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Nuclear matrix elements in the first forbidden beta decay of 198Au
1
4.c
]
hklear
Physics
A159 (1970)
Not lo be reproduced
143 - 152; @
by photoprint
North-Holland
or microfilm without
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NUCLEAR MATRIX ELEMENTS IN THE FIRST FORBIDDEN BETA DECAY OF 19’Au Department
H. A. SMITH + and P. C. SIMMS Purdue University, Lofayette, Indiana 47907tf
of Physics,
Received
30 July
1970
Abstract: Nuclear matrix elements for the 2-(0.959 MeV B-)2+ beta transition in 19*Au were extracted and comparison was made with CVC theory predictions for the vector matrix element ratio &c .= ja/jir/p. It was found that the ratio &vc was consistent with the CVC prediction based on the assumption chat the Coulomb Hamiltonian is diagonal. However, contributions from off-diagonal matrix elements of the Coulomb Hamiltonian could not be ruled out. Fairly good limits were placed on the nuclear matrix elements in spite of the fact that the Coulombenhanced combinations of matrix elements dominate the transition.
1. Introduction This paper is part of a series devoted to the analysis cf first forbidden beta transitions i-4 ). The objective of this series is to use the available experimental data on a selected group of isotopes to extract the beta-decay nuclear matrix elements and to obtain the experimental limits on the vector matrixelement ratio(Ja/fir/p). This ratio is interesting because it can be compared to theoretical predictions which are based on the conserved vector current (CVC) theory for beta decay. Furthermore the experimental value of the vector matrix element ratio can provide information on the impurities in isobaric analogue states ‘). One would expect that because of the high 2 in 198Au the r-approximation ‘e6) would hold. That is, Coulomb-enhanced terms would dominate the formulas for the observables. (For lveAu r = 16.4 > WY0= 2.87, in natural units.) If the 5-approximation were valid for this nucleus, a complete matrix element extraction would be impossible; and only the matrix element combinations V and Y could be determined. Fortunately there is a lot of accurate experimental data available, so 19’Au offers the best possibility of determining the matrix elements for a transition in which I/ and Y are dominant. The spectrum shape correction factor 7), the beta-gamma directional correlation coefficient *), and the energy dependence of the beta-gamma circular polarization correlation “) have been accurately measured. Recently a very accurate measurement has been made of the angular distribution of E2 gamma rays following the beta decay of oriented nuclei ‘). The angular dependence of the beta-gamma circular polarization correlation lo), the longitudinal ‘I), and transverse I’) polariza+ Present address: Los Alamos Scientific Laboratory, ‘+ Work supported
Los Alamos, New Mexico, 87544. by the USAEC under contract no. AT(I 1-I) 1746. 143
144
H. A. SMII’H
AND
P. C. SIhM.3
tion of the beta particle have also been measured; but the matrix elements are not sensitive to these observables. The analysis presented here is more accurate than previous studies of 19*Au for several reasons. In addition to new data from nuclear orientation experiments, finite nuclear size effects, screening of atomic electrons, and higher order matrix elements are included in the analysis.
19*A~
B-DECAY
145
The calculations of Fujita ’ “) and Eichler ’ ‘), using CVC theory, yield the following result (in natural units) for the vector matrix element ratio [see ref. “)I: n
J a
A0 cvc
s
= -
= (woT2.5)p+2.4.
&aZ,
(1)
irlp
for /?’ decay: W, is the beta transition energy, p and Z are the radius and charge of the daughter nucleus. The Fujita-Eichler result utilizes the Ahrens-Fee&erg approximation 16), wh’rch assumes the off-diagonal matrix elements of the nuclear Coulomb Hamiltonian are negligible compared to the diagonal elements. The more general approach of Damgaard and Winther I’) does not rely upon the diagonality of the Coulomb Hamiltonian. Instead, a realistic form for the nuclear Coulomb potential energy is proposed, leading to the following prediction for the vector matrix element ratio:
A cvc = s
s
a
= (WoT2.5)p++aZ(3-A),
(2)
irlp
where the parameter ;1 is defined as:
s0-
-ir -r ’
rZE
(3)
pp.
r
ir
JP
For R. = 0.6 the two expressions for A,-vc give the same numerical result. A complete discussion of this parameterization of the vector matrix element ratio is given in ref. ‘). 2.2.
