The Jπ = 92+ bands in odd-mass Sb and I isotopes: Evidence for deformed states?

THE

J’ = 3’ BAND6 IN ODD-MASS

EVIDENCE

I ISOTOPES: STATES?

Sb AND

POR DEFORMED

P. VAN BACKER +, M. WAROQUIER tt and H. VINCX tt+ Laboratorium war Kemfysica, Proeftuinstraat 86, B-9wo Gent and K. HEYDE Laboratorium voor Kemfysica, ProPfruinsiraat 86, B-9ooo Gent and Institut de Physique NucMaire (and IN2P3), Universitb Lyon-l, 43 Bd. du 11 Novembre, 69621 Vilkurbanne, France Received I1 July 1977 Abstraet:.Inthcodd-mass”3-L19Sband “7-‘nI nuclei, bands with dl = 1 spacing have been observed built on a low-lying 1’ state. We have described these bands as resulting from a band-mixing calculation at the equilibrium deformation corresponding with the $‘[4p4] Nilsson orbital. Total potential energy surfaces, energy spectra as well as electromagnetic pioperties for these nuclei are presented and compared with the experimental data. Fake in describing the observed phenomena within a harmonic core coupling model is pointed out. Therefore hole-anharmonic core coupling calculations have been performed in order to obtain the dl = I band structure as well as improved intensity rules. Results from both the band-mixing and holoanharmonic core coupling calculations are compared and similarities are pointed out.

1. Introdoedon

The odd-proton Sb and I nuclei, containing, respectively, one or three protons outside the closed 2 = 50 core, are expected to indicate a rich variety of nuclear phenomena. One can observe the interplay between the one-particle motion and the collective quadrupole and octupole vibrations, or the correlations induced via the residual n-n interaction, within the three-particle cluster. Therefore, many authors have studied this approach in describing odd-mass Sb [refs. ‘-‘)I and I [refs. 6-8)] nuclei with considerable success. Recently, however, in 113,-119Sb and 117-1z’I, the existence of AJ = 1 bands built on a low-lying 3’ state has been pointed out 9- 1s). The pattern of band spacing is very suggestive of a lg, hole state coupled to the underlying doubly even core, which is formed by the Te or Xe nuclei. In this particular case, the problem of describing low-lying (& 5 1.Oh4eV)9+ excitations becomes very difficult, if not impossible, within a spherical shell-model description of the odd-mass Sb and I nuclei. In allowing the nucleus to deform, particular high4 t IWONL fellow. tt Aangesteld navorser of the NFWO. ttt Aspirant of the NFWO. 125

126

P. VAN

ISACKER

et al.

components (G?= $‘), originating from the lg, shell-model configuration, can lower the the strong the total potential the possibility near closed many of data (variation of excitation energy of j+ state versus the total potential for oddmass Sb I nuclei, the AJ = 1 bands on top of the y [404] orbital, as well as the electromagnetic characteristics of the 4’ band structure. In sect. 3, we point out the difficulties remaining in a description within the harmonic core coupling model concerning 1h-2p calculations and discuss in some detail the possibilities of a holeanharmonic core coupling calculation. The possibility to describe AJ = 1 bands with almost the strong-coupling electromagnetic intensity rules in the particular case of the lg, single-hole configuration coupled to, respectively, Te(Sb) and Xe(1) nuclei, is obtained. Finally, we compare some of the pertinent results of both sects. 2 and 3 and indicate the similarities between both concerning the energy spectra and electromagnetic properties.

