DESCRIPTION OF THE EVEN TIN ISOTOPES IN A MIXING SHELL-MODEL AND IBA MODEL

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Progress of Theoretical Physics, Vol. 93, No. 4, April 1995

Description of the Even Tin Isotopes in a Mixing

of Shell Model and IBA Model

Der-San CHUU and S.T. HSIEH*

Department of Electrophysics, National Chiao Tung University, Hsinchu 30050 Taiwan

*Department of Physics, National Tsing Hua University, Hsinchu 30050, Taiwan

(Received January 13, 1994)

The positive parity energy levels of even 108

-124Sn isotopes are studied in terms of a mixing of shell model and interacting boson approximation model. In the calculation, the basis states consist of a pure boson configuration and Ns-1 bosons plus a fermion pair configuration, where N8 is the valence boson number outside the closed shell. The fermions are allowed to excite to the s112, d312 ,

d512, [/712 and hw, single particle orbitals. The interaction Hamiltonian contains three parts: pure boson Hamiltonian, pure fermion Hamiltonian and boson-fermion mixing Hamiltonian. The calcu-lated energy levels are compared with the observed data. The wavefunctions are used to calculate

B(£2) values. It is found that the calculated energy spectra and B(£2) values agree reasonably well with the experimental values.

§ 1. Introduction

The nuclear properties of doubly-even Sn nuclei have been investigated recently by a number of experimental and theoretical works.l)-34

J This is because Sn isotopes have been a testing ground for models and methods in nuclear structure calculations.35

J The low-lying structures of such nuclei were found to be dominated by a variety of complicated excitations. Different band structures of Sn isotopes have been observed8

J'12J contrary to the expectation from simple shell model. On the

basis of the shell model,4

J'22J various approximation methods have been developed and a quantitative understanding of level energies and transition rates in these semimagic isotopes has been obtained. It was found that shell effects might be very pronounced in the neutron-deficient Sn isotopes which have a closed proton shell and a small number of valence nucleons. Clement and Baranger4l performed a shell model calculation of even-mass Sn isotopes with mass number A= 108-124. In their work, the structures of the collective 2+ and 3- states of even Sn isotopes were studied and the importance of core excitation was investigated. Since the late seventies, deformed states have been observed in the heavier Sn isotopes.6_J-sJ,l2_J,lsJ _{These states} form a rotational-like band and can be distinguished from spherical states of quasi-particle excitation. These states have been interpreted as a consequence of two-particle two-hole excitations across the proton closed shell in terms of the spherical model. The level structures of even Sn isotopes for mass number around A= 108-124 have also been studied in terms of the broken-pair model.8_J,IOJ,22

J.23J.26J The zero- and one-broken-pair states with two-broken-pair components were mixed10

J to investigate

the spectroscopy of even Sn nuclei for mass number A=112-120. Neutron two-· quasiparticle calculations in BCS approach36

J were used to investigate the experimen-tal data of tin isotopes. Recently, odd and even Sn isotopes were also studied34

J

at National Chiao Tung University Library on May 1, 2014

http://ptp.oxfordjournals.org/

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within a quasiparticle multistep shell-model method.

It is known that the traditional interacting boson approximation37

> can describe

the nuclear collective motion for nuclei which are far away from the closed shell. Therefore, it is not suitable to study tin isotopes which are close to the doubly magic core by using the traditional IBA model. It is also not suitable to describe the level structure of tin isotopes by the shell model because the model space is too large to be manageable for tin isotopes considered in the present work. Therefore, we report here a study on even Sn isotopes by means of the mixing of shell model and the interacting boson approximation model with one boson being allowed to break into a fermion pair. This model has been applied to study the high spin states of even Pt-isotopes.38

> Because of the spin-coupling of bosons and fermions, high spin states

of Pt-isotopes are not allowed in the traditional IBA model but they are able to be well described by the extension of the IBA model as far as the level energies are concerned. The purpose of this investigation is twofold: in the first place we want to present a systematical study of the structure of positive parity energy spectra of even-mass Sn isotopes. Second and the most important, we desire to investigate to what extent the extension model of IBA can be applied to interpret the observed low lying energy states of even-mass tin isotopes. These isotopes have closed proton shell Z=50 and small boson numbers. They cannot be described satisfactorily by the shell model and IBA model. The rest part of this paper is organized as follows: In § 2 we describe the model. In § 3 we present results of this work. The fourth section presents our discussion and conclusions. A brief summary will be presented in the final section.

§ 2. The model

The even-mass Sn isotopes with Z=50 and 1242A2108 will be studied systemat-ically. Taking Z=50, N=50 as a closed shell, the boson numbers for isotopes

108

-116Sn are NB=4, 5, 6, 7 and 8, respectively. For the other isotopes which pass the

neutron midshell, Z =50, N = 82 is taken as a closed shell, the neutron boson numbers are counted as one-half of the number of neutron holes. Therefore, isotopes 118

-124Sn have valence boson (hole) numbers '1, 6, 5 and 4, respectively. In this work it is assumed that one of the bosons can be broken to form a fermion pair. The fermions are allowed to occupy the s112, da12, d51z, g11z and hll/z single particle orbitals.

Our model space includes an IBA subspace with NB bosons and a subspace with NB-1 bosons plus two fermions. Explicitly, it can be expressed as:

lnsndJ ..

