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INSTITUTODE FÍSICA

preprint

IFUSP/P-175

CHARGE-EXCHANGE COLLECTIVE MODES, AND BETA-DECAY

PROCESSES IN THE LEAD REGION

by

K. EBERT and W. V7ILD

Physik-Department, Technische Universitat Munchen,D-8046 Garching, W. Germany

and

F. KRMPOTlC

Universidade de São Paulo - Instituto de Física,

São Paulo, Brasil

UNIVERSIDADE DE SÃO PAULOINSTITUTO OE FÍSICACaixa Postal - 20.511Cidade UniversitáriaSão Paulo-BRASIL

CHAPGE-EXCHANGE COLLECTIVE MODES, AND BETA-DECAY

PROCESSES IN THE LEAD REGION

K. EBERT and W. WILD

Physik-Department, Technische Universitat Munchen,

D-8046 Garching, W. Germany

and

F. KRMPOTIC*

Universidade de São Paulo - Instituto de Física/

São Paulo, Brasil

Permanent address: Departamento de Física, Facultad de Ciências

Exactas, Universidad Nacional de La Plata, C.C. 67, Argentina.

Member of the Carrera del Investigador, CONICET, Argentina.

.1.

ABSTRACT

The first-forbidden charge exchange collective modes,

as well as the corresponding 6-decay and y-radiation processes,

have been calculated, using a large configuration space and a

density dependent interaction. The p *-l excitations, which lie

in energy around 19-26 MeV, clearly exibit the Brown - Bosterly

effect. Among the inhibited u =1 excitations only the dipole

spin wave, located at 7.3 HeV, shows a certain degree of coherence.

Fairly go / agreement with the experimental data for the 6-decay

processe: «as achieved.

.2.

1. INTRODUCTION

Multiple particle-hole correlations in nuclei may be

classified according to the properties of the corresponding

vibratonal fields, namely, by the rank of the spatial, spin and

isospin parts of the operator

M U,c,Xy;TUT) = J ±* rj [ Y ^ U ) ^ SCT(i)J Xy ij (i) (1.1)

where the spin and isospin dependence are contained, respectively,

in

í l

j 2 s (i) a=l (spin-flip)

I 1 o=0 (non-spin-flip)Su '

and

r

IT

T

1 T=0 (isoscalar)

2 t T«1 (isovector)

The quantum numbers x,A,a and T stand for the orbital angular

momentum, the total angular momentum, the spin, and the isospin

carried by the excitation, respectively. For example, T*0

corresponds to isoscalar excitations; T=1 corresponds to isovector

excitations; when o=l there are spin-flip processes, etc..

The vibrational isoscalar non-spin-flip fields, i.e. the

quadrupole and octupole vibrations have been widely studied and

their properties are well known. On the other hand, among

different isovector modes of excitation only the giant dipole

resonance (GDR), with <»1, a»0, X*l, T*1 and uT"0, is well

established.

Additional information on collective excitations in the

.3.

nucleus may be obtained from the study of the charge exchange modes

(uT=±l) by reaction processes such as (p,n) , (JHert), (ir ,ir°) , etc.

and the corresponding charge conjugate processes.

In a nucleus with N=Z, ground state isospin is T *0, and

the charge exchange modes of excitation are related to the T«1,

u =0 mode by isobaric invariance. As a consequence the transition

strengths are independent of p and the vibratioral frequencies

E(T=1, U T = ± 1 ) are given by the simple relation '

E(T-l,yT-±l) = E { T - 1 , U T - 0 ) - U T A E C O U 1 (1.2)

where E-, . is the Coulomb energy displacement.

In nuclei with T >>1 the situation is entirely different

due to an effect which arises from the Pauli principle. The

neutron excess implies a reduction in the number of proton hole-

neutron particle excitations and a simultaneous increase of

excitations of the type neutron hole-proton particle. In

particular, for 20iPb the dipole T=1, V =-1 strength is roughly

twice the dipole T=1,M =0 strength while the corresponding T=1,V *11 2)strength dies out almost completely .

Furthermore/ for nuclei which have non-zero isospin in

the ground state the GDR splits into two isospin components,

T>=T and T^T +1. The weak upper branch T> is the isobaric

analog (IA) state of the charge exchange non-spin-flip dipole mode.

The pattern of the charge-exchange excitations is schematically

illustrated in fig.l.

The study of charge-exchange particle-hole correlations

in medium and heavy nuclei is closely related with the weak

interaction processes (such as nuclear 8-decay and u"-meson

capture) and with the radiative widths of isobaric analog states.

In particular, the first-forbidden weak processes and the El

Y-decay from the IA state depend strongly on the energies and the

.4.

transition strengths of the charge-exchange vibrational fields

with K=1, u=0,l and X=0,l,2.

In the present paper we will focus our attention on the

study of these "first forbidden" collective states, in nuclei near

the c: . ly magic nucleus 208Pb, using the theory of a Finite Fermi

System (FFS) developed by Migdal . We have chosen this mass

region for our analysis because it presents the following features:

i) several first forbidden transitions occur around the

Z08Pb nucleus;

ii) there are also available experimental data on the

charge-exchange El y-moments for 207Pb and 2 0»Bi 4' 5 >;

iii) the nuclear wave functions are relatively simple and

rather well know;

lv) a set of force parameters of a density dependent

residual interaction has been recently determined

from the analysis of the electromagnetic moments and6—fl)

transition probabilities

2 . THEORY

2.1) Multipole moments and half life for 8-decay processes

When terms induced by the strong interaction and other

higher-order corrections are neglected, all the observable? for

the first-forbidden 3 transitions can be written in terms of the

six 6-moments listed in table 1. In the examination of the B-decay

processes we will employ mainly the Bohr-Mottelson representantion

shown in the third column, while in the discussion of the charge-

exchange collective modes we will use the notation displayed in

the last column.

The relativietic 3-moments are usually related to the

.5.

corresponding non-relativistic ones by defining the ratios

/3 <If

and

|M(p , A-0)|!l±

l iiM(j. ric=O, X = O ) M l . >A 1

where X is the Compton wave length and

ze212.3)

Z and R being the charge and the radius of the daughter nucleus.

In the evaluation of the relativistic vector B-moment

we take advantage of the conservation of the vector current (CVC)«

which connects the &~ vector current Jv and the electromagnetic

current J through the relationship

(2-4)

where T_ is the isospin lowering operator. From eq (2.4) one

finds for 6* decay11>l2)

where V C o u l is the Coulomb potential and E is the energy

difference between the initial and final states. The true

.6.

transition energy W is obtained from E by adding the neutron-

proton mass difference (Mft - M )c = 0.782 MeV. To estimate the

size of *Coui it is convenient to regard the charge distribution

of the nucleus as that of a uniformly charged sphere of radius R,

in which case

The axial vector current is only partially conserved

(PCAC), and in contrast to the case of vector current the PCAC

relation, when applied to the nuclear case, does not lead to any

useful relationship between the 8-moraents. Consequently, we will

use here for the ratio A , the estimate based on the Ahrens-

Feenberg-Pursey calculation ' , namely

EAQ = 2.4 + ?—— (2.7)

From relation (2.4) it follows that the 0-moment

<f:iM(pv, X=l)||i> is related to the El y-moment <f! !iM(El) :IA;i>

by

<£ ! | ±W (P V, X=l)||i> = — — (2 T j ) 1 ^ <f ! |iM(El) | ;iA;i> (2.8)

providing the (TO, - T. and that T+|f> x 0. The state

IA;i> = T_ !Í>/V5T7 is the isobaric analog (IA) state of the

state |i>. The level diagram for the relation between the

El y-decay from the IA state and the e-decay in 209Bi is show:, in

Fig. 2.