2. Deformed 9’ states: AJ = 1 bands 2.1. EVIDENCE

FOR DEFORMATION

For the Sb and I isotopes, where the number of protons nearly constitutes the Z = 50 closed-shell configuration, clear indications are available for a rapid decrease in excitation energy for the 9’ level when the neutron number approaches the mid-shell (N = 66) configuration l1 -13). The same phenomena have been observed in other mass regions near closed-shell configurations 16-‘l). The low excitation energy for the 4’ level (0.3 < E, < 1.5 MeV) is probably due to lifting a particle from the Q’ [404] Nilsson orbital with a strong positive slope for prolate deformation. Thus the nucleus can lower considerably its excitation energy compared with the spherical p-h excitation energy through the Z = 50 closed shell 22) [AE,,(spherical) x 4.8 MeV]. Explicit calculations have been made for all odd-mass Sb and I nuclei by means of the macroscopic-microscopic renormalization procedure “) in order to obtain the total potential energy surfaces. The single-particle energies were calculated with a modified harmonic oscillator potential 24)with only quadrupole degrees of freedom. The relevant parameters K and p, as well as the pairing strength G for Sb and I nuclei, were discussed by Ragnarsson “) and Heyde lg). The total energy is then calculated as E’N”;‘;= BE, +2 + EN)+ E’fs”,

(2.1)

where E, and EN+, denote the total potential energy for the adjacent doubly even nuclei and t(E, +2 + EN)+ A corresponds, within the BCS approach, to a hypothetical

127

J= = $+ BANDS

E--c

2

Fig. I. The total ptcntial emrgy surfaaa corresponding to the proton !‘[&I] Nilsson proton-hole orbi~lfort~~d_~ ~13.117.111.115.119.133sb nuclei. The odd-even background (sa text) is showo in each case by the dashed lines.

128

P. VAN ISACKER

et al.

“odd-even” system (dashed line in fig. 1). The onequasiparticle energy in a particular orbit (f&g has ken calculated for the system with an odd number of particles. In fig 1, the results for 1l3 -“‘Sb are presented. Analoguous calculations for oddmass Ii7 - 12’1 nuclei have also been performed 20), but will not be discussed in detail here. From ref. 20) and fig. 1, one can observe that when the neutron number is approaching the mid-shell N = 66 configuration a well-pronounced deformation corresponding to.the !+[404] orbital occurs. In these nuclei (113-11gSb, 117-1211) the very low excitation energy is due to the opposite effects of the spherical tendency for the proton(s) (Z = 51 and 53) and the deforming tendency of the neutron mid-shell (N = 64,66 and 68) configurations, the latter being the strongest. When the N = 82 neutron closure is approached, a gradual smooth change towards higher excitation energy (smaller deformation), caused by the coherence of both proton and neutron shell-plus-pairing corrections 20), results.

2.2. AJ = 1 ROTATIONAL

BANDS

In order to calculate the AJ = 1 rotational band, built on the 3+[404] orbital, all possible Nilsson orbitals originating from the N = 4 harmonic oscillator shell are considered in a band-mixing calculation 21*25-27) at the relevant equilibrium deformation s,(equi). The band-mixing matrix element becomes ((a, UJMIH,,+ x(-

&JQ,

r?JM) = 2

l,‘-“‘~;~~~“E,$j(6&

c (- W”c”ll;.” J.&=WWl

t?c,.(U, i’MUo,,Un,~+v,,,v,,,~)+E’~~‘.

(2.2)

Here E,= denotes the excitation energy in the Te (Sb) and Xe (I) doubly even nuclei, corresponding to the yrast angular momentum sequence J,. The Nilsson wave functions, denoted by c#,2,& the occupation probabilities vi,, and the onequasiparticle energies El’&!) have always been calculated at the equilibrium deformation for the Sb (I) nuclei, corresponding to the ++[404] orbital. If we describe states up to angular momentum J” = y+, core states up to c = 14+ can contribute although the partial contributions from the highest states lO+ 5 c s 14+ are of minor influence in determining the energy spectra up to J” = 9’. In all Xe nuclei considered (118 s A 5 128), the excitation energy up to c = lO+ is known 28.29). For 114- 12@Te,the core energy is known up to J: = 8!. For all higher angular momentum states, a least-squares fit to the lowest known states was extrapolated. The influence on the energy and wave function of the 2” 5 P s 9’ states, when small changes for the extrapolated energies were introduced is of minor importance. The wave function, at an energy E$), is then obtained by diagonalising the energy matrix (2.2X with the result (2.3)