I>EBI

nsiidDL,

Udz)];

I>,

where ns+ nd= iis+ iid+ 1 and

J

=0, 2, 4, 6, .... The model Hamiltonian can be expressed as38

>

H=HB+HF+ VBF. (1)

HB is the IBA boson Hamiltonian

(2) where cd is the single d-boson energy and pt.p, L·L, and Q·Q are the pairing interboson, the spin-dependent and quadrupole interactions respectively and can be

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expressed as: iid=d1 •

J'

and P=(1/2)(J · J)-(1/2)(

s ·

s), L=!iO (d1 _{X J)O>} (3) (4) (5)

Q= Qe+a ~(a], X ai2)<2>+ jj' ~[(a], X a].)C4> X

J-

d1 X (ah X ai2)<4_>]<2_>, ₍₆₎

Jl,J2 JI,.12

where a] is the nucleon creation operator and

(7) The octupole term T3· nand the hexadecapole term T4· T4 have been omitted in He since they are generally believed to be less important. The fermion Hamiltonian HF is

HF="i;.sr/2j+1(a] X aj)C0

>+(1/2) . ~- . VoJ2J+1 [a],xa].)lx(aj.x iL.)'](O).

J J ~Jl ,J2,J3,J4

(8) The mixing Hamiltonian VeF is

(9) which can be rewritten as

VeF=a ~ Qe·(a], X a;z)<2

>+ jj' ~ Qe·[(a], X a]2)<

4

> X

d-

d1 X(ah X ai2)<4>]<2>, (10) Jt,J2 JI,J2

where the j's run over five single fermion orbits: S112, d312, dstz, {htz and hwz. In the above expressions, as usual the boson-fermion exchange interaction is not in-cluded.38Ho> The constants a and Pin principle depend on j, and jz. For a quadrupole force one may expect that a is roughly proportional to the reduced matrix elements of Yz or r2 _Y2. _{For partially filled orbits one expects in addition a modification along the}

line predicted by BCS. Similar arguments may apply toP. However, in this calcula-tion we assume that the parameters a and Pare independent of the quantum numbers of the single particle state energies because tin isotopes considered in our work are in the vicinity of a closed shell, and therefore, the variation of single particle energies is not large. In the calculation of two-body matrix elements, the form of fermion potential is assumed to be the Yukawa type with Rosenfeld mixture and the harmonic oscillator wavefunction is adopted. The oscillator constant v=0.96A -113

fm-2 with A= 100 is assumed. The parameter

Vo

in Eq. (8) is the interaction strength of fermion-fermion interaction which is assumed to be -60 MeV. The other interaction strength parameters contained in He and single particle energies contained in the total Hamiltonian are chosen to reproduce the energy level spectra of even Sn isotopes with mass number between 108 and 124. In the shell-model calculation, energy spacings between Sl!z and d312 orbitals, and between dstz and {htz orbitals are found to be very

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small. To reduce the number of parameters, we thus assume that the single particle energies of the s112 and d312 orbitals in all tin isotopes have the same value, and the

energies of the single particle orbits ds12 and 9712 are also assumed to be the same value. The energy bands with the above-mentioned two kinds of configurations are mixed through the diagonalization of the energy matrix in the whole model space.

§ 3. Results

The interaction strengths and single-particle energies for Sn isotopes are allowed to be mass number dependent. Table I shows the best fitted interaction strengths, single particle energies, the number of levels included in the least squares fitting (n)

and the root mean square (rms) deviations of the calculated level energies from the observed data for all isotopes in the present calculation. The interaction strength a3 of the quadrupole term Q· Q, and the mixing parameters a and

/3

are found to be able to be unified as (in MeV) a3=0.033, a=O.O and /3=0.025. One can note from Table I that the interaction strength for the d -boson energy cd decreases monotonically from isotope 108

Sn to 116

Sn and then increases monotonically from the nucleus 118_{Sn to}124_Sn; the parameter a1 of the pairing term pt · P decreases from 108_{Sn to}110_{Sn and then} changes sign from 110

Sn to 112

Sn. It decreases again from 112

Sn to 116

Sn. Finally a significant change of a1 happens from ussn to ussn and then becomes a constant from ussn to 124

Sn; the parameter a2 of the L· L term increases monotonically from isotope 108_{Sn to ussn and then decreases monotonically from}118_{Sn to}124_{Sn. These parameters} have significant changes around the isotopes 116_{Sn and ussn. This is because in our} work we assume the Z=50, N=50 nucleus as a closed shell for 108

-116Sn isotopes, and the boson number for these nuclei is counted as one-half of the number of nucleons outside the closed shell, while for the other Sn isotopes (ll8

-124Sn) that pass the neutron midshell, the nucleus Z =50, N =82 is assumed as a closed shell and the neutron boson numbers are counted as one half of the number of neutron holes. Therefore, we have particle-particle to hole-hole transition from 116_{Sn nucleus to ussn nucleus.} _It_{is not} surprising that we have a significant change of the interaction parameters of cd, pt · P and L· L around the A~ 116 region. The boson pairing interaction pt ·Pis normally

Table I. The interaction parameters and single particle energies (in MeV) adopted in this work.

