ABSTRACT BETA DECAY OF ODD MASS NUCLEI ... - Yale University

A B S T R A C T

BETA DECAY OF ODD MASS NUCLEI IN THE

INTERACTING BOSON-FERMION MODEL

Fabio Dellagiacoma Yale University

1988

The accurate prediction of beta decay rates requires a detailed knowledge of nuclear structure properties of both the initial and final nuclei. Since exact calculations for medium mass and heavy systems are impossible, some approximation scheme is necessary. From spectroscopic studies it turns out that the proton-neutron interacting boson-fermion model is able to successfully describe a broad class of nuclei comprising spherical and deformed shapes. In this framework Gamow-Teller matrix elements are computed and compared with the experimental ones for many isotopic chains, ranging from tellurium to lanthanum in the N ■= 50-82 neutron shell. The mass dependence of the observed hindrance is reproduced correctly and the results show a quantitative improvement with respect to pairing theory. The strength distribution over low-lying states is also studied and it is found that a quenching factor of « 3.5 incorporates the effects due to the truncation of the large configuration space.

BETA DECAY OF ODD MASS NUCLEI IN THE

INTERACTING BOSON-FERMION MODEL

A Dissertation Presented to the Faculty of the Graduate School

ofYale University

in Candidacy for the Degree of Doctor of Philosophy

byFabio Dellaglacoma

December 1988

Acknowledgment s

The following dissertation has been made possible thanks to the support by Yale University, the A.W. Wright Nuclear Structure Laboratory and the US Department of Energy.

I like to acknowledge F. Iachello for his expert guidance, D.A. Bromley for his outstanding teaching and understanding, J. Markey, M. Schmidt and R. Shankar for their constructive criticism.

Special thanks to R. Bijker for his most valuable effort inexplaining several details of the interacting boson-fermion model, to J.M. Arias and C.E. Alonso for communicating some results prior to publication and to C.J. Lister for his input in updating theexperimental information. I am also indebted to B. Yip for reading the manuscript, to J. Baris for his assistance in solving frequentcomputer related problems and to D. Berenda for drawing the figures.

R. Bonito, L. Close, K. DeFelice, P. DiGioia, M. Scalesse andM.A. Schulz with their continuous help have also enriched the quality of my experience at Yale. Finally I express my gratitude to S. Batter for doing such a superb job.

ii

Table of Contents

Introduction......................................... 1

Chapter 2:The Interacting Boson-Fermion Model

2.1 Preliminaries............................................72 . 2 Dynamic Symmetries.......................................92.3 The Proton-Neutron Interacting Boson-Fermion Model........ 152.4 Microscopic Picture : the Link with the Shell Model.......20

Chapter 1:

Chapter 3:Nuclear Spectroscopy in the Te-Xe-Ba Region

3.1 Energy Spectra..........................................243.2 E2 Transitions and Quadrupole Moments.................... 363.3 Ml Transitions and Magnetic Moments...................... 513.4 One-Nucleon Transfer Reactions.......................... 66

Chapter 4:Beta Transitions of Medium Mass and Heavy Nuclei

4.1 General Overview.....................4.2 Fermi and Gamow-Teller Matrix Elements.4.3 Calculations for Odd-A Isotopes......4.4 Interpretation of the Results........

, .85 . 8 8

.92 .101

iii

References....................................................... 113

Chapter 5:

Summary and Concluding Remarks..................................... 110

iv

C H A P T E R 1

Introduction

Several branches of physics are affected by nuclear /3-decay and its theoretical understanding has stimulated new experimental effort to probe predictions and set new limits on fundamental quantities such as the mass of the neutrino.

On a cosmological scale the weak interaction plays a crucial role in the evolution of the early universe, in the synthesis of heavy elements and in the dynamics of gravitational collapse of stars, through neutron and electron capture rates by nuclei [K183]. Another relevant impact has been produced in determining the efficiency of solar and galactic neutrino experiments, whose detection could enlighten the related oscillation issue [Ha8 8].

Regardless of its many implications, (9-decay is on its own a remarkable process of nuclear physics, which was known long before the discovery of the neutron itself and its decay :

n ---► p + e + ur e

Since the early experimental evidence it has been possible to measure the life-time for such a process in a variety of nuclei covering a large fraction of the mass table. To account for the huge amount of

1

2

experimental data available, extensive calculations have been performed using various shell models. The corresponding microscopic description of strongly interacting nucleons in terms of an average field and a residual interaction can be implemented to compute the relevant matrix elements. Although this procedure is of practical use for light nuclei, because the model space can be restricted to a few active orbits, .it cannot be applied to the study of heavier systems, for which not only the dimension of the configurations is prohibitively large, but also collective degrees of freedom become important.

The quantity characterizing a beta transition between two nuclear states is the so-called ft value

ft- K2 2 2 2 + Ga < M g t >

( 1 . 1)

27rV ln 2 , , „ -94 2 6K = j— ^ ~ 1.23 x 10 erg cm secm c e

f = f(Z,W0) is a phase-space factor, which depends upon the charge Z of the parent nucleus and the end-point energy W0 of the emitted lepton (e‘, e+ for /}’ , fZ decay respectively), t denotes the half-

life (t]y2) and GV ’ GA are t*ie vector and axial vector coupling constants. The Fermi matrix elements

2 ' ' ( 1 . 2 )<*9“ - 'Ti(Ti+1> * TziTzf! ST,,T(i f i f

affect only isobaric analogue states, satisfying the conditions

3

= Jf, Tf - Tf and Tzf = Tzf ± 1, whereas the Gamow-Teller matrix

contain the most interesting part of the information concerning the nuclear structure.

The single particle estimate [Br77] of such quantities leads to the conclusion that typical log ft values for allowed beta transitions range between 3 and 4, while many of the measured ones are systematically larger : 4 ^ (log ft)eXp ^ 7 . Even for lightisotopes, as is the case for s-d shell nuclei, detailed calculations [Br85] show that a reduction factor = 0.9 must be included in the form of effective single particle matrix elements in order to reproduce the experimental data. The discrepancy becomes more striking in the mass region A = 100-140 because of the interplay between manyconfigurations and the occurrence of deformed shapes, arising from the proton-neutron interaction.

As far as spherical systems are concerned, the pairing theory [Ki63] appears to be an adequate starting point, because it incorporates the strong coupling between identical nucleons. Within this framework several investigations [Fu65,Ha65] have been carried out reaching a similar conclusion : that making quantitative

elements

2(2J t+l)(2Tf+1) x

(1.3)

X <afJfTf || I a(k)r(k) || a ^ T ^ 'A 2

k=l

4

predictions requires going behind the underlying assumptions. In fact2 2the microscopic dependence upon the occupation parameters Vj » uj f°r

a j-orbit describes only the overall behavior of the Gamow-Teller matrix elements, but fails to give the correct magnitude. Detailed nuclear structure calculations for many strongly interacting nucleons in several shells are not feasible - therefore one has to resort to some kind of approximation. In the last decade a new approach to the many-body problem in nuclear physics has been proposed [Ar75,Ar76,Ar78,Ar79] : instead of dealing directly with themicroscopic constituents, the nucleus is thought of as a collection of N bosons. Two major issues have been addressed since then : what are the advantages of a description in terms of the Interacting BosonModel (IBM) and what is the nature of its ingredients ?

By using group theory techniques, analytic solutionscorresponding to a variety of physical situations can be obtained including vibrational, axially symmetric rotors and 7 -unstable nuclei. Furthermore, the real examples, for which the exact symmetries are broken, can be treated conveniently by solving the eigenvalue problem for the Hamiltonian, thus providing explicit knowledge of the wave functions.

The second question has been also throughly investigated : as a result, it has been shown that the bosons can be identified with correlated nucleon pairs, establishing a link with the underlying microscopic structure. In a subsequent version, called Interacting Boson-Fermion Model (IBFM) [Ia83], fermion degrees of freedom have been incorporated as well, leading to the possibility of dealing with

odd-A nuclei.

5

The major goal of this dissertation is to present a detailedaccount of 0-decay in medium mass nuclei, using the Proton-NeutronInteracting Boson-Fermion Model (IBFM-2). It is interesting to examinethe improvements and the limitations of this treatment, by computingmany matrix elements for a selected set of isotopes with proton number53-57 and neutron number 50-82. The main concern is to study the massdependence of the hindrance observed in Gamow-Teller transitions andthe strength distribution over the low-lying states, because in mostcases the ground state to ground state transition represents only asmall fraction of the entire decay.

We begin, in the second Chapter, by reviewing the generalities ofboth the IBM for even-even systems and the IBFM for odd-A nuclei. Theimportance of the concept of dynamic symmetries is brieflyillustrated. The spectroscopy of the odd-neutron Te, Ba nuclei and52 56of the odd-proton nuclei subject of Chapter 3. Theirenergy spectra are fitted throughout the neutron shell 50-82 by usingthe same set of parameters for the boson-fermion interaction of agiven isotopic chain, the quasiparticle energies and occupationprobabilities being determined via a BCS calculation. Theelectromagnetic transitions E2, Ml and the related quadrupole Q and

Jmagnetic moments p are also studied in detail, because they contain

Jvaluable information concerning the collective and single-particle features of the wave functions. Furthermore, one-nucleon transfer intensities represent an additional tool to make a quantitative comparison with the observed stripping and pick-up reactions.

Chapter 4' is devoted specifically to studying 0-decay. After developing the necessary formalism within the IBFM-2 model, we present

6

the results of calculations for allowed Gamow-Teller transitions in the same nuclear region. Finally in Chapter 5, we draw some conclusions and outline how the ideas presented here could be extended to related topics.

C H A P T E R 2

The Interacting Boson-Fermion Model

2.1 Preliminaries

Most of the properties of nuclear structure at low energy are understood in terms of two kinds of excitations : collective andsingle quasiparticle ones. The former arise from the cooperative vibrations and rotations of the nucleus as a whole and were described originally by means of quadrupole shape variables [Bo52], whereas the latter originate from the individual motion of nucleons in an average field and find a proper treatment in the shell model [Ma49].

Among the peculiar consequences of the nuclear interaction, the pairing of identical nucleons in the same j orbit is certainly very effective, especially in medium mass and heavy nuclei, where neutrons and protons fill different shells. In this case, the seniority scheme [Ta71], is appropriate to describe spherical nuclei. However, a competing effect is the quadrupole interaction between proton and neutron quasiparticles which breaks this symmetry and is responsible for deformation.

Such characteristic structure is observed in nature and can be gauged quantitatively by studying the variation of properties, such as level energies, electromagnetic moments, transitions and reaction

7

8

rates as a function of the neutron number. As an illustrative example consider the even-even samarium ( Z=62 ) isotopes ^ 8Smgg, ^^Snigg, 1 5 2Sm90, ^^^Sm9 2 - At N=88 a remarkable change takes place : in factthe nucleus performs a transition from vibrational ( A=148,150 ) to soft rotor ( A=152 ) and even stiffer rotor ( A~154 ).

More complicated situations will occur when the coupling between vibrations and rotations is strong and cannot be treated as a perturbation. In addition to the even-even case, whose lowest excitations are described in terms of phonons, or equivalently by bosons, odd mass nuclei require incorporating quasiparticle degrees of freedom as well. In the past this was accomplished within the particle-vibration model [Bo52] for spherical nuclei, various versions of the rotor plus particle model [Me75,Le76] and the Nilsson model [Ni55] for deformed systems. However, all the previous approaches bear a limited range of validity, therefore it would be desirable to set up a framework able to provide us with a unified picture of the complete variety of known cases.

This goal is achieved in the theory of symmetries as proposed by Arima and Iachello [Ia87a] and formulated through the Interacting Boson Model (IBM) for the description of quadrupole excitations in even-even nuclei. Subsequently even-odd systems were analyzed by Iachello and Scholten [Ia79], by coupling a fermion to a bosonic core. The underlying algebraic structure needs to be extended to encompass Bose-Ferrai systems [Bi84a,Bi85], in which the two symmetries and their interaction are included explicitly.

We turn now to the description of the Interacting Boson-Fermion Model (IBFM), drawing particular attention to those aspects which are

9

relevant to our further applications.

2.2 Dynamic Symmetries

The fundamental ingredients of any abstract algebraic model are

boson and fermion creation and annihilation operators, denoted in the

usual notation as

, b bosona a

at , a. fermion3 , P J »A*

satisfying the familiar commutation and anticommutation relations,

respectively :

1 ba ’ 1 * ‘ afi

{a. , at, , } = 5. S ,J .A* J ,P j .J P>P(2 . 1)

with the additional property of commuting among one another

[ b , a 4 ] - [ bt , at ] a j ,A» J iP( 2 . 2)

. - . - l - oa J,P 1 a j ,p[ b , aj 1 = [ b[ , a. ] = 0

From these, bilinear products are formed with appropriate angular

momentum coupling

10

£ - <b I x b r >(A) (2.3)

(2.4)

where bl m = (-)1+M bl,-p and aj (A, = aj , » which

behave as spherical tensors under rotations and generate a bosonic

group Gg and a fermionic one Gp.

To be specific, we can consider quadrupole and scalar bosons only

in which case Gg = Ug(6 ) is the six dimensional unitary group of the

tion presented here, because they arise from the coupling of valence

nucleons across the major shells, but need to be considered within

this framework to explain the occurrence of negative parity states

Analogously, if we restrict the fermionic space to the simplest

case of a single j - 1/2 shell, we have Gp - Up(2).

In general, for a mixed system of N IBM bosons and M fermions,

occupying single particle levels j^, j 2 j^. the group structure

of the problem can be expressed as follows :

Ug(6) ® Up< I (2j.+l) ) D ... D GBF D ... D SU(2) D 0(2) . (2.6)

IBM. Other kinds of excitations such as the dipole ( bi = ) and

the octupole ( bj ^ ) ones have' been neglected in the approxima-

( e.g. J* - 1 \ 3' ).

All possible subgroups in the chain (2.6) can be determined by means

of group theory techniques and extensive accounts of this subject are

11

reported in the literature [Ha62,Wy74,Bi85]. Furthermore, such a group

-subgroup decomposition determines a vector space, whose states can be

viewed as a realization of the irreducible representations involved.

For illustrative purposes let us consider the special case of

two bosons with U(5) symmetry coupled to a fermion, whose angular

momentum j can assume one of the values 1/2, 3/2, 5/2. The appropriate

basis states are :

Uf i (6 ) ® Up (12) 3 Ub f (6 ) 3 UfiF(5 ) 3 S0b f (5 ) D S0fiF (3 ) D SU(2) D 0 (2 ) ' [N=2] {M=1} [N+l-i, i] (nr n2) (t^,^) L J Hjj

(2.7)

and the labels are specified in Table 2.1.