USEFUL
FORMULAS
FOR SOME
OBSERVABLES
The formulas for the general beta-gamma angular correlation functions are presented in detail in ref. ‘). The formulas for the E2 gamma-ray angular distribution following the beta decay of oriented nuclei and the electron longitudinal polarization are presented here for completeness. Consider a first forbidden beta decay from an oriented nuclear state with spin and parity li- to an intermediate state with spin and parity 2+, followed by an E2 gamma transition to a state with spin and parity O+. If the beta particle is not observed and the angular distribution of the gamma rays relative to the orientation direction is measured, the result is: _-_ w(e)
=
1 -J&(2Zi+
l)f~U,P(COS
0)-J$(21,+
l)f404P4(COS
0).
(4)
146
H. A. SMfTH AND
P. C. SIMM.3
The quantitiesf” are the statistical tensors of Fano labor), which describe the orientation of the parent nucleus. The parameters D,, describe the change in orientation of the initial system caused by the intermediate beta transition. They contain the nuclear matrix elements of the beta transition and the spins of the initial and intermediate states [see ref. ‘Of]: P,(cos 0) are the nth order Legendre polynomials, and 8 is the angle between the axis of the gamma detector and the axis of symmetry dictated by TABLE Beta-gamma
re-orientation
0,
=
parameters
2 for
two
important
spin
sequences
[ro[2+f~all’-~~~az~z
i7, = Iao12--jla,12+fla2j2
3- 4
2f
+
o+
0,
=
4~(2~all’+fl~z~2)
i7, = Jq(jlaI12
-fla212)
For bz’, see ref.2).
the orientation of the parent nucleus. The 8, are used to extract the nuclear matrix elements of the beta transition. The general expression for 8, is given below for the spin sequence Zi-(8)2+(y)O~ :
The quantities lrJ12 are the probabilities that the lepton field will carry away Junits of angular momentum, The laJj2 are normalized such that \ao12+16,12+la212
= 1.
(6)
The symbol in curly brackets is a Wigner 6-j symbol. Explicit expressions for the 8z and the U, for spin sequences 3-2-O and 2-2-O are given in table 2. The U,, are normalized so that 0, = 1.0. If, instead of the statistical tensors of Fano, one uses the orientation coefficients, f,, of Tolhoek et al. 1g*2’*22), the angular distribution in eq. (4) takes the form w(e) = i -+gj-Z
U~P,(COS
e)--
y.,f,
U,P,(COS
e),
Pa)
BETA PARTiCLE ENERGY (NM UNITS1
Fig. 1. Experimental shape correction factor from ref. 7), with theoretical values for matrix element sets A and B.
ci
L
1.0
-----*
Fig. 2. Experimental fry directional correlation coefficients from ref. ‘), with theoretical values for matrix element sets A and B.
2.0
BETA PARTICLE ENERGY (NAT. UNITS1
Fig. 3. Experimental
energy-dependent & circular pofatization correktion tical values for matrix etcment sets A and 3.
fFOm
ret *), with theore-
148
H.
A. SWTH
AND
P. C. SlMhlS
where the U,, are given by’ (5+n)! C(4-n)!
U” = W,Y
(21i-n)!
(21i+n+l)!
l+(21,+1)
’
1 +u
“.