Nucleus Ed a, a, Ein(€312) Est2 (Eon) €ttl2 n rms

particle-particle '"•sn 1.2750 0.1500 -0.0054 1.60 1.43 2.310 12 0.100 uosn _1.1167 _{0.0616 -0.0048} 1.60 1.43 2.310 14 0.094 '12Sn 0.9020 -0.0180 0.0008 1.60 1.43 2.310 15 0.097 '"Sn 0.7167 -0.0568 0.0074 1.60 1.43 2.003 18 0.113 "6Sn 0.5121 -0.0687 0.0200 1.40 1.22 1.583 17 0.112 hole-hole 118 Sn 0.7220 -0.0100 o.o:n8 1. 75 1.55 1.562 12 0.118 12oSn _0.7529 -0.0100 o.o:n5 1.57 1.55 1.454 11 0.082 '"Sn 0.9215 -0.0100 0.0048 1.57 1.60 1.365 13 0.096 '"Sn 1. 0196 -0.0100 -0.0032 1.57 1.60 1.338 10 0.085

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responsible for the corresponding fermion interaction and its strength parameter in general should not change from a microscopic point of view. One can note from Table I that our parameter a1 keeps constant for isotopes 118

-124Sn and thus it is consistent with a microscopic point of view. However, the parameter a1 decreases monotonically from 108

Sn to 116

Sn. The reason for this inconsistency with a micro-scopic point of view can be explained as follows. It is known that for a nucleus in the

0(6) limit, only pt · P and L· L terms appear in the Hamiltonian HB; and for a nucleus in the SU(3) limit, only L· L and Q· Q terms appear in HB.41

> From the correlation

of the variation of interaction parameters to the limiting symmetries, the decreasing of the parameter a1 manifests that tin isotopes have the tendency of deviation from 0(6) symmetry to become the SU(3) symmetry as the mass number A changes from A= 108 to 116. One can note from Table I that the single particle energies €1!2, e312 ,

Es12, €912 and Ew2 vary smoothly with respect to the mass number A. This is due to the

following reasons: the nuclei under considering are near the closed shell and the mass region of the nuclei considered in this work is not large. In addition to these, according to our model tin isotopes have closed proton shells, thus only n- n (neutron-neutron) interactions are involved. This makes the single particle energies vary smoothly with respect to the mass number. One can also note from Table I that the interaction parameters and single particle energies can most likely be unified for the isotopes which are close to the closed shell. For example, the parameters of the isotopes 108_{Sn, nosn and}112_{Sn are almost the same. A similar situation can be found}

also in the isotopes 122

Sn and 124

Sn.

The calculated and observed energy spectra of the Sn isotope chain are shown in Figs. 1 ~4. In these figures, we present all states which are included in the least squares fitting. For those states which are not included in the least squares fitting, we present only the lowest energy level of each angular momentum. The states with too high excitation energy are also omitted to make the figure as clear as possible. The states marked with asterisks are not included in the fitting. These states are beyond our model space because they contain core excitation configuration or the configuration of breaking two bosons into two fermion pairs. In Fig. 1 we compare the calculated energy levels with observed data for isotopes 108

Sn and 110

Sn. There have been abundant experimental data for the isotopes 108_Sn°-3

> and 110Sn3>.s>-s> observed in recent years. In the 108

Sn nucleus, the experimental data reveal that there exists a distinct positive-parity band up to the (22+) state, while in the 110_{Sn nucleus,} the positive-parity band is found up to the (20+) state. It was found2

> that a deformed

band above the

J

= 14n level can be identified in the nucleus 108

Sn and a collective behavior exists in the yrast states of nosn above the 103 + state. The even-mass Sn

isotopes have been studied by using shell-model calculation.4

> Fairly good results

have been obtained. In our calculation, one can note from Fig. 1 that energy levels for the 108_{Sn nuclei can be reproduced quite satisfactorily. For the nucleus nosn, the} large deviation of 4 + states may be due to the inclusion of 04 +, 24 + and 2s + states in the

nucleus with a small boson number (the boson number of nosn is only 5). In recent years, intruder states in nearly single-closed shell nuclei have been investigated both experimentally and theoretically.6

>-s>.12>'15> In even-even nuclei near Z =50 and 82, these states are known as the 02 + states at very low excitation energies and several

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1os₈n 108 - 161--- 141---6 - 121\ ~---· 12~.. -:: ... _ _ _ _ 4 121--- 103121---121---121---121---121---121---121---121--- 103---1a2 \ 10,, -82 ====:::::=== o j - - , Bf _ _ .:..,·2 -02--- '---"

sf--->:=

4f 2f -0 O f · · -Expt. Thea. 9 -3.0 2.8- 2.6- 2.4- 2.2- 2.0-181---· 161--- 1411 2 i -10j___ ~ 1 0 2 - - - * 101---~~~~ B f -2 5 - - ... f "' 2 ! - - ',~!, 62 '\ ~~/' ... ' -at =======' ' -43--·, oj ... _____ '~-=-=-23----_.r:_ BT __;~" 42/ 0 : \ - -.. 4+ _{1 - - , .}/ -·,/ 2 + - - - " \ 2 ' \ ,

__

2 f O f · -Expt. Thea.