The most general Hamiltonian of the Interacting Boson-Fermion

Model contains three terms describing the bosonic core, the fermion

and their interaction. When these contributions are written as a

combination of the Casimir invariants of the groups, the energy

operator is diagonal in that particular basis and is said to possess

the associated dynamical symmetry. In the Ugp(5) symmetry we are

considering

H ~ H0 + h 2 C2 (Ubf6 ) + ux Ci(Ubf5) + u 2 C ^ S ) +

( 2 . 8 )

k2 C2(SOBF5) + V2 C2(SOBF3) + W2 C2(SU2)

and its eigenvalues are given in the following expression

12

E - EQ + h 2 [(N+l-i)(N+6-i)+i(i+3)] + u 1 (n1 +n2) +

u 2 [n£(n^+ 4) + n 2 (n2 +2)] + 2k2 [r1 (r1 +3) + r2 <r2 +l)] +

(2.9)2v2 L(L+1) + 2w2 J(J+1) .

Table 2.1 Quantum numbers labelling the U £ £ (6 ) D U (5) chain.

N - 2 , M = 1

i - 0 , 1

(nr n2) - (N+1,0) + (N,0) + ... + (0,0) i - 0

(n n ) = (N,1) + (N-1,1) + ... + (1,1)i - 1

+ (N, 0) + (N-1,0) + ... + (1,0)

(r1 ,r2) = (n,0 ) + (n-2 ,0 ) + ... + (1 ,0 ) or (0 ,0 ) for ( n ^ n ^ - (n,0 )

( r - . r , ) - (n,1 ) + (n-1 ,0 ) + (n-2 ,1 ) + ... +for (n-,n_) = (n,l)

+ (1 ,1 ) or (1 ,0 )

L =■ 2r-6n., 2r-6n -2, 2r-6n -3 r-3n +1, r-3n in (r,0)A A A A A

nA= 0 , 1 , 2 , ... , 3nA < r

L = 2r-6nA+l, 2r-6nA ......r-3nA+2-(l-5n q ) in (r,l)

n A= ° ’ I ’ 1 .....................3n A " T

J - L ± | if L > 0

J - | if L - 0

|Mj| < L

13

It is customary to group the levels of the energy spectrum into

multiplets, as is done in Fig. 2.1 . For a particular choice of the

parameters several interesting limits can be obtained. If, for

instance, u 2 = 0 , the quadratic terms in (n^,n2) vanish and the

linear dependence upon those quantum numbers will give the spectrum a

vibrational character. Likewise the eigenvalues 2w2 J(J+l) of the

total angular momentum J show a typical rotational behavior.

The example reported here sheds light on the central role played

by the concept of dynamic symmetry : it is evident that, whenever such

a situation is encountered, much information can be gained by the use

of algebraic techniques, which provide one with analytic expressions

for the physical observables.

Fig. 2.1

A typical spectrum of an even-odd nucleus ( N=2, M=1 ) with

Ufi (6 ) ® Up (12) D Ugp (6 ) symmetry. The energy eigenvalues are

calculated using eq.(2.9) with Eq - -1.2045 MeV, h 2 *= 0.05 MeV,

ux - 0.4 MeV, k 2 = 0.013 MeV, w 2 = 0.003 MeV, u 2 = v 2 = 0. The

numbers in square brackets denote Ugp (6 ) labels.

O tf) O IOCVJ - - o

> LlI ©

2

15

2.3 The Proton-Neutron Interacting Boson-Fermion Model

Thus far only some basic aspects of the elegant mathematical

framework of the model have been sketched. However, any detailed

analysis of experimental data needs the introduction of proton-neutron

degrees of freedom as well as the microscopic justification of the

parameters. ,

To achieve the first goal, we begin by assuming that the core of

even-odd nuclei can be approximated by a combination of interacting

bosons Sp , dp , carrying angular momentum 0 + , 2 + respectively, with

the additional index p necessary to distinguish between neutron (i/)

and proton (7r) constituents. Furthermore, the unpaired fermion is

allowed to occupy one of the active .shell model orbits, as is shown

in Fig. 2.2 and will interact with the members of the core.

Accordingly, the structure of the Hamiltonian describing the system is

H - Hb + h f + VBF . (2.10)

The first term Hg arises from the interplay of the strong

pairing between identical nucleons, the neutron-proton quadrupole

interaction, which is an essential ingredient in establishing

collective features in the even-even core and the Majorana

contribution M,,_ , which affects the location of states with mixedi/ir 9

symmetry, relative to the totally symmetric ones.

Fig. 2.2

Neutron levels in the N=50-82 shell. The single particle

energies are relative to the lgy/ 2 orbit-

Thus, its explicit form is given by

( 2 . 1 1 )

H„ = En + c n , + £ n . + V + V +B O v a k a vv k kv K

k Q<2> • Q(2) + MV K VK

where Eq is just a constant determined by the binding energy,

£ , n, = d • d p = v,k (2.12)P dp P P

are the d-boson energy and number operator respectively, V i s the

multipole expansion of the interaction between identical pairs

is the quadrupole operator and finally the expression for the Majorana

terra is

In the language of Lie algebras the coupled system of neutron and

irreducible representations span a vector space, in which the general

IBM-2 Hamiltonian (2.11) has to be diagonalized. However, some

remarkable features emerge naturally, without requiring any numerical

effort. One of the possible decompositions of U^(6 ) ® 1^(6) leads to

the subgroup 1 ^ ( 6 )

M - - 2 X f, [df x dt](k) • [d x d ](k)vn , i o k L v 7T L vk=l, 3

+

(2.15)

proton bosons is described by the group 1^ ( 6 ) ® Uw(6 ) , whose

U (6 ) ® U (6 ) D U v n i ( 6 )(2.16)

[NJ x [NJ [N - f, f]

where

N - N + N7T(2.17)

f - 0 ,1 , min (N , N )V 7T

and introduces representations [N-f,f] in which neutrons and protons

are not coupled in a symmetric fashion. The occurrence of 1+ , 3+

states, characterized by mixed symmetry, has been confirmed

experimentally [Bo84] and is a proof of the predictive power of the

model.

When only a fermion is added to the core, its energy is accounted

for by the one-body operator

19

H„ = I E. at a. (2.18)F J J.P J ,lkJ »A*

with Ej denoting the quasiparticle energies.

The interaction Vgp can be expanded as

VB F " I I ■ 4 ” " ' <2' 19)A P , P

is the boson part, containing up to quadratic terms in B^^

( eq. (2.3) ) and A^^p is the fermion operator defined in eq. (2.4).

Although in principle all the multipoles A contribute to the boson-

fermion coupling, physical insight suggests that three ingredients

carry the essential features :

V = A n, • n + T Q ^ - q(2) + A F , (2.20)BF p d , p p V P P PP( p 9* p' , p =■ 1/.7T ; p' - i/,x )

They are the monopole-monopole interaction np ’ t*le 9uac *

rupole-quadrupole force Qp^2^’ 9 p'^2 between bosons and the unpaired

fermion and the exchange term Fpp> » whose origin has to do with the

microscopic nature of the bosons (interpreted as correlated pairs of

20

nucleons) and the consequent requirement of satisfying the Pauli

principle.

2.4 Microscopic Picture : the Link with the Shell Model

Before moving on to present the results of our numerical

calculations, it is worth pointing out that a substantial fraction of

success experienced by the IBFM lies in the ability of establishing a

direct and close connection with the underlying shell model [Ia87b].

In fact, in addition to phenomenological applications, in which

parameters were fitted to available data, a better understanding has

been obtained by studying the microscopic nature of the building

blocks and the meaning of the approximations involved in formulating

the theory.

The basic idea consists in truncating the many fermion space to

the one spanned by S and D pairs only :

Sl - 5 J “J ^ (a/, x a j t / ° >

( 2 . 21)

D* I )3. ., 1 (aj x a], )(2)P,P j,j' /2j+l J,p ,P M

where aj , £jj* are structure constants, which depend, in general,

upon the set of non-degenerate orbits considered.

The further step in the treatment of even-even nuclei is to map

[Ot78] this subspace onto the boson counterpart, generated starting

21

from s£, and their Hermitian conjugates. The justification of the

outlined procedure comes from the dominant role played by the

quadrupole excitations ( L=2 ) in low-lying collective states, where

as the introduction of scalar components ( L=0 ) takes into account

the Pauli principle. Applying the mentioned transformation allows one

to take advantage of dealing with matrix elements in the IBM space,

which are easier to compute than in the shell model context. Although

the image of one-body and two-body operators introduces higher order

contributions, the corresponding expansion converges rapidly, provided

that the low seniority scheme can be implemented. Under these

assumptions it has been shown [Sc80] that in the IBFM-2 the fermion

quadrupole operator is expressed as

( 2 . 2 2 )

with

(2.23)

and

- I10

Np(2j+1)

(2.24)

: + h . c .

with : [ ] : denoting normal ordering.

22

The definition of the parameters Uj , Vj

v2 . ° h ) ______J I a?,(j'+|)

j' J

2 i 2Uj " 1 - " j

(2.25)

is suitable to a physical interpretation : they can be identified with

the occupation and emptiness probabilities respectively of the j

shell. Therefore for practical purposes they can be estimated by

solving the BCS ( Bardeen, Cooper, Schrieffer ) equations [Ba57] :

v j[ i [l - (.. »>/. ] ] V 2 (2.26a)

E - J (e - A)2+ A 2 (2.26b)

n - I v2 (2 j+l) p - ./,* , (2.26c)j J

the quasiparticle energies Ej being determined starting from the

single particle levels €j, the Fermi energy A , the pairing gap

A = 12/P?~A2 and imposing the additional constraint on the number of

valence nucleons n^ .(2)The occurrence of the interference term qgp ' in eq. (2.22)

is the result of the composite nature of the bosons. In this

particular approximation such a quantity is characterized A by a

seniority change Av = 1 and gives rise to the exchange contribution

Fpp» in the boson-fermion interaction :

v , - q(C ■ ■ <2- 27)

Finally the coefficients /3jj ' can be expressed as a function of the

single particle quadrupole matrix elements Qjj'

V ' (ujvr + vjV V (2-28>

Qjj, * || V(2) I ■ <2 29>

provided that the D-pairs absorb all the E2 strength.

23

C H A P T E R 3

Nuclear Spectroscopy In the Te-Xe-Ba Region

3.1 Energy Spectra'

The ideas presented in the previous chapter can be applied to the

systematic study of spectroscopic properties in odd-A nuclei. Since

the parameters of the boson Hamiltonian (2.11) have already been

determined [Sa82,Pu80] for a variety of even-even isotopes throughout

the Z = 50-82 region, we will focus our attention to the odd-neutron

52^e ’ 56Ea and their odd-proton partners 5 3 I , 5 7 ^ . This account,

together with the calculation previously reported [Al84,Ar85] for the

5 ^Xe, 5 5 CS nuclei, allows one to draw quantitative conclusions about

the model by varying both quantities N and Z.

In Table 3.1 and 3.2 we summarize the non-vanishing parameters

used in fitting the energy spectra for the even-even cores. As a

result of the diagonalization, a set of collective states, whose

nature displays vibrational features approaching the magic numbers 50

and 82, is obtained. As soon as more particles or holes are added

toward the central region of the shell, a transition takes place from

spherical to deformed shape and the typical decrease in energy of the

lowest 2+ , 4+ is properly reproduced. Quasiparticle degrees of freedom

are introduced starting from the single nucleon orbits lgy^i ^d5 / 2 >

24

25

8 s1/2* 2 <*3/2’ wbose energies £j are shown in Table 3.3, as calculated

by applying the BCS procedure.

Although in principle the eigenvalue problem for the full

Hamiltonian (2.10) should be solved in the complete boson ® fermion

basis, it turns out that such a task is made impossible by the

exceedingly large dimension of the matrix. Therefore a truncation of

the model space is designed, coupling the j orbits to the most

relevant low-lying states in the core [Bi87]. As previously mentioned,

three contributions characterize the boson-fermion interaction : the

monopole, the quadrupole and the exchange term. Their strengths kp, T , have been determined phenomenologically for positive parity

states in even-odd nuclei and are given in Table 3.4.

The experimental information available is compared with the

results of the IBFM-2 calculations for the 5 2 ^© and ggl isotopic

chains, as is shown in Fig. 3.1-3.4 . Keeping in mind that several

physically different situations are encountered, ranging from weak to

strong core-fermion coupling, it is remarkable to note that the

general trends are predicted correctly. Moreover, an overall

quantitative agreement is found within an uncertainty of 200 keV in

the energy range considered. However, some discrepancies are still

present, reflecting the approximations involved in dealing, for

example, with only collective excitations in the even-even nuclei. In

fact, since the proton boson number is small (N^ = 1 for Te and I),

additional degrees of freedom, such as two-quasiparticle ones might be

important and have not been incorporated in the IBFM-2 framework.

26

Table 3.1 Parameters

even-even

of the IBM-2 Hamiltonian for

^2Te isotopes .

the

N £b > K XV

/-N* o o c 0 0

2

(Mev) (Mev) (Mev) (Mev)

52 1 . 0 2 0 -0.246 -0.680 0.308 0.14654 1.031 -0.215 -0.800 0.308 0.14656 1.040 -0.188 -1 . 0 1 0 0.308 0.14658 1.040 -0.165 -1.280 0 . 2 0 0 0.14660 1.023 -0.154 -1.320 0 . 1 0 0 -0.08562 0.966 -0.150 -1.173 -0.092 -0.15164 0.923 -0.140 -0.885 -0.154 -0.1546 6 0.885 -0.138 -0.683 -0.246 -0.1236 8 0.831 -0.138 -0.587 -0.154 -0.06270 0.800 -0.151 -0 . 2 1 0 0 . 0 0 0 0 . 0 0 0

72 0.831 -0.154 0 . 0 0 0 0.092 0 . 0 0 0

74 0.889 -0.157 0.311 0.308 0 . 0 0 0

76 0.935 -0.169 0.494 0.308 0.09578 1.046 -0.191 0.900 0.308 0.09580 1.138 -0.215 1 . 2 0 0 0.308 0.095

a) \ ■ -1-2' fr r3_ - 0 - 0 9MeV, r2= 0.12 MeV for all nuclei.