(5a)
Regardless of the spin sequence of the intermediate beta transition, t-i, = Ue. Furthermore, it is clear from eq. (5a) that if ii = 2,U” is equal to U,. Thus, for the 2-2-O spin sequence, one need not distinguish between i7, and U,. However, for the 3-2-O spin sequence, the u,, and U,, are related by U,(3-2-O) = 1.08 &(3-2-O), &(3-2-O) = 0.808 &(3-2-O). The longitudinal polarization of the emitted beta particle, with the (-p/W dependence removed, is given below for first forbidden beta transitions. 1+2
-=Pl -PiW
) energy
(1/W)(M','~mf','+M~q)m~"~)-(m(P:)2-(m~~)2 . (7) 1 t C(W
The matrix element combinations MioK,‘yandm$ are defined in ref. ‘). The quantity C(W) is the spectrum shape correction factor defined in terms of the matrix element parameters in ref. ‘). 3. Analysis and results The 2-(0.959 MeV /I-)2+(0.412 MeV 7)Of fir cascade in lg8Au was analyzed with the following experimental data being imposed: The spectrum shape correction factor ‘) (fig. 1); the beta-gamma directional correlation *) (fig. 2); the beta-gamma TABLE 3
Nuclear orientation
experiment
data summary for ‘98A~ (2- + 2+ + O+) O,(U,)
Ul(U,)
0.795f0.007
0.347 *0.030
set A
0.796
0.321
set B
0.798
0.327
circular polarization correlation “) (fig. 3); the longitudinal polarization of the electron II); and nuclear orientation data “) (table 3). The analysis of lg8Au showed that the observables were relatively insensitive to the magnitude of the B,, matrix element, except that dominance of the transition by B, was not allowed. Furthermore, all of the matrix elements were reduced by about an order of magnitude relative to their maximum possible physical size. * The re-orientation ref. 20).
parameters,
which are called V, here and in refs. t9*21*z2), are called B. in
149
19*~~ ~-DECAY
The experimental limits on the vector matrix element ratio are consistent with the assumption that the off-diagonal contributions from the Coulomb Hamiltonian are negligible (see fig. 4). For Iz = 0.6 the experimental limits and the theoretical prediction for the vector matrix element ratio are barely in agreement: &“:“,,O. = 0.6) = 0.707,
@a)
Ll;l;p,(l, = 0.6) = 0.600f0.125.
(8b)
For L = 1.4 the center value of the experimental limits agrees with the theoretical prediction for A: /I&& = 1.4) = 0.475, (9a) /I?:#
= 1.4) = 0.475+0.125.
(ob)
0.7 0.6
I
0.1
IL__0.0
Fig.
4.
Variation of the experimental
I.0
2.0
limits and the theoretical of the parameter A.
prediction
for&c
as a function
In order to maintain agreement between theory and experiment for A, the parameter ), must be restricted to the values 0.5 _I 1 _I 2.2. (10) Table 4 shows the extracted limits on the normalized matrix elements. The matrix element parameters are not presented here because the limits of error on the actual matrix elements are somewhat better than on the matrix element parameters. The tighter restrictions on the matrix elements occur because the variations in the scale factor, r~,are highly correlated with the variations in the matrix element parameters. Table 5 shows typical matrix element parameter sets for two cases: (A) The Bij matrix element approximately equal to the other matrix elements, and (B) The B,, matrix element suppressed. Plotted in figs. 1, 2 and 3 and displayed in table 3 are the theoretical values, calculated with sets A and B for the experimental observables used
150
Ii.
A. SMITH
AND
P. C. SIMMS
TABLE4 Normalized matrix elements for ‘98A~ Matrix
element
Value
0.0052*0.0021 0.0270 ~0.0070
s -ir
0.0795 kO.03 15
P
uxr s-
-0.0477
ho.0239
P 0.0302 ~0.0018
tly
-0.0433
tlv
*0.0005
-0.0379*0.0361
TABLE5 Typical matrix element parameter sets for 198A~ Parameter
Value set A B,, not suppressed
ZO
D’YO x ”
Y V wo v A
1.OOOo 1.6025 2.6710 1.984 1.073 -1.542 -1.41 0.032 0.8
in the analysis. These plots show that it would experimentally between sets A and B.
set B B,, suppressed
l.OOUO 3.9846 12.404 4.93 1 3.46 -4.93 - 10.0 0.009 1.90
be extremely
difficult
to distinguish
4. Discussion In all of the matrix element sets extracted, the combinations V and Y dominated the transition. The wealth of experimental data, however, made a more complete matrix element extraction possible.