The partial half life is given by the expression

.7.

t = 1 (2.9)/Cg(E)F(Z,E)E(Eo-E)pdE

where D = 6175 sec, Cg (E) is the spectrum shape factor, F(Z,E)

is the Fermi function and the variables E and p are the

electron energy and momentum, respectively. In the present work,

we use the exact result for the spectrum shape factor , namely

I ;CAue,xv)i2

C0(E) = 6 (2.10)

2F(Z,W)(EQ-E)2 p 2

with

V ( f-1 F-1 " 9-lplJ| /RO '

^ + fjP^) + RQy(- f ^ + ^

C1(l,-2)=(2x + u)

(-l,2) = (2x + u) (g_1F_1 - f_1F_2)/(RQ/2) , (2.11)

- u) (f2F1 - g2F_1)/(Ro/2) ,

- u) (g. 2Fi + f-2F-l)/(Ro/2") '

C2(l,-2)=/3z(- f1F_1 - \ giF_2)

C2(-l,2)=/3z(- g.1F_1 + j f_1F_2)/(2RQ)

C2(-2,l)=/Iz(g_2F1 + i f_2F_1)/(2Ro) .

The electron wave functions are represented by fxe

and g , and the neutrino wave functions by F . They arexe v

.8.

evaluated at the nuclear radius R so that the symbols employed

mean f,=f,(R ), etc..

In the ^-approximation C-(E) is energy independent and

reads

= 4ir !>U=0) + B(X=1)] (2.12a)

with

and

|<i | | | | |B(X) = i i (2.12b)

o(X=0) = M(p., X=0) - i f - M(ja, x=l, X=0) , (2.13a)

l) = M(jv, x-0, X=l) + - i — 4 — M(pv,X=l)v /2 Ae v

(2.13b)

M(j ,e A

The B(X)-values represent a measure of the destructive

interference between the 6-moments.

In order to húve at our disposal some uniform scale for

measuring the collectivity of different modes of oscillation it

is convenient to resort to the Weisskopf single-particle estimates

for the B-values of the first forbidden 6-moments shown in the

last column of table 1. They are

.9.

3RX

(o=0, X=K) = -^— ( °- ) 2

2ir X+3

3RBw (0=1, X=K-1) = - A — { m i ) ( 2w 2ir X+l X+4

(2.14a)

3RX(0=1, \=K) = -±— ( — L _ ) ( 2_ ) 2

2tr X+3 X+3

3RBu (0=1, X-K+1) = - ^ — ( 4 X ) ( 2 ) 2

w 2ir 2X+1 X+2

where the vector addition coefficients were evaluated for the

transition ^ = X+l/2, If = 1/2 and the radial integrals

<I-|rK|l.> were approximated by the values 3RK/(ic+3). Employing

a radius of RQ=1.2A ' fm and A=208, we obtain for the first

forbidden moments

Bw (o=0, X=l) = - j | — R^ = 4.53 fm2

(o»l, X=0) = — - — R^ = 4.53 fm2

Bu (0=1, X=l) = — - — nl = 2.26 fm2w 64TT °

' x=2) = "ãfr Ro = 7*24

(2.14b)

.10.

2.2) Nuclear model

The basis of the following calculations is the theory

of finite Fermi systems (FFS) . It is assumed that the single

particle model in the neighbourhood of the magic nucleus

(A-particle system) is a good fiist approximation. The lowiying

excitations in the neighbouring (A+l) nuclei are assumed to be

one guasi-particle and one quasi-hole states, which are nucleon

states with a surrounding polarization cloud. The polarization

effects are taken into account through both the "effective one-

particle operator" and the "effective interaction" which is

described by a set of constants to be taken from various

expei intents. A detailed description of FFS can be found in refs.

' ~ ' ~ ;here we simply sketch the procedure and refer to

these papers for a detailed discussion of the method.

For the residual particle-hole interaction we use the

density dependent force proposed ty Migdal '.

(2.15)

with C=380 MeV fm . The force parameters f,f , g and g' are

density dependent, i.e.

f(r> - f e x +

where f and f. denote the parameters inside and outsideex in

the nucleus. For the density p we use a Fermi distribution

p(r) - i (2.17)1 + exp((r-R)/a)

with the nuclear radius R"6.7fm and the diffuseness of the

.11.

nucleus a=0.5 fin. ?'hese parameters are taken from the nuclear

density obtained from the single particle wave functions

The residual interaction Fp depends critically on the

single-particle configuration space. Since we used here the force

F1" of ref. , we have to use the same configuration space too,

which includes two major shells above and two major shells below

the Fermi space. The number of force parameters in F01 is

reduced substantially by considering the requirements of

generalized Ward identities and by an antisymmetrization procedure

in the external region of low density. We used the parameters of

ref. for all calculations. Therefore, we had no free parameter

in order to fit experimental data.

The matrix elements of the single particle transition

operators M(A), between guasi-particle states j. and j_ of

the (A±l) particle system can be represented as '

<A±l,jJMiA±l,j,> - 6. . <A,0lM!A,0> -• T. (E ,M) (2.18)

with E - t. - c. . The symbol T is used for the so-called,

"localized vertex operator" which obeys the Pethe-Salpeter

equation

,_ eff I _ph j 3 j 4 ,_ M

(2.19)

Here, the index j denotes the single-particle quantum numbers,

t . the single-particle energies and n, the occupation proba-

bilities. The quantity M represents the effective one-

particle operator.

On the basis of CVC theory, the effective operators

.12.

Meff(pv, X=l> and Meff(Jv, x=0, X=l) are equal to the corre-

sponding bare values, i.e.

e f £ hare ,« ™,gv = gv - gv • (2.20

Since the axial voctor current is not conserved the

operators M(j , x=l, X=0,l,2) and M(p-, *=0) undergo a

renormalization due to mesonic effects. The isospin invariance

of the strong interaction gives

where E, is the parameter of spin renormalization. With the

value quoted in ref. (£ =0.13) we have

bare bareg, = 0.7^g. - - 0.918 g., (for g. = - 1.24 gtr) . (2.22)A A V A V

The different nuclear states n in 20$Tl, 20BPb and

208Bi nuclei (A-particle systems) «re described as a linear

superposition of quasi-particle-quasi-hole states. The corre-

sponding particle-hole amplitudes X, . and excitation energies

E are obtained from the renorntalized RPA-equation

(e. -e. -Ejx" . « (n. -n. ) J-. pf1, , . x" . - (2.23)31 ^2 3132 31 32 -!334 31343233 3334

From the last equation it is possible to express the

vertex operator T in the form:

.13.

•r., < <E ,M) - U"~ - I« J-iJ? n

rn\ <n|Heff|O>3!D2

E n - E o

,» , <O!Meff|n>(2.24)

where the quantity

M - i ' i F^ . . . X , (2.25)-1D2 33:J4 31:J4:I2:]3 D3:)4

measure? the quasi-particle-vibration coupling for particle-hole

phonons and

<n|Meff|0> - A MJ3J4XJ3J4

(2.26)

<0|Meff|n> = ,̂ . n f f x"

represent the amplitudes with which the phonon n is created or

annihilated by the operator Me . The graphical representation

of T. . (E .M) is shown in fig. 3. It should be noted that inD2D1 °

the case of 8 radiation processes a u_=il phonon is created and

a u.=+l phonon is annihilated.