The remaining parameters of the Hamiltonian (2 .1 1 ) are setequal to zero.

b) e =V

e = € .7T

27

Table 3.2 Parameters

even-even

o f the IBM-2 Hamiltonian f o r

^^Xe and «jgBa i sotopes

the

N , b >

(Mev)

* XV

(Mev)

c > >co(Mev)

<>>2

(Mev)

52 0.96 -0.300 -0 .70 0.30 0 . 1 0

54 0.96 -0.225 -0 .80 0.30 0 . 1 0

56 0.96 -0.215 - 1 . 0 0 0.30 0 . 1 0

58 0.96 -0.185 - 1 . 0 0 0 . 2 0 0 . 0 0

60 0.94 -0.155 - 1 . 0 0 - 0 . 2 0 -0.1562 0.85 -0.135 -0 .80 -0 .25 -0.1564 0.78 -0.130 -0 .60 -0 .25 - 0 . 1 2

66 0.76 -0.130 -0 .40 - 0 . 2 0 - 0 . 1 2

68 0.72 -0.137 . -0 . 2 0 -0.05 - 0 . 1 2

70 0.70 -0.145 0 . 0 0 0.05 - 0 . 1 0

72 0.70 -0.155 0 . 2 0 0 . 1 0 - 0 . 1 0

74 0.70 -0.170 0.33 0.30 0 . 0 0

76 0.76 -0.190 0.50 0.30 0 . 1 0

78 0.90 - 0 . 2 1 0 0.90 0.30 0 . 1 0

80 1 . 0 0 -0.227 1 . 1 2 0.30 0 . 1 0

a) X - - 0 .8 , 0.12 MeV,IK 1 Z f 3~ -0 .09 MeV f o r a l l n u c le i .

The remaining parameters o f the Hamiltonian ( 2 . 1 1 ) are set

equal to zero.

b) e = e = e. v ir $

28

Table 3.3 Single particle energies £j (MeV).

NucleusOdd

nucleon p e . Jlg7/2 2 d5/2 lhll/ 2 3 s1 / 2 2 d3/2

52T® 1/ 0 . 0 0 0.60 2 . 0 0 2 . 1 0 2.50

531n 0 . 0 0 0.40 1.50 3.35 3.00

56Ba V 0 . 0 0 0.80 2 . 0 0 2 . 1 0 2.50

57La n 0 . 0 0 0.60 1.50 3.35 3.00

Table 3.4 Boson-fermion interaction parameters for positive parity states.

Odd r A ANucleus P P P

nucleon p (MeV) (MeV) (MeV)

52Te V 0.30 0 . 1 0 -0.40

531n 0.60 0 . 2 0 1 o GJ O

56Ba V 0.50 0 . 2 0 -0.60

K 0.80 0 . 0 0 -0.30

Fig. 3.1

Comparison between calculated (lines) and experimental (points)

positive parity spectra of the odd-A 5 2 ^ isotopes. All energy

levels are plotted relative to the 1/2^ state. The experimental

data are from [B187,Au79,Ta79,Ta80,Ta81,Ha821Ha83,Au76].

2

I

0

- I

3

E(MeV)

50 54 58 62 66 70 74 78 82Neutron Number

Fig. 3.2


positive parity spectra of the odd-A 5 2 ^e isotopes. All energy


data are from [B187,Au79,Ta79,Ta80,Ta81,Ha82,Ha83,Au76].

E(MeV)

Neutron Number

Fig. 3.3


positive parity spectra of the odd-A 5 3 I isotopes. All energy



31

E(MeV)

50 54 58 62 66 70 74 78 82Neutron Number

Fig. 3.4


positive parity spectra of the odd-A g-jl isotopes. All energy



32

E(M eV)

3. -

2 . -

I. -

0 -

I. -

50 54 58 62 66 70 74 78Neutron Number

33

For the 5 gBa and 5 7 !^ isotopes the observed spectra allow one to

identify a smaller number of states, due to the neutron deficiency in

the N = 50-82 shell and the consequent shorter lifetime. Therefore

Fig. 3.5-3 . 6 cover only a smaller energy range and have the purpose of

illustrating the basic features. In addition to providing us with the

excitation spectra it is worth emphasizing that the simultaneous

knowledge of the wave functions is a necessary tool for computing both

electromagnetic deexcitation properties and one-nucleon transfer

reaction rates.

Fig. 3.5


positive parity spectra of the odd-A 5 gBa isotopes. All energy


data are from [Ha83,Au76,Se86,Se87].

34

E(MeV)

50 54 58 62 66 70 74 78 82Neutron Number

Fig. 3.6


positive parity spectra of the odd-A jyLa isotopes. All energy


data are from [Ha83,Au76,Se86,Se87,Pe83].

35

Neutron Number

36

3.2 E2 Transitions and Quadrupole Moments

It is very well known that electric quadrupole transitions are

dominant in nuclear physics. Striking evidence is given by strong

enhancements in the measured E2 strength in even-even nuclei. The

large deviations from single particle estimates , expressed in

Weisskopf units (W.u.) indicate the presence of collective features.

Since the transition rate T has a pronounced energy (E^) dependence it

is desirable to extract it from the other structure effects, leading

to the following form

T j- B(E2) - 1 (3.1)sec75 (197.32)

where the reduced B(E2) value

B(E2; Jj[- Jf) 2 (3.2)

is the quantity to be computed.

In the IBFM-2 formalism the operator is constructed

starting from the boson and fermion contributions

t (E2) _ t (E2) + t (E2)B F (3.3)

which in lowest order are given by

t (E2)B (3.4)

(3.5)

37

with

Q (2) = [s* x d + d* x s ](2) + [d* x d ](2) (3.6)P P P P P 1 P P 1

e(2)^ j ' " " J <Uj Uj ' ” vj v j * ) II Y<2) • (3 -7 )

The boson effective charges are taken to be the same for both neutron

and proton pairs [Sa82] eg v = eg ^ = 0.12 eb (lb=10~^ cm^), whereas in estimating the fermion coefficients the radial integrals <r > are

approximated by the harmonic oscillator value (N+3/2)ft/Mw, which turns

out to be the same (0.27 b) for all the positive parity orbits lgy/2>

^d5/2’ 2 ^ 3 /2 * ^sl/2 b*16 N = 4 shell. Furthermore a renormalization,

leading to ep v = 0.135 eb, ep n = 0:405 eb is adopted to account for the effects of the strong interaction among the nuclear constituents.

Even though the ability of accurately describing the observed

transitions depends in a crucial way upon details in the structure of

the wave functions, it is useful to illustrate how the essential fea

tures can be understood by dealing with a specific example. Let us127consider 5 3 I 7 4 . which is described in the model as

153I74 ~ 1252Te ® 1* - < ^ - 1, - 4 ) ® lw , (3.8)

where 4 denotes four boson holes, counted from the nearest closed

shell, which is N = 82 here. Since the first quasiparticle energy

above the Fermi level is p<i' one wouE< expect 7/2+ to be the ground

state. However, experimentally the assignment is 5/2+ with an almost

38

degenerate 7/2+ at 58 keV. If the nucleus were pure vibrational, the

lowest states could be predicted in the weak coupling limit as

|5/2+> - |0+> ® d5/2 , |7/2^> - |0+> ® g?/2 , (3.9)

followed by a multiplet around the one-phonon energy of the 2^

(E2+ ~ 700 keV) containing among its members

|3/2+> ~ ai| 2 ^ ® g7/2 + a2| 2 ^ > ® d 5/2

(3.10)

I l /2^> ~ | 2 ^ > ® d 5/2

In the corresponding symmetry SU(5) of the IBM-2 the B(E2; 2 -+0 )

is expressed as

B(E2; 2 ^ 0}> - (. N, + . N,)2 (3.11)V 7T

After inserting the appropriate values we get

B(E2; 2+- 0+) - 36 x 10'2 e2b2 = B2Q (3.12)

As a result

B(E2; 3 /2+ - 5/2+) « a2 (3 .13a)

B(E2; 3 /2+ - 7/2+) « a2 B20 (3.13b)

B(E2; 1/2+- 5/2+) « B2Q (3.13c)

showing that in this naive picture the fraction of g-j2 an<l ^5 / 2 *n

39

the |3/2f> determine the respective B(E2). The real calculation shows

that the vibrational scheme is highly perturbed, because only 63 % of

17/2f> comes from the gy component and 54 % of 15/2^> from d^ ^ 2 > the

mixing with other configurations being not negligible at all. From

Table 3 . 6 we see that the trend in observed values

B(E2; 3/2*-+ 7/2*) « 11 B(E2; 3/2*-+ 5/2*) « 6 (3.14)

suggests that a larger component is indeed in the gy/ 2 rather than in2 2the d ^ 2 » as model predicts by yielding s 50 %, s 5 % re

spectively .

From the results presented in Fig. 3.7-3 . 8 and in Table 3.5-3 . 8

it appears that the strong collective quadrupole transitions (large

B(E2) values) are reproduced quite well by the IBFM-2 model, especial

ly for the odd-proton 5 3 I isotopes, whereas some disagreement emerges

in relation to those transitions which are observed to be weak and in

the intermediate cases. The explanation of such a trend can be found

by studying the specific nature of the states involved in a de-

excitation process. As an example let us consider the first excited + 121state 3/2^ in 52^e69> which is observed experimentally at an energy

E3 / 2 - 0.212 MeV. In the IBFM-2 model the lowest 3/2^, 3/2^ states are

predicted at energies 0.160 and 0.494 MeV respectively. However, their

structure is very different. In fact the first one is of single

particle nature ( | 3/2^> - 0.98 | o£> ® d^ 2 )> whereas the second one

arises mostly from the strong coupling between the collective 2 state122of the even-even core 52^e70 and orbit plus an additional

smaller admixture with the d-j^ level ( | 3/2j> - 0.90 | 2j> ® s-^ +

40

0.40 | 2j> ® d-j^ )• Since the ground state 1/2^ is predicted by the

model to be | l/2j> - 0.99 | 0^> ® s^^2 » much smaller overlap

between the 3/2^ state and the final wave function explains why the

calculated B(E2) values ( B(E2; 3/2^ -* 1/2^) * 0.9 x 10 ' 2 e2 b2,

B(E2; 3/2^ 5 8 12 x 10" 2 e2 b 2 ) differ by an order of magni

tude. Furthermore other quantitative differences between computed and» 133observed decay rates, such as those involving the 1 /2 in gyLayg

(Table 3.8) can be understood following the same kind of analysis.

Therefore, in addition to energy spectra, the electromagnetic

deexcitation patterns constitute a valuable source of information

about the model space.

Fig. 3.7


B(E2; 1/2^-+ 5/2^) and B(E2; 3/2J — 5/2^) values of odd-A 53I

isotopes. The experimental data are from [Ta79,Ta80,Ta81,Ha82,

Ha83].

Fig. 3.8


B(E2; 3/2^ -+ 1/l\) and B(E2; 7/2^ - 5/2^) values of odd-A 53I

isotopes. The experimental data are from [1 3 8 0 ,1 3 8 1 ,1 1 3 8 2 , Ha83].

42

Neutron N u m b e r

43

Table 3.5 Experimental and calculated B(E2) values in 32^e '

Nucleus --> B(E2 ; T+ T + N Ji > Jf}

, , n - 2 2 , 2 . X (10 e b )

exp th

U 7 Te a) 52 65 5/2, -->1 /2 , 0.173 ± 0.003 0.07

121Te b) 52 693/2,

(7/2,)

-->

-->

1 /2 ,

3/2,

9.7

0.092

± 1.9 0.9

14

123Te C) 52 713/2,

3/2,

-->

-->

1 /2 ,

1 /2 ,

1 . 0

13

± 0 . 2

± 1

0.3

9

3/2, -->1 / 2 , 3.3 ± 0.3 0.06

3/22 ----> 3/2, 6 . 6 8 ± 0.67 3

3/2, -->1 / 2 , 9.3 ± 0.3 6

5/2, --> 3/2, 4.83 ±0.24 6

5/2, ----> 1 /2 , 5.3 ± 0 . 2 4

125Te d) 52 73 (7/2,) ---->5/2, (1.5 ± 2.0)E-4 0.3

(7/2,) ----> 3/2, 1.78 9

7/22 ----> 3/2, > 0.074 0.5

7 / 2 2--> 3/2, > 0 . 1 1 0 . 0 2

5/2, --> 5/2, 2.19 ±0.44 0.08

5 / 2 2 -->3/2, 3.56 ± 0.07 3.3

a) [B187], b) [Ta79], c) [Ta80], d) [Ta81]

44

Table 3.6 Experimental and calculated B(E2) values in

NucleusJ 1 B(E2; — > J+)

exp

- 2 2 2 x ( 1 0 e b )

th

121 a) 53 68 1 / 2 ! — > 5 /2 , 28 31

1 / 2 , — > 5 /2 , 23 31123 b)

53 70 3 /2 , — > 7 /2 , 27 28

3 /2 2 — > 7 /2 , (4 .7 ) 0.03

7 /2 , — > 5 /2 , 4.46 ± 0.89 3

3 /2 , — > 7 /2 , ' 13.4 22

125 c ) 53 72 3 /2 , — > 5 /2 , 7.05 ± 0.35 3

1/ 2 , — > 3 /2 , 2.2 ± 0.9 21

1/ 2 , — > 5 /2 , 13 25

7 /2 , — > 5/ 2 , 4 .4 ± 0 .8 2

3 /2 , — > 7 /2 , 1 0 . 6 17

127 d) 53 74

3 /2 , — >

1/ 2 , — >

5 /2 ,

3 /2 ,

5.9 ± 0 . 5

9.5 ± 1 .4

3

12

1 / 2 , — > V 2 , 19.0 ± 2.6 20

5 /2 2 — > 3 /2 , 4.93 ± 0.69 0 .4

continued

a) [Ta79], b) [Ta80] , c ) [Ta81] , d) [Ha82].

45

Nucleus --->j i B(E2; T+ , + v

J i > J f )- 2 2 2 x ( 1 0 * e b )

exp th

5 /2 , --- > 7 /2 , 1.18 ± 0 . 0 9 3.8

127 d) 53 74

5 /2 ,

3 /2 ,

--- >

--- >

5 /2 ,

5 /2 ,

0.82

0.27

± 0 . 1 2

± 0.06

0 . 2

3

7 /2 , --- > 5 /2 , 6.3 ± 0.5 7.5

9 /2 , --- > 5 /2 , 1.38 ± 0 . 1 2 1 . 6

9 /2 , --- > 5 /2 , 7.2 ± 3.6 11

V 2 , ---> 7 / 2 1 6 . 8 ± 2 . 2 2

129 e) 53 76

3 /2 ,

3 /2 ,

--- >

--- >

5 /2 ,

7/2 x

. 4 .9

18.3

± 1.3

± 2 . 2

2

13

5 /2 , --- > 3 /2 , 0.17 ± 0 . 0 6 0.5

5 /2 , ----> 5 /2 , 0.066 ± 0.027 0.3

d) [Ha82] , e) [Ha83],

46

Table 3.7 Experimental and ca l c u la te d B(E2) values in c ,Ba.30

Nucleus Jt B(E2; J+ — > J+)

exp

- 2 2 2 x ( 1 0 e b )

th

1/ 2 1 — > 3 /2 , 1.90 ± 0.08 11

5 /2 , — > V 2 , 1.07 ± 0.16 7

1 3 5 R* a>56 79

5 / 2 , — >

3 /2 2 — >

3/2 3

3 /2 ,

3 /2 ,

3 /2 ,

11.7 ± 0 .4

7 .4 ± 0 .4

2.9 ± 0 .4

3

3

0.7

7 /2 , — > 3 /2 , 8.15 ± 0.30 10

l / 2 2 — > 3 /2 , 4 .8 ± 0 .4I

2

a) [S e87 ] .

