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On the third forbidden β-decay of 87Rb
Volume
28B, number
7
PHYSICS
ON THE
THIRD
LETTERS
FORBIDDEN
20 January 1969
,V-DECAY
OF
87Rb
*
K. S. R. SASTRY Department
of Physics
and Astronomy,
University
Received
of Massachusetts,
29 November
Amherst,
Mass.
1968
It is shown that the shape of the third forbidden &decay of 87Rb may be satisfactorily explained by the single-particle shell model and the conserved vector current theory of weak interactions.
That the third forbidden non-unique P-decay of 87Rb ($- + !+, Wg = 275 keV, logfl. = 17.6) displays a strongly energy dependent shape [ 1,2] has been well known for a long time. Such characteristics of forbidden b-spectra were investigated earlier [3] to obtain information on the P-interaction. The purpose of this work is to seek an interpretation of the shape of the &spectrum of 87Rb in terms of nuclear matrix elements, which, in the notation of Greuling [4], are CVQ3(r, r),
CyiQda,r), CAiQQ(Ux r,r)and CAQ~(U,r).
Another aspect of particular interest is the relation [ 51 iQ3(a,
r) = - ($)AlQ3(r, r)
(1)
between matrix elements arising from the vector interaction, where < is the usual Coulomb energy factor (~2/2R. For a given decay, A is a constant but its value is estimated to be different by different investigators. The disagreement stems from the importance of nuclear potential contribution. For 87Rb, A is 2.3 according to Fujita [6] and is independent of nuclear potential as required by the conserved vector current theory. On the contrary, the semi-empirical estimate of Ahrens and Feenberg [7] gives a value of 0.9. The above estimates are believed to be accurate to about 10%. Recently Damgaard and Winther [8] questioned the accuracy of the approximation of Ahrens and Feenberg [7] used in obtaining the above estimates. Their method of evaluation of A requires specific knowledge of radial wave functions of the nuclear states involved in the transition. Since the decay of 87~b involves transformation of the 50th neutron to the 38th proton in 87Sr, it may be characterized as a lgf * 2pf or lg; d If; single-particle transition. The cor* Supported by the U.S. Army Research ham, North Carolina.
462
Office,
Dur-
responding values of A are evaluated to be 1.8 and 1.9 using harmonic oscillator wave functions. The experimental data of Egelkraut and Leutz [2] on the shape factor (see fig. 1) are used in this analysis. From the work of Greuling [4] and Pursey [9], the complete theoretical expression for C(W), the shape factor, may be written as (2)
C(W) OT
a [Z’lx2 + Tzy2+ T3 + 2T4xy + 2T5y + 2Tgx + T7z2] where x,y, z respectively represent ratios of the matrix elements CVQ3( r, r), CViQs(a, r) and CAQ4(U, r) to CAi@(aX r, r), which is determin-
6 Fig. 1. Energy dependence of the shape of the third forbidden/?-spectrum of 87Rb. The solid points (normalized to unity at W = 1.4) represent the data of Egelkraut and Leutz used in the analysis. The bars indicate estimated errors in interpolation of data. The solid curve is the shape factor predicted by eq. (2) with x = -0.8, z = 3.8 and A = 2.1.
Volume 28B, number
PHYSICS
7
ed by the fl value corrected for the nonstatistical shape of the &spectrum. The coefficients Ti are functions of neutrino momentum and combinations of electron radial wave functions. In the evaluation of these coefficients the approximation (a,2)2 <<1 is used. For convenience, the number of unknowns in eq. (2) is reduced to two (x and y) by ignoring the last term. Such a procedure is not unreasonable since the coefficient TV is negligibly small. To define the set of parameters x and y consistent with experimental data, ratios of experimental shape factors at different energies are equated to the corresponding theoretical ratios. The resulting solutions are conic sections in the x-y plane, and are shown in fig. 2. The relationship between y and x as given by eq. (1) is shown by the straight lines for different expectations of
-50 Y -25 0
5
I
:= -0.31
I :=-0.83
IO
15 /
Fig. 2. Plot of the parameters x and y allowed by experimental data. The lower figure is the mirror guadrant of the upper figure. For convenience, allowed solutions of y for positive values of x are plotted in the upper figure, while the lower figure displays allowed values of y for negative x. The shaded region represents spread of the parameters. The straight lines given by y = - ($)h
LETTERS
20 January 1969
A. The conventional estimate of Ahrens and Feen berg indicates agreement with experiment for x N +3.7, while the CVC estimates of Fujita as well as Damgaard and Winther give x = - 1. A distinction between the two possibilities requires additional information. For a ground state transition as in 87~b, the only other possible experiment is measurement of &longitudinal polarization. The very slow decay rate of 87Rb does not permit any meaningful measurement of this observable. We consider therefore expectations for the matrix element ratios in the single-particle model. For a g; + p+ transition, we have *, x = - 0.83 and z = 3.8, while for a g{ -+ f; transition, x = - 0.31 and z = - 3.4. In view of the above expectations, we may make the following observations. (i) The semiempirical estimate of A is not at all consistent with description of the decay as a single-particle transition. (ii) Even though experimental data allow description of the decay as a g+ - f; transition, the value of A in this case (-4.2) seems to be too large to account for. (iii) The range of values of x (- 1.25
Volume 28B, number 7
PHYSICS
LETTERS
The experimental value * (6.3 x 1017 set) indicates a retardation of the decay by a factor of 40. Such a reduction of the decay rate is not unusual if configuration mixing is significant. (vii) Finally we note that the physical situation in this case as represented by the above observations seems to be strikingly similar to the findings of Lipnik and Sunier [ll] on some second forbidden /I-transitions.