198~~
B-DECAY
151
A qualitative interpretation of the matrix element results can be obtained using the single-particle shell model. The reduction in the size of the matrix elements is not surprising since the transforming neutron in l 98A~ is in a different major shell from the proton which is formed in 19*Hg. The Nilsson diagram for neutrons would suggest that the 119th neutron in ‘TtAu, i9 resides in the 2ft subshell. However, the ground state spin and parity of ‘i:Hg, i&-) indicates that the 119th neutron is in the 3p, subshell. Therefore, the real wave function for the transforming neutron probably contains both 2f+ and 3p, components. The ground state spin and parity of r9’Au and 199Au (both 3’) imply that there is a proton hole in the 2d, subshell. This possibility is consistent with what one would expect from the Nilsson level diagrams. However, if the proton which is formed in the beta transition is in the 2d, subshell, the AJ = 0 matrix elements would not be permitted by the available single-particle transitions (2f* -+ 2d, and 3p+ + 2d,). Since these matrix elements make a significant contribution to the beta decay of “‘Au it is unlikely that the newly formed proton is in the 2d+ subshell. In the Nilsson dyigra’rn the next state above 2d, is 3s +. All of the experimental results are consistent with a 3s+ proton being formed in the beta transition. This proton would couple with the 2d, proton hole to form the 2+ first excited state of ‘g:Hg, is. The A J = 0, 1 matrix elements could occur in the 3p+ + 3s, transition, and the 2f+ -+ 3~ transition would permit the AJ = 2 matrix element to be present. The limits on the parameter i in eq. (10) are consistent with the assumption that the nuclear Coulomb Hamiltonian is diagonal (i = 0.6). However, since rZ can be as large as 2.2, significant contributions to the vector matrix element ratio from the offdiagonal matrix elements of the Coulomb Hamiltonian cannot be ruled out. Ref. ‘) contains an extended discussion of the implications that these off-diagonal Coulomb matrix elements have for isobaric analogue states. The values obtained for the matrix elements in the present analysis are quite different from the results of Manthuruthil and Poirier 24 ) (hereafter referred to as MP) who have reported that the matrix element JBij/p is much larger than the other matrix elements. In two previous papers, Steffen 8,23) came to the opposite conclusion when he found that the Bij contribution was very small compared to the Coulomb enhanced matrix element combinations (V and Y). It is also difficult to understand why the matrix elements presented in MP give the wrong sign for the ratio, /i. Even though the size of n cannot be predicted accurately, there is no reason to doubt the theoretical prediction for its sign. The results of the present work are in good agreement with the results of Steffen and the theoretical prediction for ,4. Since avery accurate nuclear orientation measurement and additional data on the shape correction factor were available, our analysis sets much better limits on the matrix elements than Steffen’s analysis does. The different nuclear orientation data used by MP does not explain why their results disagree with ours. When we use the same data that was used by MP, the disagreement is still present.
152
H. A. SMITH
AND
P. C. SIMMS
There are two points in the paper by MP which could explain the disagreements. First, the paper states that a negative number was used for the nuclear charge 2. The formulas of Morita ” ) which were used by MP require a+2 for negaton decay. Second, in fig. 5 of MP the fly circular polarization Pv(6su = 180°) is shown as a positive quantity, However, the original reference for the circular polarization shaws that A 1 is a positive number. Since Pu( 1SO’) = A 1P, (180”), Py must be negative not positive. All of the experimental data indicate that the 0.959 MeV transition of lg8Au is dominated by the Coulomb enhanced matrix element combinations (V and I’). Our results agree with this observation. The results of MP are in direct disagreement, and there are several aspects uf their analysis which are unclear and may be incorrect.