47

Table 3.8 Experimental and calculated B(E2) values in g^La.

Nucleus --> B(E2; J+ — > J+)

exp

- 2 2 2 x ( 1 0 e b )

th

5/2, --> 5/2, 1.77 ± 0.4 4

3/2, --> 5/2, > 24.2 5

133La a) 57 76

H H

CS cs

\

\

-->

-->

5/22

V 2 ,

2.06 ± 0.44

4.8 ± 1.2

0.5

2

d / 2 ,)-->3/2, 2 . 0 ± 1 . 2 23

d / 2 ,)--> 5/2, 0.32 ± 0.12 33

7/2, --> 5/2, 1.48 ± 0.16 0.5

5/22--> 7/2, 21.4 ± 2.1 1 0

5/2, -->5/2, 1.85 ± 0.37 0.4

135La b) 57 78 3/2, --> 5/2, > 0.29 0

3/2, -->5/2, > 0.29 4

1 / 2 , --> 3/2, . > 1.3 9

1 / 2 , --> 5/2, > 1 . 2 2 2

a) [Se8 6 ] , b) [Se87] .

48

By restricting ourselves to the diagonal matrix elements it isI

possible to define the quadrupole moments as

0 - I 16* J(2J-1) I II T (E2) II * 151QJ J 5 (J+l) (2J+1) (2J+3) ’ II II ’ . (3-15)

which are tabulated in Table 3.9.

As a function of the neutron number N the quadrupole moments of

both the 5/2^ and 7/2^ states (Fig. 3.9) in the 5 3 I isotopes decrease

monotonically to reach their minimum value toward the central region

of the neutron shell 50-82. Such a trend reflects the known behavior

of the quadrupole moment of the 2 state in the even-even 5 2 ^® cores.

A too small value is predicted by the IBFM-2 model for Q3 /2 + in12552^e73 ’ this result, together with the calculated B(E2) matrix

element (Table 3.5) for the transition to the ground state 1/2^,

highlights a substantial difference between the nature of the model

3/2^ state and the observed one, although its energy is correctly

reproduced. Even larger quadrupole moments are measured for neutron

deficient 5 gBa isotopes due to the increasing deformation of their

nuclear shape.

Fig. 3.9


electric quadrupole moments Q$/2+ and Q 7 / 2 + odd-A gjl

isotopes. The experimental data are from [Ta81,Ha82,Ha83,Au76,

Se86].

49

N eutron N u m b e r

50

Table 3.9 Experimental and ca l c u la te d e l e c t r i c quadrupole moments in

52Te ’ 531 ’ 56Ba’ 57^ *

Nucleus J+ Qj (eb)exp th

125Te a> 52 73 3 /2 , -0 .200 ± 0.023 -0 .07

125 a) 53 72 5 /2 , -0.89 -0 .89

127 b) 53 74

5 /2 ,

V 2,

-0 .79

-0 .71 ± 0.09

-0 .72

-0 .76

129 c ) 53 76

7 / 2 ,

5 /2 ,

-0 .55

- 0 . 6 8 ± 0.06

-0 .61

-0.59

131 d) 53 78 7 /2 , -0 .40 ± 0.01 -0 .45

133 e) 53 80 7 /2 , -0 .27 ± 0.01 -0 .3

1 2 3„ f ) 56 67 5 /2 , 1.52 ± 0 . 1 3 0 . 8

129IJ _ f )56 73 7 / 2 , 1.60 ± 0.13 0.4

!3 5 — f ) 56 79 3 /2 , 0.146 ± 0.016 0.3

137La «> 57 80 7 /2 , 0.26 ± 0.08 0

a) [Ta81] , b) [Ha82], c ) [Ha83], d) [Au76]. e) [Se8 6 ] , f ) [Mu83], g) [Pe83] .

51

3.3 Ml Transitions and Magnetic Moments

For even-odd nuclei, other electromagnetic transitions, such as

the Ml's, deserve particular attention, because they carry information

about the unpaired nucleon and the delicate coupling to the core.

Therefore they are complementary to the role played by the E2's,

where, as we have seen the collective features are prevailing.

The appropriate one-body operator in our language is

The bosonic g-factors gu, gff have been taken from previous studies

[Sa81,Sa84]. The coefficients of the fermionic contribution are given

and the single-particle g-fac.tors of the free nucleons are explicitly

t(M1) t (M1) + t (M1)B F (3.16)

with

t (M1 )B

(3.17)

and

(3.18)

by

t

(3.19)

52

(3.20)

° ’ gS>, - -3-8263 ( V •

Throughout the applications considered here the spin components need

to be modified to the following values

« s , , " 3 - 9 1 ° < V ’ 6 s , , " - 2 ‘ 6 7 8 ( V •

(3.21)

The calculated B(M1) values

B(M1 ; - J2T*1) <afJf II l(M1) 11 V i ^ (3>22)

are presented in Table 3.10-3.12 for the Te, I, Ba, La chains. Unlike

the E2 transitions, whose agreement with the experimental data is

qualitatively correct, the magnetic case is characterized by the oc

currence, of relatively strong deexcitations, which are predicted to be

weak and viceversa. Such a behavior can be the result of having

neglected higher order terms in the expression for the operator ,

or can reflect small configuration admixtures, responsible for large

variations in the matrix elements, without appreciably affecting the

energies of the corresponding states.

Extensive experimental evidence proves that Ml transitions among- 3 2low-lying states in even-even nuclei are extremely small ( < 1 0 )

: therefore large magnetic dipole matrix elements B(M1) in odd-A

systems are usually characterized by considerable single-particle

contributions.

«!,*- 1 (V • 5-5857 (V

53

A typical example is given by the 3/2^ -♦ 1/2^ transition in 125-121 -2 2Te, whose measured B(M1) values are « 5 x 10 p. However, the

IBFM-2 calculation yields much smaller rates, due to the different

structure of the corresponding wave functions, as already pointed out

in the analysis of the electric quadrupole transitions.

Other significant deviations can be noted in some odd-proton 5 3 I

isotopes : experimentally it is found that the first excited state in 125 +53*72 *s a 2t2 one> having a single-particle character arising from

the ^ 7 / 2 orbit; in the IBFM-2 picture the second 7/2^ state, rather

than the first one, has a similar structure, as can be understood by

looking at the composition of the wave function and at the

corresponding B(M1; 7/2^ -* 5/2^) ~ 0.18 /j , which is very close to themeasured value B(M1; 7/2^ -» 5 /2 ][)eXp.!;s 0.29 p.

Analogous considerations lead one to identify the single-particle+ 127nature /2 t*ie second 7 / 2 2 state in 5 3 1 7 4 . which give rise to a

large B(M1) ( B(M1; 7/2^ 5/2^) ss 0.17 p ) for the transition to

the ground state 5/2^, whereas the experimental value B(M1; 7/2^ -► + 25/2^) w 0.02 pfa reflects a more complicated interplay between the

available configurations.

54

Table 3.10 Experimental and calculated B(M1) values in $2^e '

NucleusJ 1

B(M1 ; J+ -

exp

-> J*) X (1 0 - 4 ,2>

th

121Te a) 52 69 3/2, — > 1 /2 , 573 ± 115 0 . 2

123Te b) 52 71 3/2, — > 1 / 2 , 412 ± 82 0 . 1

3/2, — > 1 / 2 , 394 ± 35 0

3/22 — > 3/2, 34 ± 3 13

3/22 — > 1 / 2 , 23 ± 2 0.08

5/2, — > 3/2, (1970) 1.6E3

125Te C) 52 735/2, — > 3/2, 215 ± 1 1 4

(7/2,)— > 5/2, 2 0 ± 25 291

7/22 — > 5/2, > 5.4 524

5/2, — > 5/2, 590 ± 118 1.7E3

5/22 — > 3/22 (251) 6.1E3

5/2, — > 3/2, 895 ± 18 418

a) [Ta79], b) [Ta80], c) [Ta81].

55

Table 3.11 Experimental and calculated B(M1) values in ggl.

Nucleus J* — > J* B(M1; — > J^) x (10 ^ p^)

exp th

( 7 / 2 , ) — > 5 /2 , 358 36

1 5 3 I 6 8 <° 3 7 2 . 1 7 2 . 5 3 7 6

3 /2 , — > 5 /2 , (< 90) 5.5E3

3 /2 , — > 5 /2 , (167) 5.2E3

7 /2 , — > 5 /2 , 2860 ± 573 0.02

1b3 l 72c ) 3 /2 , — > 5 /2 , 134 ± 7 4.2E3

1 /2 , — > 3 /2 , 985 ± 394 97

7 /2 , — > 5 /2 , 215 ± 3 0 8

3 /2 , — > 5 /2 , 78.8 ± 16 3.1E3

1 /2 , — > 3 /2 , 2686 ± 430 434

1 5 3 l 74d> 5 /2 2 — > 3 /2 , 394 ± 55 2.6E3

5 /2 2 — > 7 /2 , 286 ± 40 839

5 /2 2 — > 5 /2 , 1397 ± 559 320

( 9 / 2 , ) — > 7 /2 , 1056 ± 211 76

continued

a) [Ta79] , b) [Ta80] , c ) [Ta81] , d) [Ha82]

56

Nucleus B(M1; J* — > J*)

exp

X <10' 4 f a

th

5 /2 , — > 7/ 2 1 179 ± 54 14

129 e) 53 76

3 /2 , — > 5 /2 , 72 ± 18 2.1E3

5 /2 2 — > 3 /2 , 16 ± 5 2.2E3

5 /2 2 — > 5 /2 , 6 .,4 ± 2.9 322

e) [Ha83].

57

Table 3.12 Experimental and calculated B(M1) values in 3gBa and a.

Nucleus B(M1;

exp

+ + -4 2 — > J p x ( 1 0 * ppth

133o a) 56 77 3/2, — > 1 /2 , 174 ± 11 117

135,, b) 56 79

1 /2 , — >

5/2, — >

3/2,

3/2,

43 ±

75 ±

18 635

27 3.4E3

5/2, — > 5/2, 172 ± 14 145

3/2, — > 5/2, > 537 8.3E3

133La a) 57 76 7/2, — > 5/2, 11.5 ± 2.5 0.01

7/2, — > 5/2, 8 8 ± 14 21

1 / 2 , — > 3/2, 304 ± 90 327

7/2, — > 5/2, 33 ± 1 3

5/22 — > 7/2, 36 ± 2 4

135La b) 57 785/22 — >

3/2, — >

5/2,

5/22

60.9 ±

> 45

3.6 50

342

3/2, — > 5/2, > 215 5.2E3

1 / 2 , — > 3/2, > 6088 327

137La C) 57 80 5/22 — > 3/2, | 30.4 ± 1 . 6 0 . 2

a) [Se8 6 ], b) [Se87], c) [Pe83].

58

To complete our analysis we compute the magnetic moments

(3.23)

and compare them with the measured ones in Table 3.13-3.15.

An important issue, which has attracted much attention over many

years, is the understanding of deviations observed in magnetic moments

from the extreme single-particle values ( Schmidt values) :

A successfull, but yet qualitative, explanation has been given in

terms of configuration mixing, especially for those systems in the

neighborhood of doubly magic nuclei. In the transitional region we are

considering here the picture is even more complex, due to many valence

nucleons, which are responsible for strong deformation. The behavior

of the observed magnetic moments of the 1/2^ in odd-A 5 2 !® isotopes

( Table 3.13 ) shows such large deviations from the Schmidt value<fXl/2+>S “ '1-91 Pjj. aPProPri-ate to an °dd neutron in the orbit

coupled to a spherical core. The IBFM-2 calculation is able to

reproduce only a small fraction of the observed reduction. Another119typical example is 5 2 ^ 6 7 » wh°se measured magnetic moment is <^1 / 2 ^

= | 0.25 | Pfl ; in the IBFM-2 model space the ground state 1/2^ is

predominantly of single-particle nature (usi/2> therefore its

/

(3.24)( gs - gx ) J

2 j + lV

59

magnetic moment is much larger </ii/2+>IBFM-2 8 5 "1*5 ^ut at

another 1 / 2 ^ state appears, whose structure is determined to a large

extent by the coupling of the deformed even-even core to the

orbit, giving rise to a magnetic moment </i]y2+>IBFM-2 “ -0.59, much

closer to the observed one.

Fig. 3.10 and 3.11 illustrate some selected magnetic moments for

positive parity states in both odd-proton ( 5 3 I) and odd-neutron (^T®*

5 gBa) isotopic chains. A gradually increasing trend from large

negative to small positive values characterizes the magnetic moments

of the = l/2+ state in the odd ^33* ^ 3Ba isotopes [Mu83] . Within

the core-quasiparticle model this behavior has been attributed to two

key factors : the lowering of the Fermi level and the increase in de

formation going from heavier to lighter nuclei. Using the mentioned

approach, based upon the Nilsson model, one can show that the main133 131component of the wave function of ’ Ba comes from the orbital

[400^] , that of ^27-125ga ^g assocj;ated with [411^] and that of the129intermediate nucleus Ba is described as a combination of the

previous two. Here we have made use of the usual notation [NnzAfl] to

indicate the relevant Nilsson orbitals. The states [400^] and [411^]

originate from the shell model levels Sjy2 and d3/2 resP®cti-v®ly : f°r

small values of the deformation parameter fi the contribution to the

magnetic moment of — l/2+ from the orbital [400^] is large and

negative, whereas by increasing the deformation the energy of the

[411^] state decreases, thus becoming more favorable to be occupied

and its magnetic moment turns out to be slightly positive. The IBFM-2

model, on the other hand, predicts that even for the lightest isotopes

such as ^7-125ga magnetic moments of the lowest l/2 + state should

be negative. This is due to the fact that their wave functions have

not a dominamt dg^2 component. However, when this is the case, as for

l/2+ states at higher energy, then their magnetic moments become

positive.

Since in the present version of the IBFM-2 calculation, the

single-particle energies have been kept the same for a whole

isotopic chain, any improvement would require introducing a mass de

pendence, because, as we have seen, there is evidence that both B(M1)

and magnetic moments are sensitive to the relative location of the

multi-j orbits.

60

Fig. 3.10

Magnetic dipole moments of the first 5/2+ and 7/2+ states in

odd-A 53I isotopes as a function of neutron number. The dashed

lines represent the Schmidt values, whereas the solid lines

represent the results of the calculation. The experimental data

(points) are taken from [Ta81,Ha82,Ha83,Au76,Se86].

61

V - T / Z \ 5 -

( M n )4.

3.

2 .

M 5 /2 | 5 ‘

(Mn)4.

3.

2.

I.