The transition strengths B{c,X) of the excited state n

with angular momentum X are given by the relation

Bn(cr,\) - |<X,n|Meff (a,X)|0>|2 (2.27)

The calculation of transition probabilities between states

.14.

in the A and (A±2) particle systems requires an extension of

FFS-theory ~ . The matrix element for the transition from the

state m to the state n , reads

r. Mím> = ,,M) (2.28)

where E =E_~E is the transition energy and

C(j j ;mn)

(1-n. ) (1-n. )n. - n. n. (1-n. )31 32 33 31 32 33

n. (1-n )(1-n. ) - (1-n. )n. n.31 32 3 3 3 1 32 33

(1-n, )n. (1-n. )-n. (1-n. )n.31 32 33 ]1 32 33 m* n

n. n. (1-n. ) - (1-n. )(1-n. )n.31 32 33 31 32 33

(e. -E. +E )(E. -e. +E )31 33 n 32 33 m

n. (1-n. )n. - (1-n. )n. (1-n. )31 32 3 3 3 1 32 33

(1-n. )n. n. -n (1-n. )(1-n )31 32 33 31 32 33 m* n

y(2.29)

.15.

for the A particle system, and

f| (l-2nn Í (1-n -n.. ) (1-n. -n, )

C(JlJ2,n») - 2

n, (1-n. ) + (1-n. )n.-•2 O J ? J2 3 2

-n. -n,

n.j (l-nj ) + (1-nj ) n j I

+ (1-n. -n. ) — > A™** A" .. (2.30)

for the A±2 particle system. Here,

JXJ2 2 J3J4 31J2'^3

:)4 D3:)4

are the particle-vibration coupling strengths for the pairing

bosons.

All the terms in expressions (2.29) and (2.30) have the

same topological structure and differ only in the energy denomina-

tors. This statement is illustrated in fig. 4 for eg. (2.29)

which involves twelve different leading order contributions to the

matrix element <n|M|m>. Six of them, which correspond to the

situation when j3 is a hole state, are represented by diagrams

a). The graph i) corresponds to the usual shell-model

contribution, whilr the ground state RPA correlations are given

by graph * v). The remaining four diagrams contain a particle-

particle or a hole-hole scattering vertex. The polarization

mechanism enters through the second term of eq. (2.24), with four

.16.

core excitation processes corresponding to each graph of type a).

As an example, the processes which renormalize the first of these

graphs are shown in fig. 4b.

After coupling of angular momenta, the reduced matrix

elements read

|lm> = j'j c(^i32'

ImIn'X) <3 2

IIT(E O,M(À)) i|jx>

(2.28')

with

r j,+l/2 í Im+X f ^ 1 * . X j r -i(-D

3 i (-D m < . M ]33 ^ i 1

x k (-1)33

I Í Xn Jm X 1 r 1 I . I+ ("D

)

for A nucxeon systems, and

I +1 +X+1C(J1J2;IiIf,A) - (-1)

m "

:2 D 3

for A±2 nucleon systems. The symbols [ J in eqs. (2.29')

and (2.30') are the same as those of eqs. (2.29) and (2.30),

respectively.

.17.

3. RESULTS

3.1) Charge-exchange collective excitations

Let us first consider the unperturbed particle-hole

ppectrur, that is, the spectrum of particle-hole pairs, created

by an operator M (K=1,O,X,T=1, U ) acting on the ground state of

208Pb. Assuming harmonic oscillator radial wave functions whitout

spin-orbit coupling, no single-particle v -1 excitation is

allowed due to the Pauli principle, and all unperturbed spectra

of the u =-1 modes are concentrated in one line, the energy of

which is given by the expression

f _. \ M _ 9»

where h» = 40 A " 1 ^ 3 MeV =6.75 MeV and V. s 130 MeV. Theo l

change in the energy associated with the spin-orbit interaction in

the single-particle field implies both i) small but non-zero

transition strengths for P=l modes and ii) a greater spread for

the spin-flip modes of rank one and two than for the remaining

two excitations with u =-1. In order to illustrate the latter

fact we resort to a schematic model. We assume that there are

two single-particle orbitais with the orbital quantum nuirbers I

and l'"l+l, each of them being Bplit into two states by the spin-

orbit interaction. The former two states are assumed to be filled

and the latter two empty. The reduced transition probabilities

between these states are evaluated in the asymptotic limit U » l ) .

The results are displayed in fig. 3, showing the spreading of the

strengths for different first-forbidden excitations. After

including the particle-hole interactions, situation 11) basically

persists.

In figs. 4 and 5 are presented the perturbed excitation

.18.

spectra for the first forbidden modes in I0*Te(u =1) and 2#lBi

(u =-1) nuclei, respectively, measured with respect to the ground

state of *°*Pb. The strengths of the u =1 modes are mainly

associated with the transitions from the proton orbit lh.. ., t o

the neutron orbits l i1 1/2 ' li13/2 ' *97/2 and 2g9/2* T n e

spin-flip states 0~ , 1^ and ?" , which appear at the excitation

energies of 8.J6 MeV , 7.31 MeV and 8.42 MeV respectively, are

almost pure, particle-hole excitations. The first two states

arise when a nucleon in the proton orbit lh.... »_ is promoted

into the neutron orbit li.... ,_ , and the last state when the same

nucleon is lifted into the neutron 2g_ ,_ state. Their transition

strengths are, however, significantly influenced by the backward

going contributions which are of y =-1 type. These contributions

increase the spin-flip strength of the ll state by 44% and

decrease the strength of the o" and 2~ states by 32%. The

effect of the backward going graphs is also pronounced in the case

of non-spin-flip transitions. For example, they increase the

a«0 strength of the state (2gg/2 , ^ T w , ' 1 ! bv 31%*

The total transition rates in Weisskopf units are:

EB(O«1, A=0) = 3.1 W.Ü.

£B<o*0, X*l) * 7.1 W.U.

IB(a=l, \"l) * 20.1 W.U.

rB(a*l, A=2) = 3.1 W.U.

for the pT=l mode (2OITl), and

£B(c«l

SB(a-0

ZB<o-l

rBi0*i

p — 1 mode

, X-0) -

, X-l) *

, X-l) -

, X-2) «

(2B»Bi).

9.6

135

71.

33.

W.U

.2 W

3 W.

9 W.

• i

.U.,

u.

for the

It should be noted that the dipole non-spin-flip mode

obeys the sum-rule

.19.

ZB(u =-1) - EB(u =1) = - i — (N<r2> - Z<r2> ) (3.2)T T 2ir neut prot

From the present RPA calculation, the left hand side of

this expression gives 379 fm , while, with the approximation

< r 2n e u t = <r

2> r o t = | Ro2 = 30.33 fm2 , the value of the right-

hand side is 6 37 fm .

All the first-forbidden u =-1 excitations clearly

exibit the Brown-Bosterly effect , with their transition strengths

mostly concentrated in the energy region of 19-26 MeV. The

collective O~ level at .75.4 MeV absorbs 82% of the total monopole

transition strength. Sixty-nine percent of the total dipole o=0

strength is accumulated at =21 MeV. The gathering of the dipole

spin-flip strength occurs in the 1~ states at 22.8 and 24.8 MeV.

These states contain, respectively, 25% and 42% of the total

dipole o=l strength. The 2~ levels, which carry a relatively

large transition strength, are located at the energies of 6.1,

9.9, 12.0, 15.6, 19.1, 21.8 and 24.2 MeV. The first three levels

are essentially the single-particle states with Aj=2. Their

major configurations are respectively: (1 h_ ,_ , 1 1-13/2) 2~ »

(1 Í13/2 ' 1 h9/2^2" a n d *2 g9/2 ' 2 f5/2*2~* T h e w a v e functions

of the remaining four 2~ states are built up from many particle-

hole configurations with Aj=O and 1. They contain, also, large

amplitudes with A£=3 [for example (3 p&,2 , 1 h~ )2~] which

diminish significantly their transition strength. The backward

going qraphs in the case of u =-1 transitions arise from the

u =1 particle-hole excitations and, consequently, their contri-

butions are very small for heavy nuclei. Due to this partial

disappearence of the ground state correlations we are falling from

a RPA treatment into the Tamm-Dancoff approximation.