Fig. 1. Calculated and observed energy spectra for the nuclei 108_{Sn and}110_Sn.

collective states which have been found to be built on them.12l They are understood

as proton excitations through the closed shell to the next major shell.7) In the isotope

110_{Sn, the levels such as 02}

+, 12, +, 14, +, 16, + and 18, + are assigned to the members of the

intruder states.1l Since in our model we do not consider the core excitation, thus in the beginning of our calculation these states were not included in the least squares

fitting. However, large rms deviation was found in our calculation unless the 02 +

state was included in the fitting. This may be ascribed to the reason that the 02 + state

is not a pure intruder state. In fact, the wave function analysis, as presented in Table

II, manifests the 02 + consists mainly of the pure boson configuration and thus cannot

be regarded as a pure intruder state. One can see from Fig. 1 that surprisingly, our calculated results for other intruder states still agree reasonably well with observed values although they are excluded from the least squares fitting. The satisfactory reproduction of intruder states in the present calculation may be ascribed to the reason that these spurious states may not be pure intruder states, instead, they may be mixings of the core excitation configuration and the core inert configuration. In

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Table II. The relative intensities of wave functions for energy levels of isotopes

108_{Sn, nosn and}112_{Sn. The total intensity is normalized to}_1.0. _j,_(or_j,)=s112,

da12, ds12, f!ll2 or hu12, and j, =I= J,.

nucleus nosn Il2Sn

o,

1.000 0.000 0.000 0.000 0.000 0.000 0.000 02 0.000 0.000 0.000 0.500 0.500 0.000 0.000 Oa

o.

2, 4, 6, 8, 82 10, 102 lOa 12, 122 12a 12. 14, 142 16,

o,

02 Oa

o.

2, 22 2a 4, 42 4a 6, 62 8, 10, 102 lOa 12, 14, 16, 18,

o,

02 2, 22 2a 4, 42 0.997 0.000 0.000 0.000 0.500 0.500 0.998 0.000 0.000 0.080 0.000 0.000 0.000 0.000 0.000 0.001 0.000 0.000 0.262 0.000 0.000 0.012 0.000 0.001 0.000 0.000 0.177 0.259 0.500 0.012 0.000 0.000 0.000 0.000 0.000 0.001 0.000 0.001 0.000 0.001 0.000 0.002 0.481 0.740 0.500 0.033 1.000 0.000 0.000 0.942 0.000 0.000 0.500 0.000 0.500 0.003 0.028 0.920 0.049 0.000 0.000 1.000 0.500 0.000 0.500 0.135 0.735 0.130 0.025 0.941 0.034 0.000 0.000 1.000 0.000 0.004 0.996 0.000 1.000 1.000 0.000 0.000 0.000 0.000 0.995 0.000 0.000 0.001 0.001 0.000 0.000 0.000 0.003 0.000 0.000 0.990 0.998 0.996 0.000 0.946 0.014 0.030 0.000 0.949 0.961 0.000 0.000 0.954 1.000 0.996 0.998 0.000 0.500 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.997 0.000 0.000 0.000 0.962 0.000 0.014 0.000 0.000 0.500 0.000 0.000 0.000 0.001 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.500 0.000 0.002 0.000 0.001 0.318 0.010 0.001 0.319 0.000 0.009 0.007 0.000 0.500 0.000 0.002 0.000 0.001 0.318 0.010 0.001 0.319 0.500 0.009 0.004 0.000 0.000 0.000 0.000 0.000 0.000 0.026 0.002 0.003 0.008 0.000 0.001 0.001 1.000 0.000 0.000 0.006 0.002 0.002 0.337 0.032 0.981 0.324 0.500 0.032 0.027 0.000 0.000 0.000 0.500 0.000 0.500 0.000 0.008 0.008 0.001 0.029 0.000 0.000 1.000 0.000 0.000 1.000 0.000 0.000 1.000 0.000 0.000 0.000 0.000 0.000 0.001 0.000 0.000 0.001 0.002 0.318 0.000 0.005 0.000 0.002 0.000 0.001 0.000 0.001 0.318 0.003 0.000 1.000 0.000 0.000 0.000 0.000 0.021 0.000 0.006 0.000 0.000 0.002 0.002 0.001 0.341 0.030 0.978 (continued)

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4. 0.000 0.000 0.044 0.062 0.044 0.190 0.660 61 0.000 0.000 0.000 0.000 0.244 0.001 0.755 6z 0.000 0.000 0.000 0.000 0.743 0.025 0.232 6. 0.933 0.000 0.000 0.013 0.018 0.004 0.032 81 0.000 0.000 0.000 0.000 0.489 0.021 0.490 82 0.938 0.000 0.000 0.012 0.010 0.005 0.035 10, 0.000 0.000 0.000 0.000 0.000 1.000 0.000 10z 0.942 0.000 0.000 0.010 0.006 0.006 0.036 12, 0.000 0.000 0.000 0.000 1.000 0.000 12z 0.000 0.000 0.000 0.500 0.000 0.500 141 0.000 0.500 0.000 0.500 161 0.000 1.000 0.000 18, 0.000 1.000 0.000 201 1.000 0.000

Sn isotopes the intruder states were usually interpreted as a consequence of two-particle two-hole excitations across the proton closed shell in terms of the spherical model. Within the framework of IBA model these two-particle two-hole fermion configurations can be mapped onto a two-boson state making no distinction between the particle boson or hole boson.42l Therefore, it is expected that such intruder states

should contain the admixtures of NB and Ns+2 configurations for these intruder states. Hence the reproduction of these states in this work seems to support the above interpretation.