5 0 5 4 58 62 6 6 70 7 4 78 82

Neutron Number

Fig. 3.11

Magnetic dipole moments of the first l/2+ and 3/2+ states in

odd-A 5gBa and 52^® isotopes respectively, as a function of

neutron number. The dashed lines represent the Schmidt values,

whereas the solid lines represent the results of the

calculation. The experimental data (points) are taken from

[Ta80,Ta81,Ha82,Ha83,Mu83,Se86].

62

(MN)

M i / 2 1 l

( ^ N )o

- 1.

-2.

50 54 58 62 66 70 74 78 82

Neutron Number

63

Table 3.13 Experimental and calculated magneticdipole moments in 5 2 * ®'

Nucleus J+ "j < Vexp th

117Te a) 52 65 5/2, -0.75 ± 0.05 -1.3

U 9 Te b) 52A 67 1/2 , | 0.25 ± 0.05 | -1.3

123Te C) 521 71

1/2 ,

3/2,

-0.7363 ± 0.0005

0.72 ± 0.12

-1.3

0.85

V 2 x -0.88828 ± 3.E-5 -1.3

125Te d) 52 73

3/2,

3/2 2

0.604 ± 0.006

0.585 ± 0.090

0.8

2.2

V 2 , 0.37 ± 0.11 1.7

127Te e) 52 75 3/2, 0.635 ± 0.004 0.82

32Te?7f) 3/2, 0.702 ± 0.004 0.83

a) [B187], b) [Au79], c) [Ta80], d) [Ta81], e) [Ha82], f) [Ha83]

64

Table 3.14 Experimental and calculated magneticdipole moments in

Nucleus J+ "j (Vexp th

5/2, 2.821 ± 0.05a) 3.0125 53 72 r 2.2 ± 0 . 7 b >

3/2,k 1.06 ± 0.07b)

0.9

5/2, 2.8091 ± 0.0009 3.1

127t c) 53 74 7/2, 2.03 ±0.15 1.9

3/2, 1.06 ±0.17 1.2

7/21 2.6174 ± 0.0008 2.0129 d) 53 76 5/2, 2.801 ± 0.003 3.3

7/2i 2.742 ± 0.001 2.1131 e)53 78

5/2, 2.79 ± 0.50 3.5

133 f) 53 80 7/2, 2.856 ± 0.005 2.3

a) [Ta81], b) [Le78], c) [Ha82], d) [Ha83], e) [Au76], f) [Se86].

65

Table 3.15 Experimental and calculated magneticdipole moments in c,Ba and c-,La.

->0 J /

Nucleus J+ (Vexp th

123Ra a>56 67 V 2 X -0.687 ± 0.018 -0.72

125d a) 56 69 1/2 , 0.177 ± 0.012 -0.6

127Ra a) 56 71 V 2 X 0.089 ± 0.012 -0.6

i / 2 , -0.397 ± 0.006 -0.56129na a) 56 73

7/2i 0.930 ± 0.017 2.8

131n_ a) 56 75 l / 2 i -0.709 ± 0.016 -0.74

1 / 2 , -0.777 ± 0.014 -1.3133Ra b >56 77 3/2, 0.51 ±0.07 0.8

135r _ a)56 79 3/2, 0.837943i ± 1.7E-5 1.5

137La C) 37 80 7/2i 2.695 ± 0.006 2.4

a) [Mu83], b) [Se8 6], c) [Pe83].

66

3.4 One-Nucleon Transfer Reactions

Another valuable source of spectroscopic information is based on

single particle transfer, which takes place in stripping and pick-up

reactions. The various final states of the target are populated

provided the transition is compatible with the transferred angular mo

mentum. As a general remark, if a level is strongly observed in the

process, its single particle nature is manifest. On the other hand if

some state is populated in decay studies, but is absent or weakly

excited in the reaction we are considering, more complicated contribu

tions such as 2p-2h, 3 quasiparticle or phonon-particle coupling are

necessary to explain its structure. When a nucleon is removed from the

projectile and is captured by the target, the experimental stripping

cross section is expressed as

with N being a normalization factor, whose explicit value depends upon

spectroscopic factor and stands for the cross section, evalu

ated within the Distorted Wave Born Approximation. An analogous rela

tion can be derived for pick-up processes in which a particle is

emitted by the initial nucleus, leading to

(3.25)

the specific transfer ( e.g. N - 1.53 for (d,p), N = 4.42 for ^He,d),

N *= 5.06 for (t,d), N = 46 for (a,3He) and (a,t) ). S is the so-called

(3.26)

67

and we note that the statistical factor (2Jf+l)/(2J^+l) is missing

here.

To get some insight we can focus our attention to the simple case

of a single j shell, for which the spectroscopic factor is just

proportional to the number of active particles in the final state

(c.f.p.) [Sh63].

The behavior of S by filling a given orbit decreases or increases

monotonically for stripping or pick-up reactions on an initial even-

neutrons in the heavier Ca isotopes ( A = 40-48 ) . Once this trend is

associated directly with the emptiness or occupancy of the shell, its

physical interpretation becomes transparent. However, for medium mass

and heavy systems many configurations are accessible and more elabor

ate schemes need to be developed. As a matter of fact the IBFM-2

provides us with the ingredients to tackle the problem.

The one-nucleon transfer operator between adjacent nuclei having

the same number of bosons, can be written as

S ( j ) - n < j J r I) j"’1 r > 2 (3.27)

where the symbol < |) > denotes a coefficient of fractional parentage

even target, as is the case for the single ^7/2 shell occupied by

fjJ,(st x d x a9

(3.28a)

whereas

68

Pt(J>_ , t O > _ [, (.t * : , « > ] + [ i . ( d » x i ) « > ]p y i

(3.28b)

describes the process in which the bosonic part is changed by one110 110 unit. For example, in the stripping Te (d,p) Te

pt(j)112 p 11352Te60 * 52Te61 (3-29)

( N = 1 , N = 5 ) ( N = l , N ~ 5 ) ® l i /7T V TT v

the expression (3.28a) is the appropriate one, because in the IBFM-2

picture the initial and final nuclei differ by the unpaired fermion

only. On the other hand the reaction ^^ T e (d,p) ^ ZTe

pt(j)111 p 112 52Te59 » 52Te 60 (3'30)

( N = l , N = 4 ) 8 li/ ( N - l . N - 5 )it v it v

requires eq. (3.28b) to simulate the addition of a neutron.

A remarkable property is that the coefficients appearing in the

definition of have a microscopic interpretation, arising from

the correspondence between the shell model and the boson-fermion

space. The derivation [Sc80,Sc81], based upon the concept of general

ized seniority [Ta71], gives

69

^ “ bl UJ

rjj' " b2 Vj^j'j J NJ2J+1)(3.31)

b V_1 1 y n +i

u: Ib : - ■ 10jj' 2 “j "j'j 4 2j+l

for the coupling of a particle to the even-even core, while for a hole

the corresponding expressions are

f- “ b- v .J 1 J

rjj' “ " b 2 Uj J N J 2 J + 1 )

(3.32)

U 16 . = b' -J-j 1 yir

P

?jj' “ b 2 Vj V j J 2J+1

The quantities Uj, Vj, 0j*j have already been introduced in Chapter 2

( eq. (2.25), (2.28) ) in dealing with the boson-fermion Hamiltonian.

The normalization constants b^, b2 or b^' , b 2 ' are not free

parameters, but are fixed instead by the condition

b. b: , , -,1/2r - p ■ ! A y <3-33>2 2 L j . j '

70

together with the sum rules

I < a J I At(j) I a J > 2 - (21+1) u2 (3.34a)a j ° ° P 1 e e J j,Po ’ o

X <a j I Bt(j) I a J > 2 - (2j+l) v? , (3.34b)T e e ' p ' o o 1 ,Pa , J Jo o

where <*0J0 , a^Je denote all the quantum numbers necessary to specify

the states of the odd-even and even-even nuclei respectively.

If a hole is involved, the corresponding sum rules are obtained

from eq.s (3.34) by replacing Uj ^ with Vj p and vice-versa.

The matrix elements of the operator P^(j) enable one to define

the transition strength

( 2 J -+1 ) , . . . n

’str. - (2TTT) S " <“f Jf II Ptpj I' “i Ji> (3-35)

which is connected directly to the measured cross section through eq.

(3.25). Likewise, exploiting the Hermitian conjugate relation

P = (-)J 'm (Pt(^ ) ^ (3.36)m - m

(for the sake of simplicity the additional index p has been dropped in

writing eq.(3.36)), the analogous quantity for pick-up reactions can

be constructed

Vu . - s - “V f i i “ij i>2 (3-37>

71

Table 3.16 Experimental and calculated spectroscopic strengths forone-neutron stripping reactions in g2*'e'

EX(keV)

Final State Transferred1 exp

Gth

0 1/2

120Te 121Tea)

0 0.58 ±0.09 1.2

212 3/2 2 1.3 ± 0.2 2.9475 (5/2) 2 0.50 0.77681 1/2 0 0.11 0.11

809 (5/2) 2 0.19 0.001

0 1/2

122Te 123Teb)

0 0.78 1.0

159 3/2 2 2.02 2.7509 5/2 2 « 0.30 0.11

600 1/2 0 0.13 0.03691 3/2 2 0.27 0.008785 3/2 2 0.42 0.11

0 1/2

124 125„ c)Te — > Te '

0 0.84 0.8535 3/2 2 1.84 2.4

642 7/2 4 v 0.48 0.06671 5/2 2 0.36 0.006729 5/2 2 0.24 0.14

continued

a) [Li77], b) [Li75], c) [Gr69].

72

E Final State Transferred Gx +

(keV) 1 exp th

126Te 127Ted)

0 3/2 2 1.52 2.0

60 1/2 0 0.50 0.65502 3/2 2 0.08 0.008764 5/2 2 0.036 0.005784 5/2 2 0.396 0.09

128_Te — > 129_Te

r i.36e)0 3/2 2 « 4-i

<u CMr-l

O

1.5

179 1/2 0' L 0.39f)

r 0.258e)

0.46

967 5/2 2' , 0.192f)

0.02

130Te - > m ie8>

0 3/2 2 0.97 0.86

297 1/2 0 0.32 0.27643 5/2 2 0.01 0.02

944 7/2 4 0.048 0.031043 1/2 0 0.014 0.0031209 5/2 2 0.126 0.04

d) [Gr68].e) from (d,p) reaction [Mo67], f) from (t,d) reaction [Sh81].g) [Sh81].

(

Table 3.17 Experimental and calculated spectroscopic strengths forone-neutron pick-up reactions in 52^e'

EX(keV)


Gth

0 1/2

122Te _ 121Tea)

0 0.67 0.67210 3/2 2 1.24 0.8

0 1/2

124_ 123„ b) Te — > Te '

0 1.10 0.9160 3/2 2 1.70 1.1490 5/2 2 1.20 0.4

0 1/2

126_ 125_ c) Te — > Te '

0 1.4 1.134 3/2 2 2.5 1.5

444 3/2 2 0.13 0.001463 5/2 2 0.08 0.04636 7/2 4 1.9 0.05642 7/2 4 3.0 5.671 5/2 2 1.5 0.5729 3/2 2 0.08 0 .

1053 5/2 2 1.1 4.61132 5/2 2 0.82 0.091147 7/2 4 0.59 0.0071263 5/2 2 0.82 0.002

continued

a) [Ta79], b) [Ta80], c) [R^84].

74

EX(keV)

Final State

4

Transferred1 exp

Gth

1309 7/2

126 125 c) Te — > Te '

4 0.54 0.811317 1/2 0 0.01 0 .1355 7/2 4 0.39 0.11433 5/2 2 0.57 0.0041526 5/2 2 0.23 0.1

0 3/2

128Xe 127Ted)

2 2.5 1.962 1/2 0 1.5 1.3

475 5/2 2 0 . 0.007503 3/2 2 0.07 0 .687 7/2 4 0.56 0.05783 5/2 2 1.7 0.4926 7/2 4 3.9 1 .

1140 5/2 2 0.87 4.71290 5/2 2 0.53 0.241378 5/2 2 0.61 0.051405 1/2 0 0.05 0.0031429 7/2 4 0.16 0.151554 5/2 2 1.1 0.031804 7/2 4 0.21 . 2.31938 7/2 4 0.37 0.13

0 3/2

130Te 129Tee)

2 2.92 2.41180 1/2 0 1.3 1.5

c) [R< 84] , d) [R^85] , e) [Ha83].

75

Table 3.18 Experimental and calculated spectroscopic strengths for one-proton stripping reactions from g2*’e to 53*'

EX(keV)

Final State

4

Transferred1 exp

Gth

0 5/2

*20Te _ 121ia)

2 1.8 1.8

96 1/2 0 0.24 0.41133 7/2 4 4.32 2.5176 3/2 2 0.28 1.9(931) 3/2 2 1.96 0.001

951 1/2 0 0.40 0.011466 1/2 0 0.24 0 .1557 1/2 0 0.22 0.121885 1/2 0 0.16 0.32080 1/2 0 0.18 0.01

0 5/2

122_ 123Tb) Te — > I '

2 2.4 2.1

144 7/2 4 4.7 2.8

176 3/2 2 0.37 2.1

1010 5/2 2 1.6 0.071046 1/2 0 0.38 0.411152 3/2 2 1.5 0 .1240 1/2 0 0.07 0.02

1307 3/2 2 0.18 0.051338 1/2 0 0.024 0.051368 5/2 2 0.11 0.002

continued

a) [Ta79], b) [Ta80].

76

Ex Final State Transferred G(keV) 4 1 exp th

122Te — > 123xb)

1493 1/2 0 0.11 0.002

1583 5/2 2 0.11 0.0031653 3/2 2 0.14 0.02

1718 5/2 2 0.26 2.1

1862 1/2 0 0.22 0.151951 3/2 2 0.10 0.721983 5/2 2 0.14 0.002

124Te — > 125lC)

0 5/2 2 2.6 2.5111 7/2 4 4.0 3.4187 3/2 2 0.30 2.2

243 1/2 0 0.35 0.37371 5/2 2 0.20 0.11

1005 3/2 2 0.96 0.001

1066 5/2 2 1.30 0.051198 5/2 2 0.37 0.031337 5/2 2 0.47 1.71439 3/2 2 0.30 0.002

1663 5/2 2 0.20 0.161690 1/2 0 0.13 0.01

1779 1/2 0 0.07 0.10

1916 1/2 0 0.20 0.15

continued

b) [Ta80], c) [Ta81].

77

EX(keV)


Gth

126Te - > 127ld>

0 5/2 2 2.50 2.9458 7/2 4 6.27 4.1

203 3/2 2 0.27 2.3375 1/2 0 0.48 0.31418 5/2 2 0.48 0.061044 7/2 4 0.83 0.071124 1/2 0 0.16 0.001

1275 7/2 4 1.38 0.7

128Te 129le>

0 7/2 4 5.28 4.628 5/2 2 3.54 3.38

280 3/2 2 0.28 2.2

487 5/2 2 1.26 0.06561 1/2 0 0.42 0.27

1210 1/2 0 0.04 0.001

1483 1/2 0 0.42 0.061741 1/2 0 0.08 0.081823 1/2 0 0.20 0 .

d) [Sz79], e) [Au68].