2. 3. 4. 5. 6.
It is a pleasure to thank Dr. J. I. Fujita for reading the manuscript and Drs. A. R. Quinton and S. R. Lin for helpful discussions. The assistance of N. K. Gnpta and .I. R. Horgan is deeply appreciated.
7. 8. 9. 10.
* Corrected for non-statistical shape of @spectrum. 1. S. C. Curran, D. Dixon and H. W. Wilson, 84 (1951) 151;
11.
Phys. Rev.
**
464
***
20 January 1969
G. M. Lewis, Phil.Mag. 43 (1952) 1070; M. H. MacGregor and M. L. Wiedenbeck. Phvs. I Rev. 94 (1954) 138K. F. Flynn and L. E. Glendenin, Phys. Rev. 116 (1959) 744. K. Egelkraut and H. Leutz, Z. Phvs. 161 (1961) 13. M. Gorita, J. I. Fujita and M. Yamada, Progr.‘Theoret. Phys. (Kyoto) 10 (1953) 630. E. Greuling; Whys. Rev. 61’ (1942)568. M. Yamada, Progr. Theoret. Phys. (Kyoto) 9 (1953) 268. J. I. Fuj ita, Progr. Theoret. Phys. (Kyoto) 28 (1962) 338. T. Ahrens and E. Feenberg, Phys. Rev. 86 (1952) 64. J. Damgaard and A. Winther, Phys. Letters 23 (1966) 345. D. L. Pursey, Phil. Mag. 42 (1951) 1193. M. E. Rose and R. K. Osborn, Phys. Rev. 93 (1954) 1326. P. Lipnik and J. W. Sunier, Phys. Letters 7 (1963) 53; Phys. Rev. 145 (1966) 746.
28B, number
7
PHYSICS
ON THE
THIRD
LETTERS
FORBIDDEN
20 January 1969
,V-DECAY
OF
87Rb
*
K. S. R. SASTRY Department
of Physics
and Astronomy,
University
Received
of Massachusetts,
29 November
Amherst,
Mass.
1968
It is shown that the shape of the third forbidden &decay of 87Rb may be satisfactorily explained by the single-particle shell model and the conserved vector current theory of weak interactions.
That the third forbidden non-unique P-decay of 87Rb ($- + !+, Wg = 275 keV, logfl. = 17.6) displays a strongly energy dependent shape [ 1,2] has been well known for a long time. Such characteristics of forbidden b-spectra were investigated earlier [3] to obtain information on the P-interaction. The purpose of this work is to seek an interpretation of the shape of the &spectrum of 87Rb in terms of nuclear matrix elements, which, in the notation of Greuling [4], are CVQ3(r, r),
CyiQda,r), CAiQQ(Ux r,r)and CAQ~(U,r).
Another aspect of particular interest is the relation [ 51 iQ3(a,
r) = - ($)AlQ3(r, r)
(1)
between matrix elements arising from the vector interaction, where < is the usual Coulomb energy factor (~2/2R. For a given decay, A is a constant but its value is estimated to be different by different investigators. The disagreement stems from the importance of nuclear potential contribution. For 87Rb, A is 2.3 according to Fujita [6] and is independent of nuclear potential as required by the conserved vector current theory. On the contrary, the semi-empirical estimate of Ahrens and Feenberg [7] gives a value of 0.9. The above estimates are believed to be accurate to about 10%. Recently Damgaard and Winther [8] questioned the accuracy of the approximation of Ahrens and Feenberg [7] used in obtaining the above estimates. Their method of evaluation of A requires specific knowledge of radial wave functions of the nuclear states involved in the transition. Since the decay of 87~b involves transformation of the 50th neutron to the 38th proton in 87Sr, it may be characterized as a lgf * 2pf or lg; d If; single-particle transition. The cor* Supported by the U.S. Army Research ham, North Carolina.