I) 2) 3) 4) 5) 6) 7) 8) 9) IO) 11) 12) 13) 14) IS) 16) 17) 18) 19) 20) 21) 22) 23) 24) 25)
P. C. Simms, Phys. Rev. 138 (1965) l3784 H. A. Smith and P. C. Simms, Phys. Rev. Cl (1970) 1809 J. Bosken, D, Ohlms and P. C. Simms, to be published H. A. Smith and P. C. Simms, to be published T. Kotani and M. Ross, Phys. Rev. Lett. 1 (1958) 140 T. Kotani, Phys. Rev. 114 (1959) 795 M. D. Parsignault, Compt. Ren. 259 (I 964) 15 15 R. M. Steflen, Phys. Rev. 118 (1960) 763 W. P. Pratt, Nucl. Phys., to be published J. P. Deutsch and P. Lipnik, Nucl. Phys. 24 (1961) 138 G. Schwart et af., Z. Phys. 2t7 (1968) 465 P. C. Simms, Phys. Rev. 319 (1960) 768 T. Kotani and M, Ross, Prog. Theor, Phys (Kyoto) 2a (1958) 643 J. I. Fujita, Phys. Rev. 126 (1962) 202 J. Eichler, Z. Phys. 171 (1962) 463 T. Ahrens and E. Feenberg, Phys. Rev. 86 (1952) 64 J. Damgaard and A. Winther, Phys. Lett. 23 (1966) 345 U. Fano, Nat. Bur, of Standards, Washington, D.C., Report No. 1214 (1951) H. A. Tolhoek and J. A. Cox, Physica 19 (1953) 101, 673 F. M. Pipkin, Phys. Rev. 129 (1963) 2626 H. A. Tolhoek ef al., Physica 20 (1954) 1310 H. A. Tolhoek and J. A, Cox, Physica 18 (1952) 357 R. M. Steffen, Phys. Rev. 123 (1961) 1787 J. C. Manthuruthil and C. P. Poirier, Nucl. Phys. A118 (1968) 657 M. Morita and R, S. Morita, Phys. Rev. 109 (iti%) 2048
4.c
]
hklear
Physics
A159 (1970)
Not lo be reproduced
143 - 152; @
by photoprint
North-Holland
or microfilm without
written
Publishing pamission
Co., Amsterdam from
the
publisher
NUCLEAR MATRIX ELEMENTS IN THE FIRST FORBIDDEN BETA DECAY OF 19’Au Department
H. A. SMITH + and P. C. SIMMS Purdue University, Lofayette, Indiana 47907tf
of Physics,
Received
30 July
1970
Abstract: Nuclear matrix elements for the 2-(0.959 MeV B-)2+ beta transition in 19*Au were extracted and comparison was made with CVC theory predictions for the vector matrix element ratio &c .= ja/jir/p. It was found that the ratio &vc was consistent with the CVC prediction based on the assumption chat the Coulomb Hamiltonian is diagonal. However, contributions from off-diagonal matrix elements of the Coulomb Hamiltonian could not be ruled out. Fairly good limits were placed on the nuclear matrix elements in spite of the fact that the Coulombenhanced combinations of matrix elements dominate the transition.
1. Introduction This paper is part of a series devoted to the analysis cf first forbidden beta transitions i-4 ). The objective of this series is to use the available experimental data on a selected group of isotopes to extract the beta-decay nuclear matrix elements and to obtain the experimental limits on the vector matrixelement ratio(Ja/fir/p). This ratio is interesting because it can be compared to theoretical predictions which are based on the conserved vector current (CVC) theory for beta decay. Furthermore the experimental value of the vector matrix element ratio can provide information on the impurities in isobaric analogue states ‘). One would expect that because of the high 2 in 198Au the r-approximation ‘e6) would hold. That is, Coulomb-enhanced terms would dominate the formulas for the observables. (For lveAu r = 16.4 > WY0= 2.87, in natural units.) If the 5-approximation were valid for this nucleus, a complete matrix element extraction would be impossible; and only the matrix element combinations V and Y could be determined. Fortunately there is a lot of accurate experimental data available, so 19’Au offers the best possibility of determining the matrix elements for a transition in which I/ and Y are dominant. The spectrum shape correction factor 7), the beta-gamma directional correlation coefficient *), and the energy dependence of the beta-gamma circular polarization correlation “) have been accurately measured. Recently a very accurate measurement has been made of the angular distribution of E2 gamma rays following the beta decay of oriented nuclei ‘). The angular dependence of the beta-gamma circular polarization correlation lo), the longitudinal ‘I), and transverse I’) polariza+ Present address: Los Alamos Scientific Laboratory, ‘+ Work supported
Los Alamos, New Mexico, 87544. by the USAEC under contract no. AT(I 1-I) 1746. 143
144
H. A. SMII’H
AND
P. C. SIhM.3
tion of the beta particle have also been measured; but the matrix elements are not sensitive to these observables. The analysis presented here is more accurate than previous studies of 19*Au for several reasons. In addition to new data from nuclear orientation experiments, finite nuclear size effects, screening of atomic electrons, and higher order matrix elements are included in the analysis.
19*A~
B-DECAY
145
The calculations of Fujita ’ “) and Eichler ’ ‘), using CVC theory, yield the following result (in natural units) for the vector matrix element ratio [see ref. “)I: n
J a
A0 cvc
s
= -
= (woT2.5)p+2.4.