78

Table 3.19 Experimental and calculated spectroscopic strengths forone-neutron transfer reactions in c,Ba.

EX(keV)

Final State

4

Transferred G 1 exp th

0 1/2

130Ba — >

0

129Baa)

0.41 0.52109 3/2 2 0.49 0.9251 3/2 2 0.04 0 .

0 1/2

130Ba — >

0

131Bab >

0.57 ± 0.09 0.4105 3/2 2 1.01 ±0.15 1.1

364 1/2 0 0.018 ± 0.02 0.151472 1/2 0 0.020 ± 0.02 0.071820 1/2 0 0.053 ± 0.004 0.044

0 1/2

132„Ba — >

0

133Bab >

« 0.36 0.412 3/2 2 » 1.20 1.3

500 1/2 0 = 0.02 0.041247 1/2 0 « 0.07 0.01

0 3/2

134Ba - >

2

135Bab >

1.31 0.77221 1/2 0 0.30 0.23909 1/2 0 0.035 0.019

a) [Gr74], b) [Eh70].

79

On one hand the amount of experimental data collected for the

isotopic chains Te, I, Ba enables one to perform a detailed study, as

is illustrated in Table 3.16-3.19. On the other hand one-nucleon

transfer in Xe, Cs nuclei cannot be observed, due to the high insta

bility of almost all the targets.

The comparison between predicted spectroscopic strength and

measured ones GeXp is difficult for several reasons. First of all the

uncertainty in the angular distribution (do/dfl) is of the order of

15 % . Secondly the function o^y needs to be determined accurately,

because it is supposed to contain the variations observed in the cross

section when different projectiles are used in populating the same

final state for a given initial one. However, it turns out that rel

ative deviations as large as 30 % are possible. This is the case, forO

instance, in the comparative analysis of (d,t) and (He,a) performed

on Te targets [R^84,R^85], Nevertheless general properties can be

clearly identified. The expected behavior in filling single particle

orbitals with neutrons in adjacent Te nuclei is found and agrees with

the corresponding increase shown by. G going from lighter to heavier

isotopes. According to this argument one should expect an approxi

mately constant proton strength in iodine nuclei, independent of the

mass number A, provided that the single particle coupling to the core

does not change appreciably within the N ~ 50-82 shell. Since in this

region we are dealing with the unique parity orbit lh^^y2 and tbe four

positive parity levels lgy^2 , 2dg^2 , 3s^y2 , 2d3^2 , the orbital angular

monentum 1 of the transferred nucleon from the ground state of an

even-even nucleus determines the spin of the final state un

ambiguously, except for 1 - 2 , in which case both Jf = 3/2+ , 5/2+ are

80

possible and additional experimental information is needed to make the

correct assignment.

Another frequently observed feature, which emerges also from our

selected examples illustrated in Fig. 3.12-3.14, is related to the

missing strength, which can be as high as 30 % . In fact by studying

its distribution over the various excited states, several cases which

fail to satisfy the appropriate sum rule (either eq. (3.34a) or eq.

(3.34b)) have been discovered. (The sum rule identifies the total in

tensity to the number of holes or particles respectively.)

A possible explanation for such a phenomenon is that part of the

strength is shifted at higher energy, where the density of states pre

vents one from either identifying or separating them individually. In

any case the IBFM-2 calculation seems to reproduce quite well the

spectroscopic rates up to 2 MeV. Furthermore it becomes an even more

useful tool in making predictions in these regions, which cannot be

investigated within other frameworks. A good candidate reaction is the109 108proton pick-up one from 53I56 to 52ê56’ wôse mean-life has been

measured recently [G187]. The conclusion was reached that the decay

has to be attributed to a d ^ ^ proton with spectroscopic strength G5/2

= 0 .1 , because the assumption of a Z~]/2. transi-tôn would lead to a

strong enhancement, which is hard to explain. In the IBFM-2 the com

putation of the matrix elements involved is a straightforward task,

however, the picture we get is quite different. The first excited

state turns out to be a 5/2+ one at = 137 keV from the 7/2+ ground

state. Using the procedure just introduced, we obtain the results

presented in Table 3.20. It is clear that the configuration mixing

taken into account by the IBFM-2 leads to a reduction of the overlap

81

between initial and final states for both J *» 7/2+ and 5/2+ from the

single-particle estimates, but not to the extent claimed earlier.

Table 3.20 Spectroscopic strength for one-proton transfer reaction- 109 . ,, „ 108from 33I leading to 52^ •

expa) thb) s.p c)

7/25/2 « 0.1

0.610.59

0.750.67

a) [Gi87]b) IBFM-2 calculationc) single particle estimate.

1

Fig. 3.12

Experimental (Exp) [Sz79] and calculated (Th) spectroscopicion o ioistrengths G for the Te( He,d) I reaction as a function of

the excitation energy.

00ho

Ex (MeV)

Fig. 3.13

Adopted experimental [R^84] spectroscopic strength G for the 126 125one-neutron Te -♦ Te pick-up reaction as a function of

the excitation energy.

8.7 . -

6. -

5 . -

4 . -

3 . -

2. -

I . -

0.5 1 . 0

Ev (MeV)1.5 2.0

00oo

Fig. 3.14

1 9 £Calculated spectroscopic strength G for the one-neutron Te 125Te pick-up reaction as a function of the excitation energy.

E x (MeV)

C H A P T E R 4

Beta Transitions of Medium Mass and Heavy Nuclei

4.1 General Overview

Without being perturbed by external sources, a large number of

nuclei undergo spontaneous transformations characterized by emission

of a particles, electrons or positrons, provided that the binding

energy of the final state is higher than the initial one. The

fundamental interaction responsible for the decay determines the time

scale of the process, inasmuch as typical lifetimes are inversely

proportional to the square of the coupling constant. However, within

the same category, variations by orders of magnitude are observed.. o g o n

This is the case in the /J -decay of 20<"’a19 to 19^20’ whose half-lifeOOis 0.87 sec, whereas takes 2.6 years to decay into its daughter

B » 1 2 ■To explain the variety of observed transitions ranging from

allowed to forbidden [Ko66] and the further distinction between

favored and unfavored, nuclear structure effects must be invoked. To

make our argument even more convincing, we can draw an analogy with

the electromagnetic excitations of nuclei. In this process, photons

carrying a discrete amount of energy are absorbed or emitted from the

initial state. The multipole expansion of the associated field leads

85

86

us to recognize that the most important modes of excitation are the

electric quadrupole E2 and the magnetic dipole Ml ones. In fact 7 -ray

spectroscopy provides a significant amount of information concerning

the shape of the nucleus and is an essential tool in identifying the

spin and parity of its states, decay, on the other hand, arises

from the exchange of charged W^ massive bosons, mediating the weak

interaction. The electrons or positrons e^ emerging from the parent

system are emitted with a continuous energy distribution, implying the

existence of another particle, the neutrino, which ensures the

conservation of both energy and angular momentum. Even though the

fundamental nature of weak processes is by now based on solid

observational grounds, it turns out to be very difficult to make

reliable predictions regarding the. transition rates due to their

sensitivity to details of the nuclear matrix elements involved.

For light nuclei, the shell model can be applied successfully to

compute several log ft, as long as the dimensions of the space are

manageable. When medium mass and heavy elements are considered, many

complications arise from configuration mixing, reflecting the crucial

role played by the residual interaction. In the early days

calculations based upon pairing theory [Ki63] were performed assuming

a spherical shape and accounted qualitatively for the observed

general trends. However, further investigations [To85] pointed out

that in order to explain part of the systematic hindrance found in

comparing predicted matrix elements with measured ones, additional

refinements are needed, such as those ascribed to the interaction

among unlike nucleons. Even more elaborate studies [Ch83] have been

undertaken within the quasiparticle random phase approximation

87

(QPRPA), having the common purpose of understanding to what extent

nuclear structure effects are responsible for the quenching of the

Gamow-Teller strength. Since the Interacting Boson-Fermion Model has

proven to give a fairly good description of the excitation properties

of many odd-A nuclei, characterized by either spherical or deformed

shapes, it is interesting to explore if this framework is able to deal

with £-decay. The following sections present the details of

calculations throughout the neutron shell N = 50-82 for a variety of

nuclei with proton number Z = 52-57. Our attention will be focused on

allowed transitions between positive parity states and enables us to

study both the isotopic dependence of a specific decay and the

strength distribution over low-lying states. We note also that the

construction of the relevant operators is quite straightforward and

does not require the introduction of any additional parameters beyond

those already appearing in the Hamiltonian and in the spectroscopic

intensity.

88

In the treatment of nuclear /?-decay it is customary to adopt two

essential approximations. First of all, as we are dealing with states

whose energy is typically of the order of a few MeV, it is justified

to describe nucleons as non-relativistic particles. Furthermore, since

the lepton wavelengths are much bigger than the size of the nuclei

they are interacting with, their currents and wave functions can be

evaluated at r = 0 . (If such approximations are not made and the

spatial variation of the leptonic wave over the nuclear volume is

included, one obtains forbidden transitions, characterized by slower

rates.) After imposing the appropriate invariance conditions, it is

possible to show that under the previous assumptions the interaction

Hamiltonian is determined by two terms, arising from the vector and

axial vector couplings between the lepton and the nuclear charge

currents. The corresponding matrix elements between initial and final

states can be written as follows

4.2 Fermi and Gamow-Teller Matrix Elements

While the Fermi operator affects the isospin coordinates only, giving

rise to the expression

2 2 2 2 2 Mfi - + (4.1)

1 A 2I

M. ,MY t ,(k) I a.J.M.T.T .> “ , ± 1 1 1 i i zik=l

(4.2)

89

the Gamow-Teller transition involves spin degrees of freedom as well,

leading to

A<^GT> " 2J + 1 E ^ f ^ f ^ z f I £ a/i(k) Xi /i,Mi ,Mf k=l

x

(4.3)

where in the last step we have applied the Wigner-Eckart theorem to

separate the recoupling coefficient (Clebsch-Gordan) from the reduced

matrix element.

The fact that the operators have a different nature, being a

scalar (Fermi) both in coordinate and spin spaces and a vector (Gamow-

Teller) under spin rotations, has important consequences in

determining the selection rules for total angular momentum, isospin

and parity, which are summarized in Table 4.1.

Since for allowed 0-decay the outgoing leptons do not carry any

orbital angular momentum, the nucleons taking part to the process can

only change from a single particle level characterized by the quantum

numbers ( ^ .j ) to a final orbit (lf.^.jf) with l^=l£. This

restriction greatly reduces the number of possible contributions to

the evaluation of the matrix elements, but apart from light systems or

special examples near doubly magic nuclei, multi-j calculations are

not feasible if many active particles populate the valence orbits.

o<T T 1 + 1 I T T ~> i 2 i ’ zi’ 1 f’ zf

x t± (k) | “ 2(2Jt+ 1) (2Tf+ 1)

x || I a(k) r(k) ||k=l

90

As this is the case for the region of the mass table we are con

sidering here, we find it useful to apply the proton-neutron

interacting boson-fermion model to a quantitative study of transitions

among odd-A nuclei.

To construct the relevant operators we recall that 0 ' decay

occurs when a neutron ( u ) is annihilated and a proton (zr) is created.

Thus

are the IBFM-2 images of the Fermi and Gamow-Teller operators

identified with those already introduced in dealing with the one-

nucleon transfer reactions ( eq.s (3.28), (3.36) ) and contain in

lowest approximation two terms, which act on the fermionic part of the

wave function and on the bosonic one as well, without requiring any

new parameter. In eq. (4.4), (4.5) the summations run over the single

particle orbits included in the model space and the quantity is

proportional to the reduced matrix element of the spin s :

The couplings to total angular momentum 0 (Fermi) and 1 (Gamow-Teller)

are fixed by imposing the required transformation properties under

rotation.

(4.4)

(4.5)

respectively. The explicit expressions for P p ^ \ can be

(4.6)

91

Table 4.1 Selection rules for allowed /?-transitions.

Fermi Gamow-Teller

AJ = 0 Total Angular Momentum AJ = 0,1( 0 -» 0 no)

AT ■= 0 Isospin AT - 0,1

n i “ ” f Parity *1 = *f

To complete the picture we need to evaluate the observables just

introduced between the eigenvectors obtained from the diagonalization

of the Hamiltonian describing both the odd-neutron and the odd-proton

nuclei. To achieve this goal we make use of standard reduction

formulas, derived for tensor operators [Sh63], which yield

< V - I pt‘j> a' 'J"> x

(4.7)

x <a' 'J' ' II P ( || a. J .> 5 T T 11 v 11 i i J.Jj.l f

<MrT> - 1 ( . ) Ji+Jf +1+j +3/2 I 6 (2i+ l)(2 j7TI)j.j' J (2J.+1)

' 1/2 1/2 1 ' ' j j' 1 'x • - x I

. J' j 1 , a' ' ,J" ‘ Ji Jf J' \

(4.8)

,t(j)X <afJf || P ™ || a " J " > <q " J " P(j,) || a.J.> v 11 i l

92

provided that the initial and final states have the same parity.

Analogously by interchanging the labels tt and v one can also

encompass /?+ transitions in the present formalism.

4.3 Calculation for Odd-A Isotopes

Just as for the study of excitation properties, two strategies

can be pursued in analyzing nuclear beta decay within the IBFM-2

framework : the first is based on the concept of dynamical Bose-Fermi

symmetries and exploits the implications rooted in solving the group-

subgroup decomposition characterizing the problem. The second [A187]

starts from the general Hamiltonian (2.10), whose parameters are

determined by fitting the energy spectra and allows one to

simultaneously treat transitional nuclei with properties intermediate

to exactly solvable limiting cases also. The first approach has

already been applied to examine several transitions among the isotopes

77Ir, 78Pt> 79AU, 80HS A “ 195,197. The underlying symmetry

structure is defined by Ug(6) ® Up(n) ( n = 4 or 20 ) for the

odd-proton, Ug(6) ® Up(12) for the odd-neutron systems and includes

the boson 0 (6) subgroup, because this region has been extensively

investigated and is known to be a remarkable realization of such a

symmetry. Other group chains comprising the rotational limit SU(3) and

the vibrational one, described by SU(5), appear to be the natural

candidates to tackle the corresponding physical situations, although

explicit £-decay calculations have not yet been performed.