.20.

3.2) One-particle B-decays

In table 2 are presented the results for the non-

relativistic B-moments which participate in one-particle transi-

tions 3 s~*2 -• 3 p~J2 and 3 s~*2 + 3 p~*2 in 287Pb and

1 g. ,, • 1 kq/2 *n 2 ° * B i - por the sake of comparison both the

single-particle values and the renormalized ones were evaluated

with the bare axial-vector coupling constant g = -1.24 g . WeA. V

can immediately see that i) not all the matrix elements are re-

duced by the corresponding vibrational fields and ii) the

contributions of the v =1 modes (A.) are in same cases comparable

to those which arise from the corresponding charge conjugate modes

The rank-zero moment <l/2+ ||±W(jft, x=l, X=0)||l/2">

is mostly reduced by the coupling of the 0~ collective state at

25.5 MeV in 208Bi to the initial state IPT/ 2> a n d b v t h e

coupling of the final state | s^ ,-> to the particle-hole state

(1 ill/2 ' X h U / 2 ) 0 " ' w n i c n l i e s at 8.4 MeV in 2 0 8Tl. The

corresponding contributions are, respectively, 0.37 and 0.22 finct.

The collective contribution to the moment <9/2+| |UM(j ,x=l,A=0) 119/2~>

is comparatively small (0.08 fm gv) and enhances the single particle

matrix element. Due to this fact the non-collective effects are

dominant and arise mainly from the (1 i,,^ ' * hll/2*0~ a n d

(2 h ._ , 1 gl^-io" states at 8.3 and 22.1 MeV in 2O*T1 and fromp// y/i

the state (I h g / 2 , 1 g ^ *0 " a t 19'° M e V i n 2 ° ' B i '

contribute respectively with -0.14 , -0.11 and -0.13 fm g .

The renormalizations of the rank-one moments by the

collective y =-1 states are displayed in table 3. Two facts

should be stressed with respect to the 2 gg ,^ •*• 1 h- ,, transition:

first, the collective contributions to the matrix element of the

operator M(j , x*l, \=l) are of the same order of magnitude as

the single-particle contribution and second, strong destructive

.21.

interference occurs in collective contributions for both rank-one

moments. The results listed in the above mentioned table also

show that the interaction between the a=0 and <J=1 degrees of

freedom is very weak, i.e., tht single-particle o=0 field is vexy

weakly coupled to the collective CT=1 field and vice versa.

The moment <9/2~| |iM(jft, x=l, *=2M9/2+> is dominantly

renormalized (-0.12 g v fm) by the (2 <}^t2 > ! \\/2 * 2~ s t a t e at

8.4 MeV in 2 O 8 T 1 , while the most important contribution (0.58 g^ fin)

to the moment <3/2~| |i*4(jft, x=l, X=2) | |l/2+> originates in the

collective 2~ state at 19.0 MeV in 2 O i B i .

From the forgoing results and discussion it is evident

that the renormalization mechanism for the B-moments, which

participate in the transition 2 g«/ 2 "* * h9/2 ' i s s u b s t a n t i a l l Y

different from those which affect the 8-moments of the remaing two

transitions. This is due to the fact that the first transition

is of the spin-flip type and, in addition takes place between

states with a different number of radial modes.

The experimental and the theoretical partial half-lifes

t. ,_ are compared in table 4. It should be noted that the

calculated spectrum shapes for the l/2+ -• l/2~ and 9/2+ •• 9/2~

transitions do not exibit any measurable energy dependence, which

is in agreement with experiment . On the contrary the

spectrum of the l/2+ •* 3/2" deviates strongly from the statisti-ef fcal shape. It decreases by 44I and 33% when evaluated with M

or T , respectively, unfortunately, due to the small branching

ratio for this process, it should be very hard to test experi-

mentally the above mentioned theoretical results.

3.3) Charge-exchange El

The electric dipole radiation from the IA states have45been measured, in the lead region, by Shoda et al. through the

.22.

(e,e'p) reaction on Z07Pb and 209Bi and by Snover et al.51 by the

(p,Y) reaction on 20SPb. In table 5 we compare the experimental

results for the effective charge with the calculated values. In

our notation the effective charge is defined as

eff

< 3 £ I ! Í « ( P V , X - I > | ! 3 I >

In ref. two solutions equivalent in fitting accuracy

were found for each resonance: 1) 4$ s 0° and e. . < 1 , andeff D2 1

2) A4> - 90 and e. • > 1 , with A<» = • „ - <*>___ i-he intrinsicJ - J i 1* IJUK

phase difference between the interfering IA and GDR amplitudes.

The values for the effective charge obtained from the second

solution are given in parentheses in table 5.

The calculated values agree well with the (e,e'p)

measurement as well as with the inhibited-strength solution

(A<J> = 0°) of the (p,Y) study, except for the 2 gg .? • 2 f_ ..

transition. It is worthwhile to note that in this case there is

also a very serious inconsistency in the experimental data.

3.4) B-moments for the decay of 208Tl

We limit our attention here to the 3 transitions to

the 37 , 47 , 57 and 5~ states in 208pb for which the spini l l «.

and the parity are well established. The calculated wave function

of 2O*T1 is determined by the excitation of a proton from the

3 s / 2 and 2 d . states to the 2 g^/2 neutron level, namely:

; g.s.> - 0.92112 g9/2 , 3 s ^ > - 0.379 12 g9/2 , 2

Our wave function for the collective 3~ state is very

.23.

similar to one already published in ref. . Due to this fact, it

will not be given here. The predicted forward going amplitudes

for the 4.. , 5. and 5- states are listed in table 6 and

compared with those extracted from the analysis of the angular

distributions of the reaction 208Pb(p,p')20*Pb through isobaric

analog resonances

Although the 31 state is strongly correlated, there are

relatively very few particle-hole configurations which take part

in the (J-decay (see table 7), with the hole-hole 3 s.,, • 3 P3/2

transition the dominant one. The remaining hole-hole transitions,

as well as the particle-particle transitions, add incoherentely

with the dominant contribution. The destructive interference

among the one-particle contributions is even more pronounced when

the core-polarization is included.

For the 0-decays to the 4~ , 57 and 57 states the

^-approximation is valid as can be inferred from the measured

28)ft-values (ft s 5.5). Consequentely/ the moment

<If!!iW(JA, x-1, \=2) Mli> is not relevant and will not be

discussed here.

From table 7 it is seen that the single particle

transition s" ,2 •*• PT/ 2 i s t n e most important one in building up

the total transition matrix elements for the 6-decay to the 4~ ,

5* and 5_ states. The corresponding single particle B-values are:

B s P U-l'lf-4") - 88'10"4 % 2 '

B (X=0;If=5") * 110.10"4 gv

2 ,

B (X-l;If=5") = 58.10"4 gy

2 .

The configuration admixture with largest weights arise

from the transitions g9/2 *• h g / 2 , d /̂;, • p~2/2 and dj / 2 * p^

reducing the B-values for the 57 and 5~ states by a factor of

.24.

=2, while the moment B(X=1;I =4 ) remains essentially equal to

its single particle value. After ireluding the core-polarization

all the transitions are furthermore inhibited by a factor of =3

(see table 8).

The calculated shape factors for the B-decay to the

4 , 5 and 5. states do not present any observable deviation

with respect to the allowed shape.