P = 4’ BANDS

53I54 117

5364 I 110

55aa I In

5372 I 115

370

_

c-7

_____

_-em

_--

_-__ ___

_-__

___I

__I

____

PH'

374

_-mm

-D(FEnlMwT ----THEORY q$

129

----

nh’,___

q-

lyz’,___

r--

____

__-

----

-----

c--

p-m-

1

-

___s_

---

I----

_____

_____

----

----

-

-a___

-____

--B-B

---

-_-

‘*;--

-s-w

nL_--

a

** --__

I’

-----

-----

_

m-d

r--

fig. 2. Comparison of experimental and theoretical band energies (relative to the band-head) for the Al = I bands observed on the I)+ proton-hole state in odd-mass I nuclei.

I

“:%2

3ba__-

3bs3

r

r---12yJw’ I

1

/r

---___B

,m.---

1$l-

lf

w

e-s

f r---

v

~___

--‘I~-’

1$-l

';S$

B-B_

,,$u’

%,-‘----

r&-s

*_’

/----

fig. 3. Same 88 for fig. 2, but for the odd-maw sb nuclei.

“*

h

P. VAN ISACKER

130

et al.

In figs. 2 and 3, one can observe a very good description of the AJ = 1 rotational bands for the odd-mass 117-1271 nuclei. For the l l 3 - l l gSb nuclei, the calculated values are systematically higher than the experimental values when using the Xe core energies E,<. In table 1, the wave functions (2.3) for l1 ‘Sb are given if only five TABLE

The amplitudes d”(Qi; P

?+c4w

J) [see

cq.

1 (2.3)] for the nucleus “‘sb

i+c4131

++ c4221

t+c4311

f+C4401

0.121

0.062

0.044

0.041 0.057 0.039

0.010 0.036 0.008

0.055 0.059 0.069

0.023 0.015 0.037

P’ Y+

0.940 0.933

0.309 0.337

Y’ Y+

0.917 0.912

0.369 0.387

0.115 0.133 0.130

$1 Y+

0.893 0.885 0.882

0.417 0.430 0.433

0.155 0.166 0.170

Nilsson orbitals (the 1% configuration only) are considered. For the other Sb and I isotopes, analogous results are obtained 20). One clearly observes a gradual decrease in the $+ [404] amp litude for all Sb (and I) nuclei with increasing angular momentum. This same effect has also been observed in other mass regions 32*33). One can now transform the strongcoupling wave function eq. (2.3) towards a weak-coupling basis which refers to good core angular momentum (53 of the Te (Xe) nuclei and good single-hole angular momentum lg;’ [refs. 21.25-27)]. The transformation coeflicients for the holecore coupled contribution become (2.4) and similarly for the particlecore coupled contribution U,,,c(J3. This particular holecore coupled decomposition for l1 %b is shown in fig 4, for the 9’ 5 J” 5 9’ states of the AJ = 1 rotational band. Here one observes the growing importance of the ;’ C4131 admixture. In the hole-core coupled representation, the importance of two (three) contigurations for a description of the different members of the AJ = 1 rotational band becomes clear. Thus, a unified-model calculation treating explicitly lh-2p configurations 14) coupled to collective quadrupole vibrations of the core nucleus, where a cut-off at threequadrupole-phonon configurations is made (c = 6+), will be inadequate in describing the higher angular momentum states (J” 2 9’) from the AJ = 1 rotational band. 2.3. ELECTROMAGNETIC

PROPERTIES

For the odd-mass Sb, I nuclei studied some data concerning branching ratios and mixing ratios have become availablerecently 13*14*3‘), and so we also have calculated

J’ = 4’ BANDS

A

A

J: nh

a0 -

0.0 -

0.6 -

0.6 -

0.4 -

OA-

a2 -

a2

0.0

'

'

131

"U'

-

'

O'h2yp#6?+y2m

A 0.8 -

A

J”= ‘y;

J: ‘g

ae.-

06 -

06 -

a4 -

Cl6 0.2 -

J’L. 2y;

0.8 0.6 04 a2 0.0i 2 O

'I2

__LlLk 312L Y2 6 7/2 6

912IO

12

R R

Fig. 4. The strong-coupling wave function (P) (open tmrs) and the m&coupling wave functions (R) (filled bars) r[sec up.(2.3) and (2.4)] for the Q’ band in “‘Sb. The amplitudes squared are drawn.