Figure 2 shows the calculated and observed energy levels for the nuclei 112

Sn and 114Sn. These two nuclei have been intensively investigated both theoretically8l'10H 2l

and experimentally.1l-9

l'12H 6l It was found that for the nucleus 112Sn a distinct positive-parity band up to the 22+ state exists, and an intruder band which starts from the 6+ state was observed. The collective bands existing in these two nuclei can be in general reproduced by the proton excitation model. W enes et aJ. 11l studied the

nuclear structures of doubly-even Sn nuclei 112- 118Sn by constructing a model taking into account both pure quadrupole vibrational excitations as well as proton 2p-2h configurations coupled with the quadrupole vibrational excitations. Satisfactory results were obtained by them. Recently, Harada et al. 15l used a model based on the

spin-projected Hartree-Bogoliubov model to study high spin energy levels and back-bending phenomena of the nucleus 114Sn. The backback-bending phenomenon in the intruder band of 114Sn was reproduced satisfactorily. One can note from Fig. 2 that our calculation can also reproduce the energy spectra of 112

Sn reasonably well, especially the high spin states. In 114_{Sn, states 02+, 63+, 81+, 10z+, 121+, 122+, 141+, 161+,}

181 + and 201 +were assigned to the intruder bands.7l In our calculation, we include 02 +, 63+, 81+, 102+, 121+, 12z+ states in our least squares fitting in order to yield smaller rms deviation. Furthermore, the calculated energy values for 141 +, 161 +, 181 + and 201 + states agree also reasonably with their experimental counter parts although in our model we did not include the core excitation configuration. The reason of the coincidence may be ascribed to the reason that these states may not be pure intruder states, instead, they may include the core excitation configuration and the core inert configuration altogether as mentioned above. For nucleus 114Sn, the calculated energy levels agree quite reasonably well with the experimental values. Most of the

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10 8 0 -114sn 2 2 1 - - 2 2 1 2 0 f - 1Sf1 4 f 1 2 2 - 121--2 0 f - - * 1Bf--.!!" 16j--.!!" 141--* 12~---12f _ _ _ 102-- 102---1 0 102---1 - - 1 0 1 -a 1 - -

ar---2 1 - - _{2 1}

-or--

O j -Expt. Theo. 10 - 2ot--s - 1 S j " 1 6 ; ' - 141--6 0 -

or--Expt.

Fig. 2. Calculated and observed energy spectra for the nuclei '12_{Sn and}114_Sn.

2 0 f 1 a t 1 6 ; ' -141 12~---.

or--Theo.

calculated and observed levels agree within a few tens of KeVs. There are only two

levels, 02 + and 22 + which are calculated in reversed order.

The calculated and observed energy levels for the nuclei 116

-u8Sn are shown in

Fig. 3. Many works have investigated the level structures of the nuclei 116

-u8Sn

experimentally and theoretically.18

H 6> Raman and his co-workers20>·20 studied the

level scheme of u6Sn by combining the results of 115_Sn(_n,

_r

)u6Sn and u6Sn( n, n'

r )

116_Sn

experiments. They constructed a nearly complete level scheme of the 116_{Sn nucleus}

by comparing the observed data to the combined predictions of the two-broken-pair model, the interacting boson model and the deformed collective model. From a

neutron pickup reaction/9

> it can be found that the neutron shell occupations in the

nucleus u6_{Sn are not restricted to}

2ds!2 and lgm orbitals only, but other orbitals, lhnt2,

2dat2 and 3sl!2, have also sizable strengths. The broken-pair model was usually adopted to study the energy levels for these two nuclei and reasonable results were

obtained for the lower collective states. Our calculation for the nuclei u6

Sn yields satisfactory agreement with the observed data. Most of the calculated levels are in

correct order except 22 + and 3t + states. For nucleus ussn, our theoretical results

show that reasonable agreement exists between the theoretical and experimental data

except for 02 +, 22 +, 2a +, 24 +, 2s + and 61 + states.

Figure 4 shows the calculated and observed energy spectra for the doubly even

nuclei 120- 124Sn. There are abundant observed data for these three nuclei.24>'26>-aa>

The experimental level schemes of these nuclei have been compared with the level

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5 4 3

>

"'

6 w 2 0 102---?t--,, ' '

~---1====:=---===

~---:~~---

.,

_________ _

31 ~ ~ ... st,----~~---2S) _ _ _ _ _ :;.~:::~ 4~ ~-- -~--..-::::: _{2+ _ _}_, _'...:.' _ _ ~ 4~'--... /' 4t1- - -,r ... -231--~r_-- -2~}___t -~! ~---21--, o t -Expt. Thea. 5 -10~--... ,,, 4 -8 2 - - ' 3 2 2j 1 -0 _ ot -Expt. Thea.

Fig. 3. Calculated and observed energy spectra for the nuclei "6_{Sn and}118_Sn.

schemes calculated on the basis of a broken-pair model which included up to two broken pairs,26

> and reasonable agreement has been obtained. Our calculation on

these three nuclei shows that satisfactory agreement can be obtained for the lower lying levels. However, some calculated levels almost shrink together in the level spectra of nuclei 122

Sn and 124_Sn.