It should be noted also that the measurement of the S~y

29)angular correlation are consistent with the dominance of the

s7 ,« * p7/2 transition in the decays to the 4~ and 5~ states.

The same experiment also shows that the 6-transition leading to

the 5j state seems to be significantly different from predictions

for a pure si/ 2 "* P1/2 transition.

3.5) B-moments for the decay oi 206Hg and 2««T1

He discuss here the B transitions from the 0* state in

20SHg to the 1~ , 1~ and O" states in 2OtTl and the decay

of the last state to the 0+ state in 2"Pb. In the following

we used the wave functions of Kuo and Herling (approximation

2). In table 9 are shown the main single-particle contributions

for each transition. He can immediately see that the

206Hg(O+) * 2O6T1 (O* , lTj and 206Tl(o") + 20*Pb(O+) decays

are of single particle character. With, again, the s",- •• P

single-particle transition the most important one. The correspond-

ing one-particle estimate for the B(A)-values are:

Bg U=0; 0* + 0+) = 2.2.10"2

Bsp

Prom comparison of these quantities with the B(A)-values

.25.

given in tables 10 and 11 it is seen that, for the above mentioned

transitions, the core-polarization correlations are much more

important than the shell-model ones. On the contrary, in the

206Hg(O+) + 2O*T1(1~) transition, it is mostly the interplay of

different shell-model configurations which defines the magnitude

of the matrix element B(X=1).

The calculated spectrum shapes, for the transitions which

are discussed here, are almost insensitive to the core-polarization

effects and all of them deviate appreciably from a statistical

shape. By approximating the shape factor in the form

C, (E) a 1 + a EP

as is usually done in the analysis of B-spectra, the calculated

slopes correspond to the following values of the coefficient a

(in units of me ) are: -0.022, 0.036, -0.070 and -0.023 for the

transitions 206Hg(O+) * 20iTl(o") , 206Hg(O+) * 2O6T1(1~) ,

206Hg(O+) * ^ T l d " ) and 206Tl(o") •> 206Pb(O+) , respectively.

Only the shape for the decay of 206Tl has been measured ' 3 4' 3 5 )

so far, and our calculation agrees nicely with the more recent

measurement performed by Wiesner et al. :

- (0.020 ± 0.002)/mc2

3.6) B-moments for the decay of 210Pb and 210Bi

Dominant single-particle contributions, obtained with

shell model amplitudes of Kuo and Herling are shown in table

12. It is seen, from these results, that, while the zl6Pb(O+)

210Bi(o") transition is dominated by the single-particle moment

<h9/2l |0(*=O) ! |<?9/2> » t n e total matrix elements for the

210Pb(O+) + 21oBi(l") and 21oBi(l") *• 210Po(O+) transitions,

.26.

are built up from several single-particle components. More pre-

cisely, in the last two transitions! there are the vector matrix

elements which receive comparable contributions from several single-

particle processes; the axial matrix element, on the other hand,

arises mostly from the one-particle moment <ng/_| |iM(j ,x=l,X=l) |(g. •_>.

We have also evaluated the 6-decay of 210Bi with the

wave functions of Kim and Rasmussen and these results are given

in parentheses in tables 11 and 12. The main difference with

respect to the previous calculation is that, now, the single-

particle 1 1,1/2 * * h9/2 c o n t r i b u t i o n dominates all others.

The calculated B-moments for the 21oBi(l") -»• 21°Po(0+)

shown in table 11 transition compare fairly well with the phe-

nomenological ones , which are:

<O+||M(JV, X=0, X-l)||1"> = (38.7 *J||) . 103

1_ L. <O+||iM(pv, W)||l"> = (-20.1 +J;J) . 103

e

/\ f- <O+|!iM(JA, x-1, X-l)iU"> = (20.5 t j ; ^ • 10"

However, non of the sets of theoretical matrix elements

reproduces satisfactorily the spectrum shape, the longitudinal

polarization and the half-live. This apparent contradiction

arises from the strong cancellation among the 6-moments in the

leading energy-independent term <O+||0(A=l)||l"> . The above

mentioned observables depend very critically on the quantity

+||iM(pv#A-l)

.27.

The experimental value is Y/£ = -0.090 ± 0.004. The

corresponding theoretical results, given in the same order as in

table 11, are:

Y r1.03 (-0.11)

í 10.41 (0.33) .

We see that only the calculation with the wave functions

from ref. and without core-polarization give a result for the

ratio Y/Ç close to the experimental values. However, even in

this case, the above mentioned observables are not reproduced

satisfactorily. A few more details of the theoretical analysis

of the 6-decay of 210Bi can be found in ref. 19)*.

4. SUMMARY AND FINAL REMARKS

The 6-decay processes, together with charge-exchange

reactions, provide an excellent tool to probe the nucleus with

respect to the Migdal parameters g and g1.

In the present work, a detailed study of the first-

forbidden charge-exchange modes, as well as of the corresponding

S-decay and y-radiation processes, in the nuclei around 209pb,

has been performed in the framework of FFS theory.

Recently, an experiment has been done at Grenoble with

a (3He,t) reaction at 80 MeV 37) , in order to excite the y =-1

collective modes in 208Bi nucleus. Only a possible allowed

Gamow-Teller transition centered at 15.9 MeV (with respect to the

ground state of 2oePb) has been observed in this reaction study.

It should be noted, however, that due to the importance of the

* The differencie» between the values given here for the ratioY/Ç and those presented in table 2 of ref. 19' are mainly dueto the fact that there we have used g. = -0.96 g .

.28.

continuum for increasing excitation energies, the forgoing reaction

does iiOt appear favourable for studing the first-forbidden charge-

exchange modes.

Also, an attempt has been made, througth the

204Bi(ir,Y)209pb reaction38', to detect the p =1 collective

states. A collective state at 7.9 MeV was observed in this work

and interpreted as a 1 h- .. proton state coupled to the IA state

of the T component of the giant quadrupole resonance. In view

of the present analysis it also might be speculated that the

measured state corresponds to a proton coupled to dipole spin-flip

u =1 state. The unperturbed calculated energy for this state is

7.31 MeV.

In spite of the wealth of experimental and theoretical

studies of the nuclei in vicinity of 208Pb, surprisingly little

attention has been paid to the B-decay processes of these nuclei.

To our knowledge, only one thorough analysis on this subject hrs

appeared in the literature in recent years. This is the paper of

39)Damgaard et al. in which the effective one-particle moments

and effective weak charges g® and 9? have been extracted

from experimental data for the partial half-lives. In this way

they obtained two sets of values (see also ref. )

(1) g*ff = 0.5 gv , g*ff - 0.4

(2) g*ff = 0.2 gv , g*ff = 0.6 gft

It should be noted that these effective coupling

constants are to a great extent model-dependent as the analysis

39)of ref. is based on a given set of nuclear wave functions.

Our calculated values for the effective charges are

both i) considerable higher (see, for example, table 5) and

ii) appreciably different for different single-particle transi-

.29.

tions. Although no best parameter fit has been attempted we have

fairly good agreement between the calculation and the available

experimental information. It is difficult to discern if the

remaining discrepancies are due either to the evaluation of the

core-polarization effects, or to the wave functions employed or

to the approximations for the relativistic B-monents.

.30.

REFERENCES

1. A. Bohr and B.R. Mottelson, Nuclear Structure (Benjamin, NewYork, 1975) Vol. II.

2. D.F. Peterson and C.J. Veje, Phys. Letters 24B (1967) 449.

3. A.B. Migdal, Theory of finite Fermi systems (Intersci. Publ.,New York, 1967).

4. K. Shoda, A. Suzuki, M. Sugawara, T. Sctito, H. Miyase and S.

Oikawa, Phys. Rev. C3 (1971), 1999, 2006.