132

P. VAN ISACKER et al.

these observables within a deformed basis serving as a good description for the AJ = 1 rotational band. For the E2 and Ml static moments and reduced transition probabilities the operators as well as the explicit expressions have already ken discussed in some detail 21*25-27). The strength factor in calculating E2 moments and transition rates has been taken from the B(E2; 2: + 0:) reduced transition probabilities in the adjacent doubly even Te and Xe nuclei, if known 30). Otherwise an extrapolation via a least-squares fit to known E2 transition rates has been made. For the parameters entering the calculation of the magnetic properties, gR = Z/A has been used throughout. For the other gyromagnetic factors, the values g1 = 1 and g, = 2.79 (0.5 g5rcc)are used. A slightly enlarged value g, = 3.40 is used as it gives somewhat better agreement with the data. Also the sensitivity of the calculations to variations in g. can thus be studied. We discuss in some detail the electromagnetic properties obtained for “‘Sb, including the most documented case for the AJ = 1 rotational band. For the other odd-mass Sb, I nuclei, similar results are obtained. In fig 5, the static moments are given. For the pure $+[404] band, an intrinsic quadrupole moment Q, = 2.55 e - b results, corresponding to the equilibrium

Fig. 5. The titic moments (p, Q) for the nucleus ‘“Sb for the f’ band members as calculated from the band mixing (full lines) and for the puic )+ [404]band (dashed lines).

J==g+

133

BANDS

value - c2 = 0.145. One notices a sign change in the lab quadrupole moment due to the geometrical factor (Clebsch-Gordan coefficients) when the angular momentum is increasing. For B > i’ ; this sign change occurs approximately at J x int [ - 0.5 + 1.75&2](where int means the integer part). From branching ratio measurements, y-ray angular distribution measurements and conversion electron measurements l1 -14), the mixing ratios for the AJ = 1 E2, Ml transitions can also be extracted which, together with the cascade (AJ = 1) to crossover (AJ = 2) ratios, can serve as an important test for the wave functions obtained. These calculations are shown in fig. 6 and are also compared with the

/ -

-- E__ --

cd MC -4 _-

‘/ /

g.=2.m_,

8

g,=Uo g,=2.79 g, = 3.LO_g. =2.79 -

p E

_

$

!P!!?! “I2

Q

01 f ‘%

%

“h

%

‘H

‘i

Fig. 6. Cascade mixing ratios (l/P) and cascade-to-crossover ratios R, for the 4’ band in “%b. The initial angular momentum is always indicated (J,). Results for the band mixing (full line) and pure Q+[404] band (dashed line) are drawn. The experimental data are from Bron ef al. 14) (open squares) and from Foasan ‘I) (tilled circles). The lowest part of the figure applies to the left-hand scale whereas the upper part refers to the right-hand scale.

description in terms of a pure 9’ [404] band. Due to the error bars, a clear-cut difference cannot be made, although, for the band-mixing calculation, the value ga = 3.40 gives slightly better agreement. In calculating the ratio ( IlMl I I )/( I lE21I > leawde when a pure 3’ [404] band is considered, the ratio& - g,)/Q, can be obtained. For a pure band with g, = 3.40 (gn = 1.27), a value of (go-gR)/QO = 0.323 results. The explicit results for the reduced ratio, (

IWII

>I< II=11 )Ircd = < IWII

>I( II=11 >Ic~dc(~J,+lXJ,-I))*,

(2.5)

are given in fig. 7 (in table 2, the results for izlI are also indicated). In the recent study of Bron et al. 14), similar arguments have been used to obtain this reduced ratio. Finally, in table 3, the half-lifes for all members of the p rotational band are given

134

P. VAN ISACKER ef al.

E =

” I

O.l-

I

I

I

I

I

1

K

a/i

bh

‘35

9!?