&aZ,
(1)
irlp
for /?’ decay: W, is the beta transition energy, p and Z are the radius and charge of the daughter nucleus. The Fujita-Eichler result utilizes the Ahrens-Fee&erg approximation 16), wh’rch assumes the off-diagonal matrix elements of the nuclear Coulomb Hamiltonian are negligible compared to the diagonal elements. The more general approach of Damgaard and Winther I’) does not rely upon the diagonality of the Coulomb Hamiltonian. Instead, a realistic form for the nuclear Coulomb potential energy is proposed, leading to the following prediction for the vector matrix element ratio:
A cvc = s
s
a
= (WoT2.5)p++aZ(3-A),
(2)
irlp
where the parameter ;1 is defined as:
s0-
-ir -r ’
rZE
(3)
pp.
r
ir
JP
For R. = 0.6 the two expressions for A,-vc give the same numerical result. A complete discussion of this parameterization of the vector matrix element ratio is given in ref. ‘). 2.2.
USEFUL
FORMULAS
FOR SOME
OBSERVABLES
The formulas for the general beta-gamma angular correlation functions are presented in detail in ref. ‘). The formulas for the E2 gamma-ray angular distribution following the beta decay of oriented nuclei and the electron longitudinal polarization are presented here for completeness. Consider a first forbidden beta decay from an oriented nuclear state with spin and parity li- to an intermediate state with spin and parity 2+, followed by an E2 gamma transition to a state with spin and parity O+. If the beta particle is not observed and the angular distribution of the gamma rays relative to the orientation direction is measured, the result is: _-_ w(e)
=
1 -J&(2Zi+
l)f~U,P(COS
0)-J$(21,+
l)f404P4(COS
0).
(4)
146
H. A. SMfTH AND
P. C. SIMM.3
The quantitiesf” are the statistical tensors of Fano labor), which describe the orientation of the parent nucleus. The parameters D,, describe the change in orientation of the initial system caused by the intermediate beta transition. They contain the nuclear matrix elements of the beta transition and the spins of the initial and intermediate states [see ref. ‘Of]: P,(cos 0) are the nth order Legendre polynomials, and 8 is the angle between the axis of the gamma detector and the axis of symmetry dictated by TABLE Beta-gamma
re-orientation
0,
=
parameters
2 for
two
important
spin
sequences
[ro[2+f~all’-~~~az~z
i7, = Iao12--jla,12+fla2j2
3- 4
2f
+
o+
0,
=
4~(2~all’+fl~z~2)
i7, = Jq(jlaI12
-fla212)
For bz’, see ref.2).
the orientation of the parent nucleus. The 8, are used to extract the nuclear matrix elements of the beta transition. The general expression for 8, is given below for the spin sequence Zi-(8)2+(y)O~ :
The quantities lrJ12 are the probabilities that the lepton field will carry away Junits of angular momentum, The laJj2 are normalized such that \ao12+16,12+la212
= 1.
(6)
The symbol in curly brackets is a Wigner 6-j symbol. Explicit expressions for the 8z and the U, for spin sequences 3-2-O and 2-2-O are given in table 2. The U,, are normalized so that 0, = 1.0. If, instead of the statistical tensors of Fano, one uses the orientation coefficients, f,, of Tolhoek et al. 1g*2’*22), the angular distribution in eq. (4) takes the form w(e) = i -+gj-Z
U~P,(COS
e)--
y.,f,
U,P,(COS
e),
Pa)
BETA PARTiCLE ENERGY (NM UNITS1
Fig. 1. Experimental shape correction factor from ref. 7), with theoretical values for matrix element sets A and B.
ci
L
1.0
-----*
Fig. 2. Experimental fry directional correlation coefficients from ref. ‘), with theoretical values for matrix element sets A and B.
2.0
BETA PARTICLE ENERGY (NAT. UNITS1
Fig. 3. Experimental
energy-dependent & circular pofatization correktion tical values for matrix etcment sets A and 3.
fFOm
ret *), with theore-
148
H.
A. SWTH
AND
P. C. SlMhlS
where the U,, are given by’ (5+n)! C(4-n)!
U” = W,Y
(21i-n)!
(21i+n+l)!
l+(21,+1)
’
1 +u
“.