5. K.A. Snover, J.F. Amann, W. Hering and P. Paul, Phys. Letters37B (1971) 29.

6. P. Ring and J. Speth, Nvcl. Phys. A235 (1974) 315.

7. R. Bauer, K. Ebert, P. Ring, W. Theis, E. Werner and W. Wild,Z. Phys. A274 (1975) 41.

8. K. Ebert, P. Ring, W. Wild, V. Klemt and J. Speth, Nucl. Phys.A298 (1978) 285.

9. E.J. Konopinski and G.E. Uhlenbeck, Phys. Rev. 60 (1941) 308.

10. A. Bohr and B.R. Mottelson, Nuclear Structure (Benjamin, New

York, 1969) Vol. I.

11. I. Fujita, Phys. Rev. 126 (1962) 202.

12. J. Eichler, Z. Physik 171 (196 2) 463.

13. J. Damgaard and A. Winther, Phys. Letters £3 (1966) 345.

14. T. Ahrens and B. Feenberg, Phys. Rev. 8_6 (1952) 64.

15. D.L. Pursey, Phil. Mag. 4_2 (1951) 1193.

16. H.E. Bosch, M.C. Cambiaggio, L. Szybisz, F. Knnpotic and J.

Navaza, Phys. Rev. Ç7 (1973) 760.

17. J. Speth, Z. Phys. 239 (1970) 249.

18. K. Ebert, Ph.D. thesis, Thechnischen Universitât Munchen, 1975

(unpublished).

19. K. Ebert, W. Wild and F. Krmpotic, Phys. Letters B58 (1975) 132.

20. V. Klemt and J. Speth, Z. Phys. A278 (1976) 59.

21. J. Speth, E. Werner and W. Wild, Phys. Reports 33 (1977) 127.

22. G.E. Brown and M. Bolsterli, Phys. Rev. Letters 2 (1959) 472.

23. J.M. Trischuk and E. Kankeleit, Nucl. Phys. A9£ (1967) 33.

.31.

24. M.R. Schmorak and R.L. Auble, Nucl. Data 5 (1971) 207.

25. B.I. Persson, I. Plesser and J.M. Sunier, Nucl. Phys. A167

(1971) 470.

26. H. Behrens, M. Kobelt, W.G. Thies and H. Apple, Z. Phys. 252

(1972) 349.

27. A. Heusler and P. Von Bretano, Annals of Physics J75 (1973) 381.

28. M. Kortelahti, A. Pakkanen and J. Kantele, Nucl. Phys. A240

(1975) 85.

29. N.K. Gupta and S.R. Sastry, Radioactivity in Nuclear Spectroscopy

Modern Techniques and Application, vol.2, ed. J.C. Manthuruthil, p.1279.

30. T.T.S. Kuo and G.H. Herling, NRI Memorandum Report 2258 (1971).

31. Y.E. Kim and J.O. Rasmussen, Nucl. Phys. 4_7 (1963) 184; 6_1

(1965) 173.

32. K.K. Seth, Nuclear Data B7 (1972) 161.

33. M.B. Lewis, Nuclear Data B5 (1971) 631.

34. D.A. Howe and L.M. Langer, Phys. Rev. 12£ (1961) 519.

35. W. Wiesner, D. Flothmann, H.J. Gils, R. Lohken and H. Rebel,

Nucl. Phys. A191 (1972) 166.

36. O. Civitarese, F. Krmpotlc and L. Szybisz, Phys. Lett. 48B

(1974) 199.

37. A. Willis, D. Ovazza, M. Morlet, N. Marty, P. Martin, P. de

Saintignon and M. Buenerd, J. Phys. Soc. Jap. Suppl. 4_4 (1978)

211.

38. H.W. Baer, J.A. Bistirlich, N. de Botton, S. Cooper, K.K.

Crowe, P. Truol and J.D. Vergados, Phys. Rev. Ç3 (1974) 1140.

39. J. Damgaard, R. Broglia and C. Riedel, Nucl. Phy3. A135 (1969)

310.

TABLE 1. Correspondence between cartesian and spherical notations for the matrix elements: a) cartesian

representation of Konopinski and Uhlembeck ; b) normalization factor; c) spherical represen

tation of Bohr and Mottelson (= cartesian/normalization factor) and d) reduced matrix ele-

ments of the operators defined in eq. (1.1) . The upper and lower signs in the last column cor-

respond to B~(yT=-l) and B+(P T=1), respectively. It should be noted that 2t±1=± SI t±.

a) b) c) d)

-v = gA<Y5> -(4Tr)1/2(2I±+l)~1/2 -(4n)1/"2(2Ii+l)"1/2 -(4it)1/2(2Ii+l)"1/2 ( 8 T T / 3 ) 1 / 2 ( 2 I Í + 1 ) "1 / 2 <l f | | iM ( J A , K = 1 , A=l) | | Ii> ;g A ( 4 T T / 3 ) I / 2 ( 2 I Í + 1 ) "1 / 2 ;g y ( 1 6 T I / 3 )1 / 2 ( 2 I Í + 1 ) "

1 / 2 <l f | | iM ( J A,K = 1 , X = 2) |

Table 2. One-particle 8-moments in units of gv fm. We indicate with A and A , the total contributions of the charge -

exchange vibrational fields with u "1 and y =-1, respectively. For the sake of comparison all the axial -vector

matrix elements were evaluated with gA=-1.24 gy.

K-l,K=l) | | ji> <jil |iM(JA,K=l,X-2t | | j£>

TRANSITION M*ff A A T M ! " & á T M * " A fl_ T M*" A A. TJ f J i * l Jf ] i 3fJi l l JfJi JfJi * ' }t}i Jf : i l l J | J i

-2.«0 0.20 0.53 -1.67 1.93 0.00 0.62 1.31 -3.39 0.1* 0.87 -2.35

207Tl(l/2*)»*07Pb<3/2~) -2.68 -0.0S 0.76 -1.97 -2.3S 0.07 0.52 -1.7c -5.26 0.03 1.62 -3.60

2"Pb(9/2*)^""Bi(9/2") 2.15 -0.31 -0.10 1.74 0.30 0.08 0.05 0.43 2.65 -0.37 -0.23 2.05 1.50 -0.28 -0.23 0.99

TABLE 3, Partial contributions of the 1 collective states to the rank-one p-moments. The symbols I,

II, III and IV stand for the levels at 2X.2, 21.5, 22.8 and 24.« MeV in *u"Hi, respectively.

The axial-vector coupling constant is g = -1.24 g .

< 3 f I |M(pv,X =

TRANSITION

II III IV I II III IV

i 0 7 T l ( l A + ) + ? 0 7 P b d / 2 ) - 0 .22 -0 .26 -0 .03 0.06 0.02 0.09 0.25 0.18

20 7ml I 1 /->*\ j . 2 0 7TK1/2 ) -• 207Pb(3/2 ) 0.11 0.27 0.01 0.14 -0.01 -0.09 -0.04 0.41

Pb(9/2 ) - 20SBi(9/2 ) -0.23 0.17 0.04 0.08 -0.02 -0.06 -0.29 0.24

Tal >.e 4. Results for the S- lecay of *"T1 and **'»i. The reduced aatrix elaaents of the aoaents listad

The reduced transition probablliaa B(A) ara inin the first colia i arc given In unit» ofunits of g 2 and the partial half lifat in seconds. Tha first value in «ach row was obtainedwith the local vertex function M*ff and «fc»1.24 Oy and tha second ona with the renonullxedvertex T and gA»0.»18 gv. The syafaol b stands for the branching ratio.