2&

Ji

Fig. 7. The reduced (IIMI ll>/<.lE211)ratios for the $’ band members in “%b [see eq. (2.511as calculated in a band-mixing calculation (full lines) and from a pure $+[4@4] band (dashed lines). The experimental data are taken from ref. 14).

TABLE 2

Thereductdratio(I(MIII>/(IIElll)lnd

[sceeq. (2.5)lfortheAJ

= I transitionsin ““Sband values of ga, Q, and (Qn- gr)/Qo for the pure ?+CW band

“‘I,andthe

,211

“‘Sb J; + J;

y+ + 4’ y+ +y+ y++y+ y-y+ ‘ip+-ry+ Y+ -9’ ga

Q, (e . b) ba-srl/Qo

g, = 3.40

g, = 2.79

g, = 3.40

g, = 2.79

0.23 0.26 0.26 0.27 0.27

0.21 0.24 0.24 0.25 0.25 0.26

0.21 0.21 0.21 0.22 0.22 0.22

0.19 0.19 0.20 0.20 0.20 0.20

1.27 2.55 0.32

1.20 2.55 0.30

I.57 3.34 0.25

I.20 3.34 0.23

‘0.27

TABLE

3

The half-life T1,* (ps) for the different members of the AJ = 1 rotational band in ” “Sb and “‘1 Ill*

‘I%&

Jf g, = 3.40

g, = 2.79

g, =’ 3.40

8, = 2.79

gi

2.78 0.98 I .58

3.29 I.14 1.80

3.45 0.83 I .47

3.97 0.92 1.70

o+ Y+

0.46 0.67 0.41

0.52 0.76 0.44

0.39 0.56 0.33

0.42 0.62 0.36

Y+

I” = 9’ BANDS

135

for “‘Sb as well as for 1211.It is seen that in all cases this half-life is of the order of a few ps and gradually becomes smaller for the higher angular momentum members of the band.

3. unifIed-nKt&l caIalIatIon?3 3.1. HARMONIC

CORE

The harmonic core coupling model has been widely used to describe the interplay between the single-particle motion lm5) [or of the three-particle cluster “-s)] with the quadrupole and octupole degrees of freedom. In order to be able-to describe low-lying 4’ states, one has to introduce an explicit treatment of 1h-2p configurations together with the lp core coupled configurations. Such calculations have been performed recently by Bron et al. 14) but only the lowest members (up to J” = y+) of the 9’ band have been calculated. Thus, one cannot make a full comparison between both calculations. From their results concerning the lowest three band members however, one observes that the anharmonicities calculated by treating the lh-2p configurations explicitly are unable to change considerably the conventional E2, Ml intensity rules as obtained in the harmonic core coupling model. Thus one still finds: (i) the AN = 1 B(M1) values remain small; (ii) the AN = 0 and AN = 1 (AJ = 1) &(E2) values remain small; (iii) the Ji + Ji- 1 E2/Ml mixing ratios, starting from J, 2 y’, show an oscillatory behaviour. The experimental data on ” ‘Sb, however, contradict these intensity rules strongly. As was discussed in sect. 2, the rotational model E2 and Ml intensity rules are fulfilled to a very high degree. The reason for this failure within the harmonic core coupling model can be due to a too-small anharmonicity in describing the doubly even Te (Xe) nuclei by means of 2p (4~) cluster-harmonic core coupling calculations. Furthermore, truncating at three quadrupole phonon excitations could also be one of the reasons for this failure. We have clearly seen (fig. 4) in decomposing the strongcoupling wave functions in a weakcoupling basis (j, J,) that values of the core angular momentum Jl= 8+, lo+, 12+ can be of importance.