(5a)
Regardless of the spin sequence of the intermediate beta transition, t-i, = Ue. Furthermore, it is clear from eq. (5a) that if ii = 2,U” is equal to U,. Thus, for the 2-2-O spin sequence, one need not distinguish between i7, and U,. However, for the 3-2-O spin sequence, the u,, and U,, are related by U,(3-2-O) = 1.08 &(3-2-O), &(3-2-O) = 0.808 &(3-2-O). The longitudinal polarization of the emitted beta particle, with the (-p/W dependence removed, is given below for first forbidden beta transitions. 1+2
-=Pl -PiW
) energy
(1/W)(M','~mf','+M~q)m~"~)-(m(P:)2-(m~~)2 . (7) 1 t C(W
The matrix element combinations MioK,‘yandm$ are defined in ref. ‘). The quantity C(W) is the spectrum shape correction factor defined in terms of the matrix element parameters in ref. ‘). 3. Analysis and results The 2-(0.959 MeV /I-)2+(0.412 MeV 7)Of fir cascade in lg8Au was analyzed with the following experimental data being imposed: The spectrum shape correction factor ‘) (fig. 1); the beta-gamma directional correlation *) (fig. 2); the beta-gamma TABLE 3
Nuclear orientation
experiment
data summary for ‘98A~ (2- + 2+ + O+) O,(U,)
Ul(U,)
0.795f0.007
0.347 *0.030
set A
0.796
0.321
set B
0.798
0.327
circular polarization correlation “) (fig. 3); the longitudinal polarization of the electron II); and nuclear orientation data “) (table 3). The analysis of lg8Au showed that the observables were relatively insensitive to the magnitude of the B,, matrix element, except that dominance of the transition by B, was not allowed. Furthermore, all of the matrix elements were reduced by about an order of magnitude relative to their maximum possible physical size. * The re-orientation ref. 20).
parameters,
which are called V, here and in refs. t9*21*z2), are called B. in
149
19*~~ ~-DECAY
The experimental limits on the vector matrix element ratio are consistent with the assumption that the off-diagonal contributions from the Coulomb Hamiltonian are negligible (see fig. 4). For Iz = 0.6 the experimental limits and the theoretical prediction for the vector matrix element ratio are barely in agreement: &“:“,,O. = 0.6) = 0.707,
@a)
Ll;l;p,(l, = 0.6) = 0.600f0.125.
(8b)
For L = 1.4 the center value of the experimental limits agrees with the theoretical prediction for A: /I&& = 1.4) = 0.475, (9a) /I?:#
= 1.4) = 0.475+0.125.
(ob)
0.7 0.6
I
0.1
IL__0.0
Fig.
4.
Variation of the experimental
I.0
2.0
limits and the theoretical of the parameter A.
prediction
for&c
as a function
In order to maintain agreement between theory and experiment for A, the parameter ), must be restricted to the values 0.5 _I 1 _I 2.2. (10) Table 4 shows the extracted limits on the normalized matrix elements. The matrix element parameters are not presented here because the limits of error on the actual matrix elements are somewhat better than on the matrix element parameters. The tighter restrictions on the matrix elements occur because the variations in the scale factor, r~,are highly correlated with the variations in the matrix element parameters. Table 5 shows typical matrix element parameter sets for two cases: (A) The Bij matrix element approximately equal to the other matrix elements, and (B) The B,, matrix element suppressed. Plotted in figs. 1, 2 and 3 and displayed in table 3 are the theoretical values, calculated with sets A and B for the experimental observables used
150
Ii.
A. SMITH
AND
P. C. SIMMS
TABLE4 Normalized matrix elements for ‘98A~ Matrix
element
Value
0.0052*0.0021 0.0270 ~0.0070
s -ir
0.0795 kO.03 15
P
uxr s-
-0.0477
ho.0239
P 0.0302 ~0.0018
tly
-0.0433
tlv
*0.0005
-0.0379*0.0361
TABLE5 Typical matrix element parameter sets for 198A~ Parameter
Value set A B,, not suppressed
ZO
D’YO x ”
Y V wo v A
1.OOOo 1.6025 2.6710 1.984 1.073 -1.542 -1.41 0.032 0.8
in the analysis. These plots show that it would experimentally between sets A and B.
set B B,, suppressed
l.OOUO 3.9846 12.404 4.93 1 3.46 -4.93 - 10.0 0.009 1.90
be extremely
difficult
to distinguish
4. Discussion In all of the matrix element sets extracted, the combinations V and Y dominated the transition. The wealth of experimental data, however, made a more complete matrix element extraction possible.