We also analyze our wave function for each state of these even-even Sn isotopes to study the relative intensities of the pure Ns bosons configuration and the Ns-1 bosons plus two fermions configuration. For illustration, we list the relative inten-sities of the wave functions for isotopes 108_Sn,110

Sn and 112

Sn in Table II. For the nucleus 108_{Sn only} _{Ot+, Oa+, 2t+} _and_8z+ _{states are dominated by the pure boson}

configuration. All other states contain significant components of the configuration of Ns-1 bosons coupled with two fermions. Most of the states with Ns-1 bosons coupled with two fermions occur with a fermion pair in the same orbit. Table II also shows that states Ot+, Oz+, Oa+, 2t+, 2z+, 3a+, 4t+, 44+ and 6z+ in 116_{Sn; states Ot+, Oz+, Oa+,} 2t+, 2a+ and 4z+ in 118_{Sn; states Ot+, Oz+, Oa+, 2t+, 2z+}_{and 4z+}_in120_{Sn; states Ot+, Oz+, 2t+,}

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1 -~-11" 6~ -4 5 - - , 4~ - -.... ~,, 4- ... ''----.1!.. 3 - 3'l - - , , I ' / ==:; 1ct\ -·~ 2~~:---<·

---8;

- =

~ ;;._'-'==-ojt _ _ _ __ 61 ~~r---2~ - - : , / 4~~ ... 22 -4-1-/-====:,:...-:..:.= 2 - ; I o2 -2i - - - _{2i ·} -124sn 3 - 3j 4~ o!j =~- ~ 10j _ _ _ ::./ 8;+ _ _ :;:~/ 2~/-- ,' 61" 2~)===~~-~ 0~ -2-41/ 2i -:: o

L

or --·---

o

I_

or

Ex pt. Theo. Expt Theo. Ex pt. Theo.

Fig. 4. Calculated and observed energy spectra for the nuclei 120_Sn,122_{Sn and}124_Sn.

2z+, 24+ and 41+ in 122_{Sn; and states}_01+,_Oz+,_{21+, 2z+}_and₄₁₊_in124_{Sn are all dominated}

by pure boson configurations. One can note from Table II that the Oz + state in 108Sn

consists mainly of Nn-1 bosons configuration, while in other nuclei the Oz + state consists of pure boson configuration. This can be attributed to the reason below. The isotope 108Sn lies closer to the doubly closed shell and its Oz + state is higher than

the 61 + state. However, in isotopes 110Sn and 112Sn, their Oz + states are all lower than

the corresponding 61 + states. Hence, the Oz + state in 108Sn possesses relatively higher

excitation energy than those Oz + states in other two isotopes. And this is why it

consists mainly of Nn-1 bosons configuration. One can see, from Table II, that, in those states which have both Nn boson and Nn-1 bosons plus two fermions configura-tions, the intensities of the former component are in general close to either 1 or 0. This means the mixing between the two configurations is usually smalL However, the inclusion of the configuration of boson breaking is still necessary in some cases although the mixing is smalL Because of the spin-coupling of bosons and fermions, high-spin states are not allowed in the traditional IBA model but they can be well described by the extension of the IBA model as far as the level energies are con-cerned.38> In Sn isotopes, the proton shell is closed, the level structure thus cannot be well described by the traditional IBA modeL However, as we have seen from Tables II and III, our extension of the IBA model with a small mixing between the configurations of the pure boson and Nn-1 bosons plus two fermions can well

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describe the level energies and the B(E2) values of the Sn isotopes.

Some experimental B(E2) values for tin isotopes1_>·3_>·5_>.s>.13_>·16_>·25_>·27_>·26_>·30_>-33_>_have

been observed. A study of B(E2) values will give us a good test of the model wave functions. The electric quadrupole operator can be written as

T(E2)= e8Q8

+

eF a

::E

(aJJih)(Z)

+

/3e 8

::E

[(a],a].)<4>

J-

d t

U.h

ajz)(4)](Z), (11)

h.h ii,h

where Q8

is taken as

QB=(dt s+std)(2)_K(dt d)(2). ₍₁₂₎

In our calculation the fermion effective charge eF is assumed to be 0.5 eb and the

boson effective charge e8

is assumed to be 0.1 eb. It was found that a different

fermion effective charge cannot yield a significant change in B(E2) values. The

parameters a and /3 are assumed to be the same as those used in the mixing

Table III. The calculated and observed B(£2) val-ues for Sn isotopes. The results listed in the column abbreviated as "cal. I" are obtained by using K=O.O and those listed in the column ab-breviated as "cal. II" are obtained by using K

=-1.322. A 112 114 116 118 120 122 n

expt. cal. I cal. II 4 . 87 4 . 02 4- 23 2 1.25 20.3 1.57 1.79 6.82 6.64 1.65 0.00 0.00 5.25 3.80 4.07 7.22 9.25 8.20 6.59 6.64 2.07 2.18 3.90 3.47 3.80 6.05 8.93 8.07 1.65 9.05 0.91 1.14 0.13 0.23 7.39 0.72 0.20 1.31 5.62 7.84 2.00 2.92 0.25 6.38 5.93 6.12 4.33 4.15 4.45 6.53 9.77 8.41 5.94 3.80 4.06 6.53 8.88 7.44 2.11 5.90 8.22 o, 7.73 26.2 2.42

o,

3.84 3.60 3.77 2, < 2. 19 7. 98 6. 42 2, 3.59 5.71 5.45 2, <2.16 0.32 0.31 4 2 3 2 2 2 3 2 2 3 1 2 2 3 2 2 2 2 2 2 4 2 2 2 2

Hamiltonian. Two different values of K

are chosen in our calculation; one is

chosen as 0.0 (calculation I) and the

other is chosen as -/7/2 (calculation II)

which is one of the generators of the

SU(3) group. It is found that the

different values of K can only yield some

change in the B(E2) values of the inter-band transitions. For illustration, we plot the calculated B(E2) values for the yrast band transitions for the nucleus

114_{Sn by using two different values of}

K.