3.2. ANHARMONIC

CORE

Recently, Alaga and Paar 34) have shown that by coupling the single-particle (hole) motion with an anharmonic core, decoupled as well as strongly coupled bands on top of the single-particle (hole) configuration can result. Exactly this situation occurs in describing the 3’ bands in the odd-mass Sb and I nuclei. If the interaction describing the anharmonic core is taken as in ref. 34),the total Hamiltonian describing

136

P. VAN

ISACKER

et al.

the single-hole motion coupled with the anharmonic core is given as H =

hoA,,CCb:b:b:l,b,l,+h.c.+H,,,,,

(3.1)

where Hp_o is the harmonic core coupling interaction 3s-37). The anharmonicity parameter A, i is related to the quadrupole moment of the underlying core nucleus by the relation

Q(2:) = - 16(~)*~,,WX

2: + O:);tsmnie.

(3.2)

If we limit the description to odd-mass Sb, I nuclei, as far as the available experimental data indicate 38*3g), the core quadrupole moment Q(2:) is negative in all cases. Thus, due to the introduction of positive A,, values, the anharmonic core coupling changes the harmonic core coupling level scheme such as to induce a strong coupling level ordering for the yrast levels J” = Q+, y’,q+, . . .. More explicitly, up to second-order perturbation theory in the coupling strength and anharmonicity parameter (<, A&, for the one quadrupole phonon multiplet (l& i, 2+)5, the energy

-lD -

0

2

L

-al”L

STREIL”

c;

-

xl

Fig. 8. The calculated energies for the lg,: hole configuration for the holc-anhmnonic core coupling cakulation as a function of the coupling strength C (anhammnkity parameter A,, = 0.6) in the case of lzlI. The phonon energy ho is taken a8 the 2+ excitation energy in ‘anXe (h, = 0.335 MeV).

splitting LIE(J) becomes

-- 35

Q@:)’

256~ B(E2;2;

(3.3)

+ O$

Here, we have made use of the relation (3.2) in substituting A,, for Q(2:); Mlg;‘) denotes the quadrupole moment of the lg; l single-hole configuration and r the conventional holecore coupling strength ‘) for a harmonic core.

Aa=0 A&t6 t =a0 g.ao

qO.6 eaD.O

i /

! I’

En@mmt

\ \ \

\\

I

I \ I I--

B?yJ-‘ng

-zyz ‘95

Fig. 9. Comparison of the anharmonic core coupling calculated p’ band structure (A,, = 0.6). the harmonic core coupling calculation (A,, = 0) as well as the band-mixing result with the experimental data for “II. Energies are given relative to the 9’ band-head.

P. VAN

138

HACKER

et al.

33.2. Level scheme. We would like to discuss in some more detail the 9’ band in ’ 2‘I although the situation for all other I as well as Sb nuclei is completely analogous.

Se&s.e in ‘22Xe only the B(E2*, 2+r + 0:) value is known, one cannot determine uniquely the A,, parameter. Therefore, we have taken A,, = 0.6 as a reasonable value, being able to relate B(E2; 2: + 0:) and Q(2:) in the neighbouring doubly even Te, Xe and Ba nuclei where, in some cases, both values are known. In fig. 8, the results of diagonalizing (3.1) for 1211, up to threequadrupole-phonon states, is shown explicitly as a function of the coupling strength t and A,, = 0.6. Here, the more general statements of subsect. 3.2 concerning the (lg;‘,2+)5 multiplet can be verified numerically. In fig. 9, we compare for dilferent values of the holeare coupling strength C$close to the value obtained from the B(E2; 2: + 0:) value in 122Xe, the resulting 9’ band structure with the result obtained for A,, = 0 and with the results of the bandmixing calculation (subsect. 2.2). One notices the typical narrowing of thej+ R, j+ R - 1 states in the yrast sequence (for A, 1 = 0) as compared with the anharmonic core as well as with the band-mixing results. The introduction of an anharmonic core thus enables the reproduction of a strongcoupling band structure. 3.2.2. Electromagnetic properties. The anharmonic core coupling model also alters the harmonic core coupling intensity ruleS considerably such that for the Sb and I nuclei, where the product Q(2:)Q(lg; ‘) becomes negative, the strongcoupling intensity rules are approached. The corrections originating from the anharmonic core are such that one has to make a distinction between AN = 0 and AN = 1 transitions. (i) For AN = 0 transitions, the reduced E2 matrix element, up to first-order perturbation theory in the coupling strength and anharmonicity parameter (& A,,) becomes (lg&NR

