reproduce the ex-

perimental energies. Especially the 3’ 16241 band came out too high in energy, indicat-

ing that the effect of the hexadecapole deformation plays an important role.

3.2. TRANSFER REACTION CROSS SECTIONS

The (d, p) and (d, t) cross sections are calculated using eqs. (4)-(6) together with

the wave function given in table 1. In all cases the agreement with the experiment is

improved when the Coriolis coupling is included as seen from table 2.

3.3. TRANSITION PROBABILITIES

Only a few branching ratios in the ground state band are known experimental-

1Y 14* ’ “). These values together with the calculated ones are given in table 3. The

quadrupole moment Q, is taken 16) to be 8.76 b, and for the gyromagnetic ratios

gS = gs,frpe and gR = 0.3 are used. As seen from table 3, the Coriolis coupling has a

pronounced effect on the calculated branching ratios and the agreement with the

experiment is surprisingly good. Usually due to polarization effects, a reduced value

of the gyromagnetic ratio gS is used. With this in mind, it is surprising that the agree-

ment between theory and experiment is worse when gS = 0.6 gs,free is used (see table

3). However, similar behaviour has been found for other odd Er isotopes ‘). On the

other hand, a reduction of the gyromagnetic ratio improves the results if pure K = 3

wave functions are used. However, a reduction factor equal 0.6 is insLI~cient to

increase the calculated branching ratios to match the experimental ones.

672 1. KANESTR0M AND G. LBVHOIDEN

4. Conclusions

The calculation performed has been successful in reproducing several properties of the positive parity states in 167Er; e.g. the level structure, the stripping and pick-up cross sections and the branching ratios for the ground state band. The inclusion of the hexadecapole deformation is essential in order to explain the level energies. As seen from the results, the particle-rotation coupling plays and important role in the rotational model description of ’ 67Er. However, we will stress that only about half of the theoretical coupling strength is used, and this phenomenon ought to be an object for further investigations.

The authors are indebted to Dr. B. Nilsson for copy of his computer code for the Nilsson model energies. Fruitful discussions with Dr. P. 0. Tjarm are gratefully acknowledged.

References

1) S. A. Hjorth, H. Ryde, K. A. Hagemann, G. Lsvhcriden and J. C. Waddington, Nucl. Phys. Al44 (1970) 513

2) P. 0. Tj0m and B. Elbek, Mat. Fys. Med. Dan. Vid. Selsk. 37, no. 7 (1969) 3) W. Michaelis, F. Weller, U. Fanger, R. Gaeta, G. Markus. H. Ottmar and H. Schmidt, Nucl.

Phys. Al43 (1970) 225 4) M. Rozkos, F. Sterba, J. Sterbova, 9. Elbek and P. 0. Tj@m, Int. Symp. on nuclear structure,

(Dubna, USSR, July 1968) 5) C. M. Lederer, I. M. Hollander and I. Perlman, Table of isotopes, 6th ed. (Wiley and Sons,

New York, 1967) 6) I. Kanestrom and P. 0. Tjsm, Nucl. Phys. A138 (1969) 177 7) M. E. Bunker and C. W. Reich, Phys. Lett. 2% (1968) 396 8) J. Borggreen, G. Lravhaiden and J. C. Waddington, Nucl. Phys. A131 (1969) 241 9) I. Hamamoto and T. Udagawa, Nucl. Phys. Al26 (1969) 241

10) I. L. Lamm, Nucl. Phys. A125 (1969) 504 11) B. Elbek and P. 0. Tj@m, Single nucleon transfer in deformed nuclei, Advances in nuclear

physics vol. 3 (Plenum Press, New York) 12) S. G. Nilsson, Mat. Fys. Med. Med. Dan. Vid. Seisk, 29, no. 16 (1955) 13) G. Lsvheiden, J. C. Waddington, K. A. Hagemann, S. A. Hjorth and H. Ryde, Nucl. Phys.

AX48 (1970) 657 14) H. R. Koch, Z. Phys. 187 (1965) 450 15) A. Tveter and B. Herskind, Nucl. Phys. Al34 (1969) 599 16) B. Elbek, Determination of nuclear transition probabilities by Coulomb excitation (Copenhagen,

1963)