- ---- x~o.o

10

2 4 6 8 1 0 12 14 16 18 20 22

Fig. 5. Calculated and observed B(£2) values of transitions from L state to L;-2 state for the yrast band transitions of the nucleus 114_Sn.

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As shown in Fig. 5, one can note that the different values of K (the solid line is obtained by using K=-.fi/2 and the dotted line is obtained by using K=O.O) cannot yield an important change in the calculated B(E2) values for the yrast band transitions of the nucleus u4Sn. The deviation becomes prominent only for ]; being equal to or larger

than 16. Table III lists the calculated and available observed B(E2) values for Sn isotopes. Two different values of K are used to calculate the B(E2) values. As

shown in the table, the values listed in the fifth column (abbreviated as cal. I) were obtained by using the value of K=O.O, while the sixth column (abbreviated as cal. II) are obtained by using the value K=- .fi /2. One can note from Table III that the calculated B(E2) values for the yrast band transitions are almost insensitive to the different values of K, while the calculated B(E2) values for the inter-band transitions

depend on the choice of the K value. However, most of our calculated B(E2) values

agree with the experimental data within the same order of magnitude. § 4. Discussion and conclusions

As we mentioned above, we have studied the structure of the positive parity energy spectra and B(E2) values of Sn isotopes. In a preliminary study of tin isotopes, we found that high spin states of tin isotopes were not able to be reproduced if only pure boson configurations were considered. For example, the highest value of the angular momenta for the states in nuclei 112

Sn and 114

Sn can be reproduced up to L= 12 and 14, respectively if only pure boson configurations were taken into account. In the present work we find that accurate results for the energy spectra and B(E2) values can be obtained by considering a mixing of the shell model with the extended IBA model which includes a pure boson configuration and a configuration of a boson being broken into a fermion pair. To compare our results with those obtained by previous works, we list the available calculated results of level energies of even mass tin isotopes in Table IV. Clement and Baranger4

> used shell model to calculate the

energy levels of the first 2+ and 3- states in u6Sn and 120Sn with the Tabakin

interac-tion as the residual interacinterac-tion. In their calculainterac-tion twelve single-particle proton and neutron levels and the BCS quasi-particle approximation were included. Although their calculated 2+ level energies agree with the experiment within a few tens of KeV s, however, there is no evidence that the level energies for other higher spin states can also be reproduced well in their model. Van Poelgeest et al.8

> investigated high-spin

(J:::;: 10) neutron quasi-particle excitations in the even-mass 110

-118Sn nuclei using

Cd(a, 2nr)Sn reactions. The experimental data were interpreted within the frame-work of the broken-pair model including proton 1p-1h excitations. Although the excitation energies calculated by them can fairly be accounted by their model, however, the experimental transition rates were not able to be reproduced well. Weenes et al.ll) investigated the collective bands in doubly-even Sn nuclei u4-u8Sn.

They treated proton two-particle two .. hole excitations coupled with spherical qua-drupole vibrations in interaction with the low-lying quaqua-drupole vibrational states. The regular L1]=2 band structure on top of excited ]"=0+ states were able to be accounted. One can note from Table IV that although the low lying states for these nuclei were accountable by their model, however, calculations of states in non-ground

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Table IV. Comparison of available calculated level energies Ecai and

observed data Eexpt.

Nucleus states Eexpt Ecal

This work Ref. 11) Ref. 20) Ref. 26)

uosn 2, 1.212 1.106 1. 011") 6, 2.477 2.582 2. 248"1 Il2Sn 2, 1.257 1.159 1. 082•> 6, 2.549 2.733 2. 422"1 , .. Sn ₀₂ _1.953 _2.140 _1.901 Oa 2.155 2.999 2.862 2, 1.300 1.164 1.381 1.49"1 22 2.239 2.016 2.324 2a 2.454 2.266 2.693 4, 2.188 2.262 2.581 6, 3.189 3.049 2.970 8, 3.871 4.001 3.582 10, 4.140 4.145 3.983 12, 5.183 5.263 4.901 116_Sn 02 1.757 1.893 1.940 1.535 Oa 2.028 2. 710 2.871 2.242