Our main objective is to give a systematic survey, rather than

specializing to a particular case, therefore we will concentrate on

the second procedure. We can take advantage of the detailed

spectroscopic study reported in Chapter 2 , where, starting from the

even-even cores, energy spectra, electromagnetic transitions and one-

nucleon transfer rates for the odd-neutron Te-Xe-Ba and the odd-proton

I-Cs-La have been considered. In addition to the wave functions, the

IBFM-2 model provides us with the single nucleon creation and

annihilation operators P p ^ 3\ Pp^^> whose coefficients are

determined microscopically in terms of the parameters uj , vj ,

introduced in the quasiparticle transformation of the pairing theory

and which are estimated from a BCS calculation. Based upon eq. (4.7)

and (4.8) a computer code has been written [De87] to evaluate the

Fermi and Gamow-Teller matrix elements and enables us not only to make

comparisons with the observed transitions, whenever possible, but also

to predict their behavior for systems far from the stability line. In

Table 4.2-4.6 we summarize the results of our calculations.

The theoretical ft values are expressed as

ft *= j---- -------- z----- j - sec (4.9)“V + (V V ^

and the ratio of the coupling constants is assumed to be the one inOfree space (G^/Gy) = 1.59 ± 0.02, even though a renormalization

should incorporate part of the physical modifications taking place in

nuclear matter. A variety of cases are contemplated In this scheme by

combining the quasiparticle degrees of freedom built from the positive

parity orbits lgy^* ^ 5 / 2 > 3sl/2 ’ ^d3/2 to t*ie l°w-lying collective

excitations of the even-even cores.

93

94

Table 4.2 Experimental and calculated log ft values along withGamow-Teller matrix elements for 0-decay in = 5 2^e'

0 -decayJ+i "

j+log ft <MGT> 2

Jftha)exp tha) exp

117I -U 7 Te b) 53 64 52 65 5/2 - 3/2, 4.75 4.79 0.069 0.064

119I -+119Te C) 53 66 52 67 5/2 - 3/2, 4.96 5.13 0.043 0.029

121I ->121Te d) 52 68 52 69 5/2 - 3/2, 5.20 5.23 0.025 0.023

123I -»123Te e) 53 70 52 71 5/2 - 3/2, 5.23 5.36 0.023 0.0175/2 - 3/2, 7.31 5.8 2.E-4 0.006

125I -»125Te f) 53 72 52 73 5/2 - 3/2, 5.37 5.42 0.017 0.015

127Te -127I 52 75 53 74 3/2 - 5/2, 5.47 5.30 0.013 0.020

3/2 - 5/22 6.05 5.85 0.0035 0.0055

129 -,129 j h) 52 77 53 76 3/2 - 5/2, 5.82 5.50 0.006 0.012

3/2 - V 22 6.18 6.04 0.0026 0.0035

131Te -131I 52 79 53 78 3/2 - 5/2, 6.15 5.84 0.0027 0.00563/2 - 5/2, 6.17 6.25 0.0026 0.0022

3/2 - 5/23 5.85 4.6 0.006 0.11

3/2 - 5/24 6.25 5.99 0.0022 0.0040

a) IBFM calculation including a renormalization factor 3.5 .b) [B187], c) [Au79], d) [Ta79], e) [Ta80] •f) [Ta81], g) [Ha82], h) [Ha83], i) [Au76].

95

Table 4.3 Experimental and calculated log ft values along withGamow-Teller matrix elements for /3-decay in 54Xe s* I.

log ft <M > 2a A T + T + GT/3-decay J -*■ J_ ---------------- -----------------

^ a> «-»»> exp th exp th

121Xe -121I b) 54 67 53 68 5/2 -► HCM 6.85 6.69 6.E-4 8.E-45/2 7/22 6.86 6.0 5.E-4 0.004

123Xe -123I c) 54 69 53 70 1/2 -►1/2 , 5.59 5.38 0.010 0.0161/2 3/2, 6.21 7.8 0.002 7.E-51/2 3/22 6.56 7.3 0.001 2.E-41/2 l/2a 6.42 6.55 0.0015 0.0011

125Xe -125I d) 54 71 5372 1/2 3/2, 6.22 7.6 0.002 9.E-51/2 -►1/2 , 5.90 5.39 0.005 0.0161/2 -►1C3/22) 6.93 6.78 5.E-4 7.E-41/2 l/22 6.69 6.65 8.E-4 9.E-4

127Xe ^127I e> 54 73 53 74 1/2 3/2, 6.61 7.6 0.001 9.E-51/2 -►1/2 , 6.22 5.4 0.002 0.015

131I -f131Xe f) 53 78 54 77 7/2 5/2, 6.65 7.23 9.E-4 2.E-47/2 7/2 , 6.86 5.3 5.E-4 0.0207/2 -►5/22 6.98 7.6 4.E-4 l.E-4

133I ->133Xe «) 53 80 54 79 7/2 -►5/2, 6.82 7.8 6.E-4 6.E-57/2 7/2, 7.59 5.2 l.E-4 0.0267/2 -►7/22 6.90 7.6 5.E-4 9.E-5

a) IBFM calculation including a renormalization factor 3.5b) [Ta79], c) [Ta80], d) [Ta81].e) [Ha82], f) [Au76], g) [Se8 6].

96

Table 4.4 Experimental and calculated log ft values along withGaraow-Teller matrix elements for £-decay in 5 5CS = 54^ •

-+ -+ 106 £t ^p -decay i " Jfexp tha) exp tha)

b)l 1/2 - 1/2, 5.28 5.48 0.020 0.013

c) 1/2 - 1/2 , 5.56 5.14 0.011 0.028

d)1 1/2 - 1/2 , 6.53 4.9 0.001 0.052

e) 1/2 - 1/2 , 6.22 4.6 0.002 0.0891/2 - 1/22 5.60 5.48 0.010 0.013

f)1 5/2 - 3/2, 5.55 5.60 0.011 0.010

123_ 123„55 68" 54 69

125 125„Cs,.-* . Xe.

127_ 127v5 5 7 2 " 54 73

129. 129rPCs,,-* P,Xe.

131 131vprCs„-* P,Xe.

133Xe79-133Cs78g) 3/2 - 5/2, 5.62 5.48 0.009 0.0133/2 5/22 7.33 6.4 2.E-4 0.002

a) IBFM calculation including a renormalization factor 3.5b) [Ta80], c) [Ta81], d) [Ha82].e) [Ha83], f) [Au76], g) [Se86].

97

Gamow-Teller matrix elements for /3-decay in c,Ba = ccCs.J O D D

Table 4.5 Experimental and calculated log ft values along with

f i - decaylog ft ^ G T ^ 2

i °fexp tha) exp tha)

1/2 - 1/2 , 5.20 5.92 0.025 0.0051/2 - 3/2, 6.82 7.16 6.E-4 3.E-4

1/2 - 1/2 , 8.10 5.9 3.E-5 0.0051/2 - 3/2, 7.40 7.27 1.5E-4 2.1E-41/2 - 3/22 7.31 6.56 2.E-4 0.001

1/2 - 3/23 8.22 6.97 2.E-5 4.E-41/2 - l/22 6.62 6.59 9.E-4 0.001

1/2 - 3/24 8.50 9.8 l.E-5 6.E-7

1/2 - 3/2, 8.06 8.00 3.4E-5 3.8E-51/2 - 1/2, 6.68 6.57 8.E-4 0.001

127. 127- b)56Ba? r 55 72

131 131 c)56 75 55 76

133 133 d)56 77 55 78

a) IBFM calculation including a renormalization factor 3.5b) [Ha82], c) [Au76], d) [Se86],

98

Table 4.6 Experimental and calculated log ft values along withGamow-Teller matrix elements for 0-decay in gyLa » 5 6®a-

0 -decaylog ft ^ G T *

exp th exp th

129. 129 a)57 72 56 73

131, 131. b)57 74 56 75

133, 133 c)57 76"* 56 77

135, 135. d)57 78"* 56 79

3/2 - 1/2 , 6.03 7.1 0.004 3.E-4

3/2 - 1/2 , 6.26 6.9 0.002 5.E-43/2 - 1/2 2 5.85 7.9 0.006 5.E-5

5/2 - 3/2, 5.47 5.44 0.013 0.0145/2 - 3/2, 6.95 6.10 4.E-4 0.003

5/2 - 3/2, 5.66 5.17 0.009 0.0275/2 - 3/2, 7.88 7.26 5.E-5 2.E-45/2 - 3/23 7.75 5.8 7.E-5 0.006

a) [Ha83], b) (Au76), c) [Se8 6], d) [Se87].

Fig. 4.1

Fractional deviation of log ft values versus neutron number

for 0 transitions between adjacent nuclei. Both theoretical (th)

and experimental (exp) values are taken from Table 4.2, 4.3, 4.4

and 4.6 .

99

50 54 58 62 66 70 74 78 82Neutron Number

Fig. 4.2

Fractional deviation of log ft values versus neutron number

for P transitions between adjacent nuclei. Both theoretical (th)

and experimental (exp) values are taken from Table 4.3, 4.4, 4.5

and 4.6 .

©

©■r*

©00

IoIX

Z® 0> E. roo3

©0>

C3c r - j CD O

->1

->lCO

CDro

ioro

io

L o g f t

L o g f tth

- 1exp

o o— roT------- T

io — +

Uro — +

-0.3

ioro

iO

L ° g f t th L o g f t g x pO O

ro

-I

o©

100

101

No attempt has been made to deal with intruder states arising

from admixtures with the next shell 82-126. However, it is interesting

to stress that nuclear structure properties such as deformation can be

properly accounted for. This is crucial especially in 0-decay

processes, because small changes in the wave functions may produce

large variations in their transition rates.

4.4 Interpretation of the Results

Over many years several analyses have been developed motivated by

the need to explain striking discrepancies between observed and

calculated matrix elements in 0-decay. It is evident that even in the

simplest cases single particle (s.p.) estimates fail to be quan

titatively correct, because they ignore configuration mixing [Ar87]

and disregard other kinds of excitations, whose origin must be found

in the details of the residual interaction. It is the first natural

step to introduce the strong coupling between identical nucleons via

the pairing theory, which is adequate to treat spherical nuclei. In

this context Gamow-Teller matrix elements connecting one quasiparticle

states are expressed as follows

for 0 transitions of the type ( v odd, n even ) -► ( v - l , n + l )

(4.10)

102

2 2 2 2<M > . - v7 v7, <M„_> (4.11)GT pair. j , u J'.tt GTs.p.

2for processes like ( v even, jt odd ) -+ ( v - 1 , tt+1 ), where vJ » Vo(uj ^) are the BCS parameters counting how full (empty) a neutron

orbit is.

Focusing our attention on a specific example such as the

transition 5/2^-+ 3/2^ between tj-jl and 52Te illustrated in Fig 4.3,

one discovers that the pairing theory is able to predict only a small

fraction of the observed suppression from the single-particle value o<Mq2i> s p = 1.6 describing the transformation of a proton from the

d5/2 level to a neutron in the d g ^ orbital. A factor ® 70 is still

unaccounted for and indicates the importance of the neighboring

configurations, whose contribution turns out to be significant.

The IBFM-2 calculation shows a remarkable improvement, reducing2 2

the magnitude of the quenching IBFM-2 / <^GT> exp to ab°ut 3-5 .

To understand why this is the case we stress that most of the

complex interplay between nuclear degrees of freedom Is taken into ac

count in the model through the collective excitations, represented by

the bosons, the quasiparticle features and their interactions. As a

consequence the wave functions contain much physical information con

cerning the different couplings, ranging from weak to strong. Other

isotopic chains , including s^Xe, 5 5 C S , 5§Ba, syLa , have been consi

dered in the attempt of establishing a more complete picture. The

mentioned transition 5/2^ -► 3/2^ has also been computed for those

nuclei and the results are summarized in Fig.4.4 using the same

quenching factor 3.5. For the heavier species it seems to be more

difficult to reproduce the correct behavior.

Fig. 4.3

Comparison of Gamow-Teller matrix elements predicted by pairing

theory (dashed line), results of the calculation (rectangles)

using the interacting boson-fermion model (IBFM) and

experimental data (triangles) for I -*■ Te. The experimental data

are from [B187,Au79,Ta79,Ta80,Ta81,Ha82,Ha83,Au76].

103

( m g t )

N e u t r o n N u m b e r

Fig. 4.4

Comparison between Garaow-Teller matrix elements, calculated

(continuous lines) using the interacting boson-fermion model

(IBFM), including a renormalization factor 3.5, and experimental

data (triangles,squares and circles) for I -*• Te, Cs -» Xe,

La-+ Ba. The experimental data are from [B187,Au79,Ta79,Ta80,

Ha82,Ha83,Au76,Se86,Se87,Pe83].

104

Neutron Number

105

A possible reason can be the role played by the configurations

arising from orbitals in the next shell 82-126, which have been

neglected in our approach.

Furthermore, we note that a departure takes place between the

IBFM-2 predictions and the results expected from the pairing theory

for the decays 5 3 ! “ * 5 4 ^ and 5 5 C S -* 5 g B a ( Fig. 4 . 5 ). According tooeq. (4.11) the matrix elements should increase monotonically like v 4J » V

as more neutrons are added toward the closed shell at N = 82. However,2the IBFM-2 exibits a sharp change in tendency and <MqX> drops

abruptly, due to the different nature of the model states, which are

no longer of one quasiparticle character. Unfortunately no experi

mental information is available to provide further insight.

Thus far we have been concerned with the whereabouts of a

particular transition, but in practice the Gamow-Teller (GT) strength

is distributed over many states and much effort has been spent in

understanding the related mechanism responsible for its fragmentation

and the even more intriguing property of its observed hindrance.

Moreover, (p,n) reaction studies [Do75] indicate that in medium mass

and heavy nuclei a considerable fraction of GT strength is

concentrated as a resonance at higher energy. Our method deals with

only low-lying excitations, therefore it is a reasonable choice to

restrict the study below 2 Mev.

In Fig. 4.6 we compare the GT distribution predicted by the IBFM+ 121 121 calculation with the observed one for the /? -decay I -*• Te. As

we have explained earlier, the operators simulating the proton

destruction and the consequent creation of a neutron contain

normalization coefficients and b2 > which have been fixed by the

106

requirement of exhausting the one-nucleon transfer sum rule.

Although this is the most meaningful choice, the missing strength

gives a quantitative measure of the violation of such a constraint.

Following the same ideas, one can easily determine distributions which

are of increased interest, because they are involved in solar neutrino

detection. As a matter of fact the capture rate from the lowest 5/2+127 127in I leading to Xe has been suggested as a possible candidate to

build more efficient counting devices [Ha88]. We show in Fig. 4.7 the1 9 7 +outcome of our prediction. Since the ground state of Xe is 1/2 ,

such a transition is forbidden ( AJ - 2 ) and most of the strength is

shared by the various 3/2+ and 5/2+ states, while a small portion goes

into the 7/2+ . This is only an example which illustrates the power of

the present technique, which is amenable to further applications,

especially for neutron rich elements, whose importance in

astrophysical processes is well recognized.