TRANSITION l t7Tiri/2*)-»*"PbU/2") » i 7 Tl( l /2 + >- f "Pb<3/2~) l*»Pb(9/2*)~I#*Bi<9/2~)

H(pm,l-0).103 " 2*9

- 124217

130

M(j.,ic=i.l=0).103

509154

BO-OKIO4 1 1 0

2815.85.7

1 0 3

72136103

11

19

/3 * e

6548

7.411

11660

8044

91

52

1 4 8

480.380.64

7.61.9

i — T- «ij.,x-l,»-2).103

/3 Ae *253181

73

35

"cal

42

142

300

1O0

71.10

216.IO2

286 t

(MeV) 1.436b> 0.534b>

(118.9 ± 1.2).10(117.1 ± 0.5).10

0.634 í 0.004C>

0.6446 t 0.0012*

2d>

d>

99.76b) 0.24b» 100c'd)

a) Ref.23» , b) R.f.24' , c) *.f.2S> , d) «af.2().

Table 5. Effective charges for the El charge-exchange y trans i t ions , lhe

matrix elanent <f j j i t í í ^ X ^ ) | |i> i s given in units o f fin ^ and

i s related t o the charge exchange El ncnent fay mans of eq. ( 2 . 8 ) .

2

1

2

2

3

2

1

3

2

3

3

4

3

4

3

3

3

2

2

2

1

2

2

2

1

1

1

a

TRANSITION <

89/2 *

hi/:87/2 *

89/2 "*

d 5 / 2 -

S 7 / 2 "

j15/2

d , -»5/2

g7/2 "

d 3/2 •*

d 5 / 2 " *

S l / 2 "

d 3 / 2 *

S l / 2 " *

d 3 / 2 "

•1)2 ̂8I/2 "d 3 / 2 "

<?/2«^ 2 *

hIl/2

S/2"dsVd •*

R •*

g 7 / 2 -

) Ref

l h 9 / 2

* l h9/2

1 h9/2

2 f 7 / 2

2 f 7/2

2 f 7 / 2

* X x13/2

• 2 f ,5/2

• 2 f 5 / 2

• 2 f 5 / 2

' 3 P3/2

3 p3/2

• 3 P3/2

• 3 p l / 2

• 3 P l / 2

3 p-; 2

• 3 p~ l

• 1 p-J2

• 2 f i /2

' 3 P"3/2

-• 1 i " 1

" 2 f5/2

' 3 P3/2

• 2 f~

• 2 f"1^ *5/2

• 2 f"1

Z f 7/2

. A ; b)

:f j |iM(pv,A«l)! [

0.302

6.643

-2.765

5.877

-3.517

1.009

8.549

0.736

5.264

-2.958

4.175

-2.704

1.417

1.904

3.179

1.931

-2.681

2.122

-3.942

0.903

-7.616

1.057

2.873

-4.722

2.358

0.368

-5.698

Ref. 5

i> (e. . )theoryJ2J1

1.43

0.64

0.76

0.76

0.80

0.66

0.71

0.93

0.74

0.84

0.82

0.86

0.78

0.89

0.91

0.68

0.73

0.75

0.68

0.82

0.70

0.60

0.71

0.72

0.70

1.22

0.65

-o0 .

0 .

0 .

~c

0,

. eff .(e , ) exp.

J2J1. 6 ( 1 . 4 ) » , 2m*>

45(1.48)b ); 0.46+0.

55(1.34)b)

63(1.22)b)

1.4(1. l ) b )

.56 ± 0.08a)

TABLE 6. Comparison between the calculated and experimental Forward qomq amplitudes and energies of Che 4j,

5. and 5l states of "Pb. The phenomenoloqlcal wave functions were derived from a least squares

fit of the experimental data for the inelastic proton scattering via lsobaric analog resonances. Aa

the errors obtained in rf»f. certainly understimate the true uncertainties of the amplitudes they

are omitted here. Only those components that contribute more than 1% are listed.

State

Energy (MeV)

Cal.

3.58

Exp.

3.47

Cal.•

3.37

Exp.

3.20

Cal.

3.83

Exp.

3.71

Protons

1 h 9 / 2 2 dj^ 2 0.119 - O.XJ» 0.173 - 0.193 0.409

1 h , / 2 3 s ^ 2 - 0.202 - 0.226 - 0.3Í» - 0.51?

d~*2 - 0.130 - 0.133

h U/2 " ° - U 0 - ° - 1 0 S

Neutrons

2 g 9 / 2 3 Pjy2 0.157 0.24S 0.101 • 0.273 - 0.130

2 g 9 / J 2 fj*j 0.176 • 0.158 0.306 0.514 0.518

2 g 9 / 2 3p^J2 -0.970 - 0.921 0.878 0.786 - 0.425 • 0.480

f5/2 0.146

" J 0.227 - 0.13J 0.415 - 0.302

1 Í15/2 l h U ••"»

3 d5/2 J '"l/|

0.

0.

0.

0.

101

306

786

131

- 0.273

0.514

- 0.425

0.165

0.415

0.138

0. 111

Table 7. Dominant single-particle contributions and the total matrix elements for the 8 decay of "*T1. The matrix elementsare given in units of fm g »-1.24graph contributions, recpectively.are given in units of fm g --1.24 gy. With RPA, CP and SG we label the total RPA, core-polarization and scattering

Transition S •» 3j 5 •» 4^ 5 •» 5j 1 • 5̂

iM(pu,X«l) iM(i, ,»-l ,A-l) iM(j.,x-l,»-O) iM(p1,,\-l) iM(j ,*V ^ A V A

(-l h - 0.021 0.290 0.042 0.309 0.591 0.084 0.735

0.471

2g -»2f . 0.820

S*í/2*3pl/2 " 2 ' 4 9 5 ' 3 - 1 3 3 S ' 4 9 3 " 4 < S 3 8 2 t 3 1 2 " 4-06i 2 ' 2 0 3 " LA2° 1.963

3s~'2*3p~*2 0.2S3 0.300 0.263 - 0.28S - 0.250 - 0.866 0,759

2d"J,-3p'}, 0.092 0.060 0.213 0.441 - 0.019 - 0.068 0.429 - 0.064 - 0.224

2dj},*2("J, 0.134 - 0.139 0.123 - 0.111 0.097

ÜÍ . - ÍPTL 0.818 0.71? 0.905 - 0.794 -0.445 - 0.391

0.290

RPA

CP

SG

TOTAL

- 0.663

0.220

0.024

- 0.418

- 2.146

0.726

0.000

- 1.420

6.908

- 2.014

0.001

4.896

- 4.063

1.283

0.004

- 2.776

2.899

- 0.796

- 0.012

2.091

- 3.012

0.961

0.017

- 2.034

3.508

1.018

0.006

2.496

- 3.169

0.968

- 0.010

- 2.212

2.274

- 0.728

0.016

1.562

Table 8. Results for the 6 decay of 2 > (T1. The symbols have the same meaning as in table

4. All the experimental results were taken from ref. .

TRANSITION

. io3

. 103

B(,«0) . 104

3,A>1) . 103

Tf f ««v*-1» • lo3

,«r-l,X-l) . 103

B(X-l) . 104

5f f; «<V«-i.»

'cal

^exp

(MeV)

- 32

- 15

337.103

1540.103

:610.1C3

2.378

:0.03

- 118

- 77

52

34

- 236

- 124

83

25

- 84

- 45

130

420

843

1.548

21.7

- 429

- 217

170

86

6115.6

- 156

- 109

70

50

- 103

- 51

32

11.1

- 14

- 7.8

48

359

1.795

51

214190

- 147

- 77

4211.6

169

116

- 77

- 53

78

39

26

9.5

45

28

215

691

803

1.285

22.8

Table 9, Dominant single-particle contributions and the total matrix t-lements for the Jecay of • u ' Hg and ?I)6T.. All the

symbols have the same meaning as in table 7, The wave funet ions emuloved in t-.hc- calculations are chose t f Kuo -ma

Herling30».