= 2N;J((E21&‘,NR

= 2N;J’)

= (-l)‘++((W+l@J’+l))*

(3.4)

Here Q(I& ‘) = QUG ‘)[e,+t(%)*B(E2

2: -, W*/)/<~>l

(3.5)

contains the lowest-order quadrupole core-polarization corrections. The 6j symbols in (3.4) are such that for AN = 0, AJ = 1 transitions between yrast states the

Jr = 1’

139

BANDS

contributions from Q(2:) and Q(lg; ‘) add up coherently if Q (2:)Q(lg; ‘) -Z 0, which is the case in all odd-mass Sb and I nuclei. (ii) For NV = 1 transitions, the reduced E2 matrix element up to second-order perturbation theory in the coupling strength and anharmonicity parameter (5, A,,) becomes (~g;‘,~-1R-2;JIJE2lllg;‘,NR X I:2

;

;]BIEZ:2:

x
= 2N;J’) + O:)+-(-

= I-1)‘+*((2J+1)(2J’+l)N(2R+l))*

V’+~($C)~~{;

Rf2

:}((ZR+

‘, NR; JI(EZlllg; ‘, NR; J’).

l)N)f (3.6)

Here again, the phases of the 6j symbols are such that for Q (2:)Q(lg;

‘) < 0, for

AN = 1, AJ = 1 transitions, the zero-order term adds coherently with the anharmonic core and lowest-order quadrupolecore-polarization corrections. For AN = 1, AJ = 2

transitions, however, the anharmonic core adds incoherently with the zeroth-order and lowest-order core-polarization contributions. J+R+Z

J*R*2

kR+l J+R

J+R-1

J+R

AR-1

Fig. 10. Schematic figure indicating the harmonic and anharmonic core coupling E2 intensity rules (in the case of a strong-coupling band system). Here R( R = 2N) and j denote the core angular momentum and the single-hole angular

momentum,

respectively.

Only the yrast sequence of levels is indicated.

We have schematically indicated rules (i) and (ii) in fig. 10, in order to show the modifications of the intensity rules when introducing the anharmonic core. If, for the particular case of “‘1, one calculates the E2 moments and transition rates, one observes the rules (i) and (ii) verified in a numerical way in fig. 11, where also the pure harmonic core coupling situation (A2 1 = 0) as well as the results from the bandmixing calculation are shown. The alternation in the E2 intensity within the strongcoupling model (C(: 3, f,) is rather well reproduced starting from the hole-anharmonic core coupling calculation. As in the anharmonic core coupling calculation only three quadrupole phonon states were considered, truncation effects show up rather soon so as to give for J” >=y.’ unphysical results.

140

P. VAN BACKER

er

al.

BAND-MIXING

\

5 0

----____________

ANHARMONIC CORE HARMm,C CORE

-

=i t i-

-

r

-

Fig. I I. The reduced E2 transition probabilities in, respectively, an anharmonic core (thick-dashed line) harmonic core (thin-dashed line) coupling and the complete band-mixing calculation (full lines) for the 4’ band structure in the odd-mass “II nucleus.

... .

BAND -MIXING .-

--_ $-..._..~~____ \ .

Fig. 12. Static quadrupole

PERT. THEORY ANHARMONIC CORE HARMONIC CORE

moments for the 3’ band members in 12’1 from: (i) an anharmonic core coupling calculation (thick-dashed line): (ii) a harmonic core coupling calculation (thin-dashed line); (iii) a perturbational treatment with anharmonic core (dotted line); and (iv) a complete band-mixing calculation (full line).