462
Office,
Dur-
responding values of A are evaluated to be 1.8 and 1.9 using harmonic oscillator wave functions. The experimental data of Egelkraut and Leutz [2] on the shape factor (see fig. 1) are used in this analysis. From the work of Greuling [4] and Pursey [9], the complete theoretical expression for C(W), the shape factor, may be written as (2)
C(W) OT
a [Z’lx2 + Tzy2+ T3 + 2T4xy + 2T5y + 2Tgx + T7z2] where x,y, z respectively represent ratios of the matrix elements CVQ3( r, r), CViQs(a, r) and CAQ4(U, r) to CAi@(aX r, r), which is determin-
6 Fig. 1. Energy dependence of the shape of the third forbidden/?-spectrum of 87Rb. The solid points (normalized to unity at W = 1.4) represent the data of Egelkraut and Leutz used in the analysis. The bars indicate estimated errors in interpolation of data. The solid curve is the shape factor predicted by eq. (2) with x = -0.8, z = 3.8 and A = 2.1.
Volume 28B, number
PHYSICS
7
ed by the fl value corrected for the nonstatistical shape of the &spectrum. The coefficients Ti are functions of neutrino momentum and combinations of electron radial wave functions. In the evaluation of these coefficients the approximation (a,2)2 <<1 is used. For convenience, the number of unknowns in eq. (2) is reduced to two (x and y) by ignoring the last term. Such a procedure is not unreasonable since the coefficient TV is negligibly small. To define the set of parameters x and y consistent with experimental data, ratios of experimental shape factors at different energies are equated to the corresponding theoretical ratios. The resulting solutions are conic sections in the x-y plane, and are shown in fig. 2. The relationship between y and x as given by eq. (1) is shown by the straight lines for different expectations of
-50 Y -25 0
5
I
:= -0.31
I :=-0.83
IO
15 /
Fig. 2. Plot of the parameters x and y allowed by experimental data. The lower figure is the mirror guadrant of the upper figure. For convenience, allowed solutions of y for positive values of x are plotted in the upper figure, while the lower figure displays allowed values of y for negative x. The shaded region represents spread of the parameters. The straight lines given by y = - ($)h
LETTERS
20 January 1969
A. The conventional estimate of Ahrens and Feen berg indicates agreement with experiment for x N +3.7, while the CVC estimates of Fujita as well as Damgaard and Winther give x = - 1. A distinction between the two possibilities requires additional information. For a ground state transition as in 87~b, the only other possible experiment is measurement of &longitudinal polarization. The very slow decay rate of 87Rb does not permit any meaningful measurement of this observable. We consider therefore expectations for the matrix element ratios in the single-particle model. For a g; + p+ transition, we have *, x = - 0.83 and z = 3.8, while for a g{ -+ f; transition, x = - 0.31 and z = - 3.4. In view of the above expectations, we may make the following observations. (i) The semiempirical estimate of A is not at all consistent with description of the decay as a single-particle transition. (ii) Even though experimental data allow description of the decay as a g+ - f; transition, the value of A in this case (-4.2) seems to be too large to account for. (iii) The range of values of x (- 1.25
Volume 28B, number 7
PHYSICS
LETTERS
The experimental value * (6.3 x 1017 set) indicates a retardation of the decay by a factor of 40. Such a reduction of the decay rate is not unusual if configuration mixing is significant. (vii) Finally we note that the physical situation in this case as represented by the above observations seems to be strikingly similar to the findings of Lipnik and Sunier [ll] on some second forbidden /I-transitions.
2. 3. 4. 5. 6.
It is a pleasure to thank Dr. J. I. Fujita for reading the manuscript and Drs. A. R. Quinton and S. R. Lin for helpful discussions. The assistance of N. K. Gnpta and .I. R. Horgan is deeply appreciated.
7. 8. 9. 10.
* Corrected for non-statistical shape of @spectrum. 1. S. C. Curran, D. Dixon and H. W. Wilson, 84 (1951) 151;
11.
Phys. Rev.
**
464
***
20 January 1969
G. M. Lewis, Phil.Mag. 43 (1952) 1070; M. H. MacGregor and M. L. Wiedenbeck. Phvs. I Rev. 94 (1954) 138K. F. Flynn and L. E. Glendenin, Phys. Rev. 116 (1959) 744. K. Egelkraut and H. Leutz, Z. Phvs. 161 (1961) 13. M. Gorita, J. I. Fujita and M. Yamada, Progr.‘Theoret. Phys. (Kyoto) 10 (1953) 630. E. Greuling; Whys. Rev. 61’ (1942)568. M. Yamada, Progr. Theoret. Phys. (Kyoto) 9 (1953) 268. J. I. Fuj ita, Progr. Theoret. Phys. (Kyoto) 28 (1962) 338. T. Ahrens and E. Feenberg, Phys. Rev. 86 (1952) 64. J. Damgaard and A. Winther, Phys. Letters 23 (1966) 345. D. L. Pursey, Phil. Mag. 42 (1951) 1193. M. E. Rose and R. K. Osborn, Phys. Rev. 93 (1954) 1326. P. Lipnik and J. W. Sunier, Phys. Letters 7 (1963) 53; Phys. Rev. 145 (1966) 746.