Post on 23-Aug-2020
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.
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2. D.F. Peterson and C.J. Veje, Phys. Letters 24B (1967) 449.
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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.
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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.
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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 <I f | | M <PA, X=0) | 11±
-y = g y <*> -(4n)1/"2(2Ii+l)"1/2 <I f | | iM (jy,K=0 , X=l)
w = gA<io.r> -(4it)1/2(2Ii+l)"1/2 <I f | | iM( JA,ic=l,X=O) | 1 1 ^ ;g A<I f I |M (o=l,X=0)
-u = gA<oxr> ( 8 T T / 3 ) 1 / 2 ( 2 I Í + 1 ) "1 / 2 <l f | | iM ( J A , K = 1 , A=l) | | Ii> ;g A<I f | |M(a=l,X=l)
-x = gv<i r> ( 4 T T / 3 ) I / 2 ( 2 I Í + 1 ) "1 / 2 <I f | |iM(pv,X=l) | 11±> ;g y<I f | |M(a=0 ,X=1)
z = g A <iBlj> ( 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
A
.1
(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