Transition Hg(O+) - Tl(l~ ) Hg(O+) • TKljJ Hg(O*; T1I0 Tl>o") * i'blcf )

A=l) iM(JA.x=l.A=l) iM{pv,X-l)

* 3 pj^2 0.328 - 0.288 0.522 - 0.458

-» 3 p ^ 2 0.114 - 0.100 0.558 - 0.489

* 2 f~*2 - 0.168 - 0.148 0.295 0.259

2 djtj * 2 f?p2 0.086 - 0.074

-» ^ - 1 •» « - 1

3 pjy2 - 1.457 - 3.236 0.202 0.353 1.792 1.362

- 0.052 0.080 - 0.280 0.206 0.175

0.236 - 0.207

0.100 - 0.200

RPA - 1.239 - 3.079

CP 0.413 0.921

SC - 0.012 - 0.058

TOTAL - 0.838 - 2.215

2.041

0.559

0.025

1.457

- 0.831

1.168

- 0.014

- 0.677

2.104

- 0.642

0.072

1.534

1.736

- 0.532

0.054

1.258

Table 10. Results for the 6-decay of the 0 •* O transitions.

All the symbols have the same meaning as those of table

4. The wave functi

of Kuo ana Herling

4. The wave functions used in the calculation are those.30)

Transition 206Hg + 206Tl JO6T1 * 20SPb 2l0Pb * 2"»Bi

M(fA,À=0).103

i

B(A-0).104

Ccal

£exp

215116

- 87

- 47

163

47

83

285

802 -llf

18197

- 73

- 39

118

34

62

215

256 ± 3 a )

83

43

- 36

- 19

22

6.1

52.107

189.107

V 1.307 ± 0.020a) 1.534 ± 0.005a) 0.017 i 0.001b)

o

b(%) 61 i 12a) 100 80 ± 10b)

a) Ref. 32) ; b) Ref. 3 3 )

Table 11. Results for the B-decay of the O »1~ and 1~-»O transitions. All the symbols have the same meaning as

those of table 4. The wave functions used in the calculation are those of Kuo and Herling . For the

decay of *'°Bi we have also employed the wave functions of Kim and Rasmussen and the results are given

in parentheses.

Transition206Hg - Hg °Pb JBi DPo

k-

— f- *<ja,x=l,*=l).103

ccal

65

43

- 30

- 20

104

55

193

62

2.5.102

7.7.102

(14.0i » )

- 102

- 72

49

35

28

17

6

4

12.7.

21.7.

(16.3 I33

.4

.2

10

10

).

3

3

103

- 27

- 21

14

11

40

21

7

1

0.83

7.0.

.2

.3

.10

108

1.10'Db)

35.9 (39.9)

30.1 (26.5)

- 17.3 (-18.6)

- 14.5 (-12.4)

- 36.5 (-19.3)

- 21.4 (-18.2)

1.072 (0.014)

0.115 (0.055)

6.3.104 (11.6.104)

476.104 (905.104)

(43,7 í 0,01).10'

1.002 ± 0.020a) 0.657 ± 0.020a 0.061 ± 0.002 1.1610 ± 0.00111b)

36 ± 7* 7a' 3 ± 2° 20 ± 10£ 99%b)

a) Ref.32' ; b) Ref.33>.

Table 12. Dominant single-particle contributions and the total matrix elements for the g-decay of 2 l 0Pb

and l I 0Bi. The symbols correspond to those of table 7. The results obtained with the wave

functions of Kim and Rasmussen are given in parenthesis.

Tansition Pb(O Pb(O ) -• Bid") Bid") * Po(0 )

iM(JA,x=l,A«O) iM(pv,X-l) iM(JA,x=l,X=l) iM(pv,X=l) iM(JA,x=l,X=l)

g9/2 9/2

1 lll/2 X h9/2

2 g 7 / 2

2 *9/2 2 f 7/2

15/2 X i13/2

2 f7/2

0.785

0.048

0.109

0.108

0.027

0.221

0.087

0.954

- 0.095

0.024

0.194

0.076

- 0.108 (- 0.116) - 0.951 (- 1.016)

- 0.234 (- 0.576) 0.205 (0.505)

- 0.115 (- 0.113) - 0.101 (- 0.099)

- 0.110 (0.041) - 0.096 (0.036)

- 0.121 - 0.106

RPA

CP

SG

TOTAL

0.866

0.165

0.098

O.fiOj

0.572

- 0.085

- 0.048

0.439

1.160

- 0.243

- 0.083

0.835

- 0.712 (- 0.763) - 1.058 (- 0.559)

0.122 (0.162) 0.198 (0.014)

- 0.008 (0.000) 0.019 (0.067)

- 0.797 (- 0.601) - 0.841 (- 0.475)

FIGURE CAPTIONS

Fig. 1 - Schematic diagram of the charge-exchange excitations in

nucleus with neutron excess. The T +1 , T and T -1o o o

are the total isospins of the (N+l.Z-1), (N,Z) and

(N-1,Z+1) nuclei, respectively. The energies of the

corresponding T=1 excitations with u =l#0 and -1 are

E(p ) and the transitions feeding these states are

indicated by thick lines. The particle-hole transitions

are represented by thin lines. The states |IA> and

T > are the isobaric analog states of the ground and

collective charge-exchange states, respectively, in the

(N+1,Z-1) nucleus. The isobaric analog state of |T >

state in the (N,Z) nucleus is the collective state with

T=T in (N-1,Z+1) nucleus.

Fig. 2 - Schematic diagram of 6~ decay of 209Bi nuclei and the IA

El y process chowing notation and energy relationships.

Fig. 3 - Graphical representation of the vertex operator T ,

given by eq. (2.24).

Fig. 4 - Diagrams contributing to the matrix element <n|M|m> for

the particle-hole phonons. The diagrams in (a) represent

the contributions of the effective operator Me . There

is a one-to-one correspondence between these graphs and

the six terms of eq. (2,29). The renormalization of the

first diagram in (a) by the charge-exchange field is

illustrated in (b).

Fig. 5 - The B(o ,X)-values in units of (2X+1) |<r >| /4 ir , between

two single particle orbitais with quantum numbers t and

l'=l+l , evaluated in the asymptotic limit (£>>1). The

thick: ess of the vertical lines represent the transition

strengths given in the circles.

Fig. 6 - Perturbed excitation spectra for the first-forbidden

modes in 2O0T1.

Fig. 7 - Perturbed excitation spectra for the first-forbidden

modes in 2i0Bi.

= T M T = T 0 M T = T o - 1

[ N + 1 , Z - 1 ) N , 1 - 1 , Z + 1

1

o.o»

- o»

AA

o*

IIA

I]A

1,

X

M t f l

JlJt

<O|Meff|n>

X

(VI

II

Ã

II«<

CO

A>_

V

A

V

oII

to

ÃV

I

"iift

•¥

50-

B( Ü* I ,X*0) I -Xtm1

50

0

B(^0,J

f10 20

E-32fm2

E(MtV)

150

too

50

0

fm2 '

50

10 20 E(MeV)'

- | 1 120 E(MeV)

i

10

B((T»1,X«2)

. I I

10 20 E(MeV)

«O-

Btf-I.X'O)

L.

L Ie « w n

zoo-

CMKO

300

ZOO

00

.1 . . L L

cnwi

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