198~~
B-DECAY
151
A qualitative interpretation of the matrix element results can be obtained using the single-particle shell model. The reduction in the size of the matrix elements is not surprising since the transforming neutron in l 98A~ is in a different major shell from the proton which is formed in 19*Hg. The Nilsson diagram for neutrons would suggest that the 119th neutron in ‘TtAu, i9 resides in the 2ft subshell. However, the ground state spin and parity of ‘i:Hg, i&-) indicates that the 119th neutron is in the 3p, subshell. Therefore, the real wave function for the transforming neutron probably contains both 2f+ and 3p, components. The ground state spin and parity of r9’Au and 199Au (both 3’) imply that there is a proton hole in the 2d, subshell. This possibility is consistent with what one would expect from the Nilsson level diagrams. However, if the proton which is formed in the beta transition is in the 2d, subshell, the AJ = 0 matrix elements would not be permitted by the available single-particle transitions (2f* -+ 2d, and 3p+ + 2d,). Since these matrix elements make a significant contribution to the beta decay of “‘Au it is unlikely that the newly formed proton is in the 2d+ subshell. In the Nilsson dyigra’rn the next state above 2d, is 3s +. All of the experimental results are consistent with a 3s+ proton being formed in the beta transition. This proton would couple with the 2d, proton hole to form the 2+ first excited state of ‘g:Hg, is. The A J = 0, 1 matrix elements could occur in the 3p+ + 3s, transition, and the 2f+ -+ 3~ transition would permit the AJ = 2 matrix element to be present. The limits on the parameter i in eq. (10) are consistent with the assumption that the nuclear Coulomb Hamiltonian is diagonal (i = 0.6). However, since rZ can be as large as 2.2, significant contributions to the vector matrix element ratio from the offdiagonal matrix elements of the Coulomb Hamiltonian cannot be ruled out. Ref. ‘) contains an extended discussion of the implications that these off-diagonal Coulomb matrix elements have for isobaric analogue states. The values obtained for the matrix elements in the present analysis are quite different from the results of Manthuruthil and Poirier 24 ) (hereafter referred to as MP) who have reported that the matrix element JBij/p is much larger than the other matrix elements. In two previous papers, Steffen 8,23) came to the opposite conclusion when he found that the Bij contribution was very small compared to the Coulomb enhanced matrix element combinations (V and Y). It is also difficult to understand why the matrix elements presented in MP give the wrong sign for the ratio, /i. Even though the size of n cannot be predicted accurately, there is no reason to doubt the theoretical prediction for its sign. The results of the present work are in good agreement with the results of Steffen and the theoretical prediction for ,4. Since avery accurate nuclear orientation measurement and additional data on the shape correction factor were available, our analysis sets much better limits on the matrix elements than Steffen’s analysis does. The different nuclear orientation data used by MP does not explain why their results disagree with ours. When we use the same data that was used by MP, the disagreement is still present.
152
H. A. SMITH
AND
P. C. SIMMS
There are two points in the paper by MP which could explain the disagreements. First, the paper states that a negative number was used for the nuclear charge 2. The formulas of Morita ” ) which were used by MP require a+2 for negaton decay. Second, in fig. 5 of MP the fly circular polarization Pv(6su = 180°) is shown as a positive quantity, However, the original reference for the circular polarization shaws that A 1 is a positive number. Since Pu( 1SO’) = A 1P, (180”), Py must be negative not positive. All of the experimental data indicate that the 0.959 MeV transition of lg8Au is dominated by the Coulomb enhanced matrix element combinations (V and I’). Our results agree with this observation. The results of MP are in direct disagreement, and there are several aspects uf their analysis which are unclear and may be incorrect.
I) 2) 3) 4) 5) 6) 7) 8) 9) IO) 11) 12) 13) 14) IS) 16) 17) 18) 19) 20) 21) 22) 23) 24) 25)
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