From the detailed spectroscopic study performed in Chapter 3 we

have learned that both collectivity and single particle features play

important roles in nuclear physics. 0 -decay represents an additional

test for any theory, because of its sensitivity to the basic

ingredients determining the structure of the states. The IBFM-2 model

gives in part a quantitative account of the main observed property,

that is, most of the hindrance exibited by the matrix elements is

reproduced. The truncation of the space , affecting both the bosonic

part and the fermionic configurations is presumably responsible for

the remaining discrepancy, together with the coupling of nuclear

degrees of freedom to A-resonances [Ar87,To87].

Fig. 4.5

Calculated Gamow-Teller matrix elements in the interacting

boson-fermion model (IBFM) for I -» Xe and Cs -*• Ba.

107

Neutron Number

Fig. 4.6

Comparison between Gamow-Teller strength distribution, calcu

lated (Th) using the interacting boson-fermion model (IBFM),

including a renormalization factor 3.5, and the experimental one

(Exp) [Ta79] for the decay 121I -► 121Te.

( m g t )

( m g t )

0 . 5 1.0

Ex (MeV)1.5 2.0

108

Fig. 4.7

Predicted Gamow-Teller strength distribution by the interacting

boson-fermion model (IBFM), including a renormalization factor

3.5, for 127I -+ 127Xe.

2 0 .0 8 -

0 .07 -

0 .06 -

0 .05 -

0 .0 4

0 .03

0.02 -

0.01 -

E x (MeV)

C H A P T E R 5

Summary and Concluding Remarks

The application of the interacting boson-fermion model to the

study of nuclear spectroscopy has been extended to comprise single 0

transitions. The results are encouraging and support the usefulness of

the procedure, which enables one to cover a broad spectrum of

phenomena in a unified picture.

The assumption of replacing proton-proton and neutron-neutron

pairs by proton and neutron bosons respectively has been previously

justified as an excellent approximation to the complicated microscopic

problem of many nucleons interacting in an open shell and is regarded

as a physically meaningful choice, as long as protons and neutrons

occupy different orbitals. This is the case of the mass region we have

extensively compared to the known experimental data, where a few

valence protons are present and neutrons fill more than half of their

shell. Another kind of collective excitation, arising from proton-

neutron pairs, has been neglected so far, but becomes necessary

whenever unlike nucleons populate the same single particle levels.

Consequently, a new type of boson has been suggested to complete the

isospin triplet (T~l), enlarging the symmetry structure to U^(3) ®

Usd(6) and the associated model is denoted as IBM-3 [E180].

The formulation can be further expanded, if isoscalar (T=0) bosons are

110

Ill

introduced along with the concept of intrinsic spin S, giving rise to

what is known as IBM-4, characterized by the product group Ugg,(6) ®

Ugd(6) [E181].

In our calculations we have confined ourselves to the set of

states having isospin purity. This argument is supported by the known

evidence that the isobaric analogues appear at higher energy, well

above the ground state. Although numerical results have been obtained

for as many as 84 isotopes, we feel that all the essential ingredients

are now ready and it should be straightforward to apply our method to

achieve a complete compilation of the matrix elements appearing in 0-

decay. As a matter of fact their reliable determination is required in

nucleosynthesis reactions, in which the formation of heavy elements is

induced by neutron, capture. When their rates are fast we are dealing

with r-processes, while if electron decay rates become dominant, as

occurs in the vicinity of the valley of beta stability, we talk about

s-processes. The former conditions are met in supernovae, due to high

neutron fluxes, while the opposite case is realized in the burning of

red-giant stars.

Another outstanding problem, which has attracted increased at

tention in the past few years, concerns double 0-decay [Ha84]. For

certain even-even nuclei the Q-value favors reactions in which a pair

of neutrons is transformed into two protons or viceversa, rather than

allowing the conversion of a single nucleon, leading to the20neighboring element. Typical half-lives are of the order of 10 years

and their measurements could only be performed using geochemical

methods until recently, when laboratory experiments have detected the

two-neutrino double 0-decay of 82Se [E187].

112

On the other hand, there is no experimental evidence as yet for

the neutrinoless process, which violates the conservation of lepton

number, but represents a key test to answer the question, whether

neutrinos are Majorana or Dirac particles.

From a nuclear structure viewpoint, the corresponding Gamow-

Teller matrix elements, extracted from shell model calculations are

overestimated by up to a factor 10, therefore a retardation of the oorder 10 remains unexplained in the attempt of reproducing the

observed transition rates.

The proton-neutron interacting boson model has been used

elsewhere [Sc85] to analyze what progress can be made by taking into

account important effects such as deformation, particularly in those

systems for which the condition of being spherical is not satisfied.

However, it appears that the closure approximation, adopted to avoid

the summation over the intermediate states, is too drastic an

assumption and some important physics might be missing. Going behind

this picture requires the explicit knowledge of odd-proton and odd-

neutron nuclei, which have been successfully described in the

present thesis within the IBFM-2. One should also mention that

several cases of practical interest involve odd-odd systems, whose

structure is complicated further by the occurrence of an additional

unpaired fermion. More work must be done in this direction, before any

definite conclusion can be reached.

R E F E R E N C E S

A184 C.E. Alonso, J.M. Arias, R. Bijker and F. Iachello,

Phys. Lett. 144B (1984) 141.

A187 C.E. Alonso and J.M. Arias, private communication.

Ar75 A. Arima and F. Iachello, Phys. Rev. Lett. 35 (1975) 1069.

Ar76 A. Arima and F. Iachello, Ann. Phys. (N.Y.) 99 (1976) 253.

Ar78 A. Arima and F. Iachello, Ann. Phys. (N.Y.) Ill (1978) 201.

Ar79 A. Arima and F. Iachello, Ann. Phys. (N.Y.) 123 (1979) 468.

Ar85 J.M. Arias, C.E. Alonso and R. Bijker, Nucl. Phys. A445

(1985) 333.

Ar86 J.M. Arias, C.E. Alonso and M. Lozano, Phys. Rev. C33 (1986)

1482.

Au68 R.L. Auble, J.B. Ball and C.B. Fulmer, Phys. Rev. 169 (1968)

955.

Au76 R.L. Auble, H.R. Hiddleston and C.P. Browne, Nucl. Data

Sheets 17 (1976) 573.

Au79 R.L. Auble, Nucl. Data Sheets 26 (1979) 207.

Ar87 A. Arima, K. Shimizu, W. Bentz and Hyuga, Adv. Nucl. Phys. 18

(1987) 1.

Ba57 J. Bardeen, L.N. Cooper and R. Schrieffer, Phys. Rev. 108

(1957) 1175.

B184a R. Bijker, Ph.D. Thesis, University of Groningen, 1984.

Bi84b R. Bijker, program package IBFM-2, KVI Groningen, 1984.

B185 R. Bijker and F. Iachello, Ann. Phys. (N.Y.) 161 (1985) 3.

113

114

Bi87

B187

Bo52

Bo84

Br77

Br85

% Ch83

De87

Do75

Eh70

E180

E181

E187

Fu65

Gi87

Gr68

R. Bijker, private communication.

J. Blachot, G. Marguier, Nucl. Data Sheets 50 (1987) 63.

A. Bohr, Mat. Fys. Medd. Dan. Vid. Selsk. 26 (1952) No. 14.

D. Bohle, G. Kuechler, A. Richter and W. Steffen, Phys. Lett.

148B (1984) 260.

P.J. Brussard and P.W.M. Glaudemans, "Shell-Model

Applications in Nuclear Spectroscopy", North-Holland

Publishing Co., Amsterdam, 1977.

B.A. Brown and B.H. Wildenthal, At. Data Nucl. Data Tables ^3

(1985) 347.

D. Cha, Phys. Rev. C27 (1983) 2269.

F. Dellagiacoma, program package ODDBETA, WNSL Yale

University, 1987.

R.R. Doering, A. Galonsky, D.M. Patterson and G.F. Bertsch,

Phys. Rev. Lett. 35 (1975) 1691.

D. von Ehrenstein, G.C. Morrison, J.A. Nolen Jr. and

N. Williams, Phys. Rev. Cl (1970) 2066.

J.P. Elliott and A.P. White, Phys. Lett. 97B (1980) 16?.

J.P. Elliott and J.A. Evans, Phys. Lett. 101B (1981) 216.

S.R. Elliott, A.A. Hahn, M.K. Moe, Phys. Rev. Lett. 59 (1987)

2020 .J. Fujita and K. Ikeda, Nucl. Phys. 67 (1965) 145.

A.Gillitzer, T. Faestermann, K. Hartel, P. Kienle and

E. Nolte, Z. Phys. A326 (1987) 107.

A. Graue, E. Hvidsten, J.R. Lien, G. Sandvik, W.H. Moore,

Nucl. Phys. A120 (1968) 493.

115

Gr69 A. Graue, J.R. Lien, S. R^yrvik, O.J. Aar^y and W.H. Moore,

Nucl. Phys. A136 (1969) 513.

Gr74 R.D. Griffioen, R.K. Sheline, Phys. Rev. CIO (1974) 624.

Ha62 M. Hamermesh, "Group Theory and Its Application to Physical

Problems", Addison-Wesley Publishing Co., Mass., 1962.

Ha65 I. Hamamoto, Nucl. Phys. 62 (1965) 49.

Ha82 A. Hashizume, Y. Tendow, K. Kitao, M. Kanbe, T. Tamura, Nucl.

Data Sheets 35 (1982) 181.

Ha83 A. Hashizume, Y. Tendow, M. Ohshima, Nucl. Data Sheets 39

(1983) 551.

Ha84 W.C. Haxton and G.J. Stephenson, Prog. Part. Nucl. Phys. 12

(1984) 409.

Ha88 W.C. Haxton, to be published.

Ia79 F. Iachello and 0. Scholten, Phys. Rev. Lett. 43 (1979) 679.

Ia83 F. Iachello, Prog. Part. Nucl. Phys. 9 (1983) 5.

Ia87a F. Iachello and A. Arima, "The interacting boson model",

Cambridge University Press, Cambridge, 1987.

Ia87b F. Iachello, I. Talmi, Rev. Mod. Phys. 59 (1987) 339.

Ki63 L.S. Kisslinger and R.A. Sorensen,Rev. Mod. Phys. 35 (1963)

853.

K183 H.V. Klapdor, Progr. Part. Nucl. Phys. 10 (1983) 131.

Ko66 E.J. Konopinski, "The Theory of Beta Radioactivity",

Oxford University Press, London, 1966.

Le76 G. Leander, Nucl. Phys. A273 (1976) 286.

Le78 C.M. Lederer and V.S. Shirley, Table of Isotopes, Seventh

Edition, J. Wiley & Sons, Inc., New York, 1978.

116

L177

Ma49

Me75

Mo67

Mu83

Na88

Ni55

Ot78

Pe83

Pu80

R^84

R^85

Sa81

Sa82

Li75

Sa84

J.R. Lien, J.S. Vaagen and A. Graue, Nucl. Phys. A253 (1975)

165.

J.R. Lien, C. Lunde Nilsen, R. Nilsen, P.B. Void, A. Graue

and G. L^vhîden, Can. J. Phys. 55 (1977) 463.

M.G. Mayer, Phys. Rev. 75 (1949) 1969.

J. Meyer-ter-Vehn, Nucl. Phys. A249 (1975) 111.

W.H. Moore, G.K. Schlegel, S. O'Dell, A. Graue and J.R. Lien,

Nucl. Phys. A104 (1967) 327.

A.C. Mueller, F. Buchinger, W. Klempt, E.W. Otten, R. Neugart

, C. Ekstrom, J. Heinemeier, Nucl. Phys. A403 (1983) 234.

P. Navratil and J. Dobes, to be published.

S.G. Nilsson, Mat. Fys. Medd. Dan. Vid. Selsk. 29 (1955)

No. 16.

T. Otsuka, A. Arima and F. Iachello, Nucl. Phys. A309 (1978)

1.

L.K. Peker, Nucl. Data Sheets 38 (1983) 87.

G. Puddu, 0. Scholten and T. Otsuka, Nucl. Phys. A348 (1980)

109.

T. R^dland, J.R. Lien, J.S. Vaagen, G. L^vhîden

and C. Ellegaard, Physica Scripta 29 (1984) 529.

T. R^dland, J.R. Lien, G. L^vhîden, J.S. Vaagen, V. Oygaard

and C. Ellegaard, Physica Scripta 32 (1985) 201.

M. Sambataro and A.E.L. Dieperink , Phys. Lett. 107B (1981)

249.

M. Sambataro, Nucl. Phys. A380 (1982) 365.

M. Sambataro, 0. Scholten, A.E.L. Dieperink and G. Piccitto,

Nucl. Phys. A423 (1984) 333.

117

Sc80 0. Scholten, Ph.D. Thesis, University of Groningen, 1980.

Sc81 0. Scholten and A.E.L. Dieperink in "Interacting Bose-Fermi

systems in nuclei", F. Iachello ed., Plenum Press,

New York, (1981) 343.

Sc82 0. Scholten and N. Blasi, Nucl. Phys. A380 (1982) 509.

Sc85 0. Scholten and Z.R. Yu, Phys. Lett. 161B (1985) 13.

Se86 Yu.V. Sergeenkov, V.M. Sigalov, Nucl. Data Sheets 49 (1986)

639.

Se87 Yu.V. Sergeenkov, Nucl. Data Sheets 52 (1987) 205.

Sh63 A. de Shalit and I. Talmi, "Nuclear shell theory",

Academic Press, New York and London, 1963.

Sh81 M.A.M. Shahabuddin, J.A. Kuehner and A.A. Pilt, Phys. Rev.

C23 (1981) 64.

Sz79 A. Szanto de Toledo, H. Hafner and H.V. Klapdor, Nucl. Phys.

A320 (1979) 309.

Ta71 I. Talmi, Nucl. Phys. A172 (1971) 1.

Ta79 T. Tamura, Z. Matumoto, A. Hashizume, Y. Tendow, K. Miyano,

S. Ohya, K. Kitao and M. Kanbe, Nucl. Data Sheets 26 (1979)

385.

Ta80 T. Tamura, Z. Matumoto, K. Miyano and S. Ohya, Nucl. Data

Sheets 29 (1980) 453.

Ta81 T. Tamura, Z. Matumoto and M Ohshima, Nucl. Data Sheets 22(1981) 497.

To85 I.S. Towner, Nucl. Phys. A444 (1985) 402.

To87 I.S. Towner, Phys. Rep. 155 (1987) 263.

Wy74 B.G. Wybourne, "Classical Groups for Physicists", J.Wiley

& Sons, New York, 1974.

ABSTRACT BETA DECAY OF ODD MASS NUCLEI ... - Yale University

Documents

Transcript of ABSTRACT BETA DECAY OF ODD MASS NUCLEI ... - Yale University