141

Jx Q 3’ BANDS

In studying the quadrt@ole moment Q (j), otkderives from the diagonal matrix element in (3.5) that for the yrast sequence s” = j+, y+,y’, , . ., the anharmonic corrections always add up coherently with the single-hole quadrupole moment e a numerical application in the case of izlI for the value A, 1 = 0.6 Q(lg;‘).Ifwemak [thus Q(2:) = -0.85 e * b-j and Q(lg;‘) = 2.3 e * b [see eq. (3.5) with eP = l.Se] (fig ll), we observe a pronounced similarity between both the anharmonic core coupling (diagonalization and perturbation theory) and the band-mixing results (fig 12). This equivalence points towards the basic resemblance of both descriptions. For the magnetic properties, the anharmonic core correction does not induce such pronounced regularities. The resulting magnetic dipole moments do not differ substantially from the band-mixing calculations but are generally a factor x 2 bigger then the latter. The same conclusions have been obtained in studying the f+[431] band in the odd-mass In isotopes ‘I). The B(W) values resulting from the band-mixing calculation show a gradual increase with J. An approximate expression (assuming a single-hole configuration 1% ’ and a pure $+[404] band structure) is obtained as

9 (JrzMJ*+Q)

(l&$‘llwl1~‘>*1,o

B(Ml;J,+J,-I)=

JAw+l)

*

I

(3.7)

In the anharmonic core coupling calculation, almost the same values result. In table 4, detailed values for p and B(M1) are compared for both a deformed and a vibrational (harmonic, anharmonic) representation. TABLE 4 The WMI) values and magnetic dipole moments for the different members of the f’ band structure in lzlI, making a comparison bctwuo the band-mixing cakulations, the harmonic core and the anharmonic

core

coupling(A,, = 0.6) calculations

&Ml) (n.m.)2 Jr

9+

P (n.m.) J=

band mixing A,, = 0.6

Y+ g

0.29 0.48 0.66 0.71 039

9’

0.65

0.46 0.58 0.32 0.64 0.50

A,, = 0.0

band mixing

0.43 0.39 0.47 0.24

Y+

4.99 5.13 6.02 5.68 5.38

1.54 8.18 11.33 10.06 9.11

7.58 8.15 11.41 10.02 9.33

Y’

6.38

12.06

12.09

A,, = 0.6

AZ, = 0

A coadlaioa

In studying surfaces with obtained. The of the AJ = .l

the odd-mass ‘13-ligSb a pronounced minimum subsequent band-mixing rotational band structure

and 11’-1271 nuclei, total potential energy corresponding to the $+[404] orbital are calculations resulted in a good reproduction observed in all cases on top of the 3’ state.

142

P. VAN ISACKER et al.

Wave functions describing these particular states have been used to calculate static moments as well as transition rates. Extensive comparison for the l1 %b nucleus, where mixing ratios as well as cascade-tocrossover ratios have been measured, indicates good agreement with a deformed basis. The same band structure can be described in a lh-2p core coupling calculation, as has been performed by Bron et al. 14) recently. Here, the intensity rules resulting for the lowest band members (J” 5 q.‘) deviate considerably from experiment and are still consistent with the harmonic core coupling intensity rules. Introducing explicitly an anharmonic core through cubic terms in the collective Hamiltonian, the energy spectra and intensity rules are modified such as to resemble very much the strongcoupling model intensity rules in the specific situation of the dJ = 1 band structure on top of the 8’ state in odd-mass Sb and I isotopes. Finally, we have noticed that the resulting calculations concerning energy spectra, E2 and Ml transition rates and moments do not differ substantially when calculated in a deformed basis or starting horn a hole-anharmonic core coupling calculation. These equivalences can be explained rather well (sect. 3) through the induced effect of the anharmonic core (aside from the quadrupole core-polarization effects induced within the harmonic core coupling model) which always acts so as to modify the harmonic core coupling model towards the strongcoupling description in the specific case of the lg; ’ hole configuration in the odd-mass Sb and I isotopes. The authors are indebted to Prof. Dr. A. J. Deruytter for his interest during the course of this work and acknowledge the private communication from Drs. V. Paar, R. A. Meyer and D. Fossan. One of the authors (K.H.) is most grateful to Prof. R. Chery for the hospitality at the IPN, Lyon where this study was finished. References i) K. Hey& and P. J. Bruaswd,

2) 3) 4) 5) 6) 7) .8) 9) 10) 11) 12) 13) 14) 15)

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