o,

2.546 2.753 2.367 Os 2.791 3.113 3.598 1, 4.252 2.753 2.730 2, 1.294 1.147 1.402 1.549 1.245b) 22 2.112 2.019 2.403 2.200 2a 2.225 2.174 2.702 2.742 2, 2.650 2.560 3.734 2.846 2s 2.844 2.813 3.176 3, 2.996 2. 711 2.920 32 3.180 2.753 3.190 3a 3.315 2.969 3.264 3, 3.371 3.059 3.697 4, 2.391 2.238 2.651 2.501 42 2.529 2.709 3.065 2.977 4a 2.801 2.786 4.058 3.071 4, 3.046 3.076 3.247 4s 3.212 3.186 3.642 6, 3.032 2.803 3.142 3.345 62 3.277 3.391 3.970 3.697 8, 3.493 3.478 3.437 3.535 82 3. 714 3.838 4. 712 5.380 10, 3.547 3.478 4.552 3.443 3. 750"1 102 4.507 4.563 5.500 4. 768"1 118_Sn 02 1.758 1.908 1.764 1.999

o,

2.057 2.847 2.694 2.375

o,

2.497 3.048 3.167 Os 3.137 3.072 3.622 2, 1.230 1.189 1.287 1.236"1 _1.230 22 2.043 1.819 2.247 2.045 2a 2.328 2.145 2.174 2.181 (continued)

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120 Sn t22Sn a) Ref. 8). b) Ref. 4). 2.403 2.677 2.280 2.489 2.734 2.963 2.879 2.999 3.052 3.692 3.108 4.495 5.379 1.875 2.160 2.587 1.171 2.097 2.355 2.194 2.466 2.644 2.698 3.058 2.836 2.902 2.088 2.674 1.140 2.153 2.415 2.735 2.775 2.797 2.142 2.331 3.082 3.233 3.305 2.555 2.690 2.765 2.754 2.897 2.345 2.477 2.867 2.969 3.048 3.072 3.072 4.134 3.072 4.225 4.225 1.863 2.763 2.834 1.168 2.097 2.507 2.289 2.438 2.467 2.692 2.982 2.834 2.834 2.093 2.629 1.126 2.155 2.509 3.009 3.059 2.847 2.144 2.474 2.895 3.080 3.093 2.629 2.629 2.629 2.485 2.883 2.833 2.868 2.906 5.242 3. 501") 2.550 2.835 2.389 2.749 2.934 3.053 2.856 3.798 2.997 3.675 3.034 4.431 4.415 2.066 2.761 3.324 1.171 2.146 2.364 2.214 2.817 3.174 3.416 3.469 2.823 2.831 2.273 3.023 1.140 2.248 2.519 2.826 3.409 3.080 2.175 2.735 3.303 3.536 3.682 2.645 2.807 2.809

state bands usually yielded prominent discrepancies between the theoretical values and observed ones. On the contrary, our calculation for the corresponding levels can yield more accurate results than theirs. Raman and his co-workers studied the nuclear excited states of even-even nuclei 116Sn20

> and 118-122Sn26> by combining the

results of 115

Sn(n, r)116_{Sn and}116

Sn(n, n'r)116

Sn experiments and by means of the decays of 118

-122ln isomers. Their observed data were compared with calculated level

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energies obtained on the basis of two broken-pair model. One can note from Table IV that, in general, their calculations for the higher excitation states in non-ground state bands yielded larger deviations from the observed data when compared with our calculated results. Especially, the states Os, 23, 2s, 43, 44, 4s, 6z, Bz and 10z of 116Sn; 05 , 43 and 6z of 118Sn; 04, 4z, 43, 44 and 4s of 120Sn; 03, 2s, 43, 44 and 4s of 122Sn. All of these states were calculated by them with deviations around several hundred KeVs or even larger when compared with the experimental data. In our calculations, there are several levels, e.g., 1t and 33 of 116Sn; 82 and 12t of 118Sn; 23 of 120Sn; and 24 of 122_{Sn which}

are reproduced worse than the previous works. However, one can note from Table IV that our calculated results are superior to those of previous works for most of other levels. Therefore, one can conclude that positive parity states of even-even Sn isotopes with mass number A= 108-124 can be well accounted systematically by the interacting boson approximation model provided the configuration of a boson being broken into a fermion pair is included in the traditional pure boson configuration.

§ 5. Summary

In summary, we have investigated the structure of the positive parity energy spectra of tin isotopes with mass number between 108 and 124. We mixed the shell model with the IBA model and include the configuration of a boson being broken to form two fermions which can occupy the s112, d3tz, dstz, !fm and hwz single particle orbitals. The calculated energy levels and B(E2) values are in satisfactory agree-ment with the observed values for the whole chain of tin isotopes. The analysis of wave functions shows the mixing between the pure boson and NB-1 bosons plus two fermions configurations is small. This is similar to the result obtained in the calcula-tion of high spin states of Pt isotopes.38> Because of spin-coupling of bosons and

fermions, high-spin states are not allowed in the IBA model but they can be well described by the extension of the model. In the present study the level energies of Sn isotopes cannot be well described by the use of IBA model because these nuclei are close to doubly closed shell. However, our calculation shows a small mixing between the pure boson configuration and the NB-1 bosons plus two fermions configuration can well describe the low-lying energy levels in Sn isotopes. The same philosophy is also able to apply to the coupling of three fermions to IBA bosons to explain odd-mass tin isotopes.

Acknowledgements

This work is partially supported by the National Science Council, Taiwan, under the contract of number NSC-82-0208-M009-020.

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DESCRIPTION OF THE EVEN TIN ISOTOPES IN A MIXING SHELL-MODEL AND IBA MODEL