Nuclear electromagnetic moments of the ground states of148Pm and210Bi calculated with phenomenological wave functions derived from analyses ofβ-decay experiments

The magnetic dipole and electric quadrupole moments of the ground states of148Pm and210Bi are evaluated with phenomenological wave functions derived fromβ-decay studies published in previous works. It is found that these wave functions account satisfactorily for the experimental data of both nuclear moments of the210Bi ground state. In the case of148Pm, while the calculated value of the electric quadrupole moment is not inconsistent with the experimental data, a strong disagreement between theory and experiment is found for the magnetic dipole moment. We attribute this failure to the use of a too small configuration space for the expansion of the nuclear wave function of148Pm.


Introduction
Some time ago phenomenological nuclear wave functions for the ground states of the odd-odd l~Spm and 21~ nuclides were derived from studies of the corresponding ground-state-to-ground-state first-forbidden fi-transitions [1,2].The wave functions obtained in [1,2] are capable of describing very well the ground-state-to-ground-state fi-decays of these isotopes.More recently, the wave function singled out in Ref. [1] has been used to calculate fi-observables corresponding to the decay from the groundstate of 14SPm to the first excited state 2 + in the even-even l~SSm nucleus [3].tn this latter work [3] the success was only partial.Although the predictions for energy dependent observables like the spectrum shape factor, the longitudinal polarization of the electrons and some data on the fi-7 angular 1 Fellow of the Consejo Nacional de Investigaciones Cientificas y T~cnicas (CONICET) of Argentina 2 Fellow of the Alexander yon Humboldt Foundation.Member of the Carrera del Investigador Cientifico of CONICET of Argentina.On leave of absence from the Departamento de Fisica, Facultad de Ciencias Exactas, Universidad Nacional de La Plata, Argentina correlations are in agreement with experimental data, the calculation fails to reproduce the relatively large logft value of this 1--->2 + fi-transition.However, a priori, this failure cannot be attached exclusively to the description of the ground state 1-of 14Spm since some degree of uncertainty can also be ascribed to the model adopted for the first excited state 2 + in 148Sm.Hitherto no attempt to use these nuclear wave functions in order to estimate any other property of the ground states of 14Spm and 21~ has been done.Indeed, since the phenomenological wave functions derived from experimental data of fi-observables should already contain information about some residual processes, e.g.core polarization effects, it is very interesting to calculate the static electromagnetic moments using these wave functions.For both nuclides there are data of magnetic dipole, #, and electric quadrupole, Q, moments [4][5][6][7] (cf.Table 1).In the case of 21~ there are also shell-model predictions [7,8] evaluated using the Kim-Rasmussen's [9] and Kuo-Herling's [10] wave functions in conjunction with empirical gyromagnetic factors and E2 effective charges.These wave The error of # in the calculation comes from the uncertainties in the core polarization correction, which is taken as 20 % in each case, and the errors of the orbital g-factors g~ff quoted in the text functions [9,10] were obtained considering complex admixtures of various shell-model states into the 1ground state.Thus the aim of the present work is to perform an additional test of the phenomenological wave functions reported in Refs.[1,2].To this purpose we investigate the extent to which these wave functions are capable of accounting for nuclear moments of the ground states of ~48pm and 21~ and also compare the theoretical results with other available calculations.In Sect. 2 the relevant formulas are given.The analysis and the results are provided in Sect.3. Finally the main conclusions are summarized in Sect. 4.

The Relevant Formulas
We assume that the wave function of the ground state J=l-of a doubly odd nucleus can be expressed as [Nucleus; J = 1 -, M) = .~. Cj, j.IJp,J,; J = 1 -, M) JpJn =~ Cp. lp, n; J=l-,m), pn (1) where jp and j, denote proton and neutron quasipartides states and single-particle states in the case of 148pm and 21~ respectively.The amplitudes Cv,  2. The static nuclear electromagnetic moments of a state with spin J described by the nuclear wave function of (1) can be evaluated according to where p(n) stands always for jp(j,) and the brakets ( ) and { } are the usual 3-j and 6-j symbols.The one-body matrix elements are given by m~= l(]'J) for the magnetic dipole moment and  2 has, for neither protons nor neutrons, contributions of both spin-orbit partners (j = 1 + 1/2 and j = l-1/2) of any single-particle orbital.This fact insures that there exists no off-diagonal contribution to the magnetic dipole moment given by ( 2) and (3).Therefore in this case the expression for the total magnetic dipole moment reduces to /~(,4Spm)= ~ 2 " ' Cpn~J = l(.Jp,Jn). ( pn Here the magnetic dipole moment of a system consisting of a proton and a neutron whose angular momenta, jp and j,, are coupled to resultant J is given by the shell-model additivity relation where /~(]p) and #(j.) are the magnetic dipole moments, in the single-particle states indicated, of a proton and a neutron respectively.It is usual to write down the single-particle #(]) following the Schmidt model [12] #(J) = gsJ = [gt -+ (gs -gt)/(21 + 1)]j, (7) where gj is the single-particle gyromagnetic factor (g-factor), gz is the orbital g-factor which takes the value 1 (0) for the proton (neutron) and g, is the intrinsic or spin g-factor being 5.586 (-3.826) for the proton (neutron), all quantities are expressed in units of nuclear magnetons (nm) #N=eh/2MpC.The plus (minus) sign corresponds to j = 1 + 1/2 (j = 1 -1/2).The residual effects due to the core polarization and the mesonic exchange current can be described in terms of renormalized gt and g,.The reader can find several papers on this subject in the proceedings of the topical conference, Osaka 1972, Ref. [13].For 21~ however, there are, in addition to the diagonal terms, contributions containing off-diagonal matrix elements like, e.g., (v 2g9/211 J{(M1) ][v297/2}.It is possible to demonstrate that this kind of off-diagonal matrix element can be calculated according to rnx= l(]'J) = (4=/3) */2 (,J' =J + 1 I1V//(M1) IkJ) = g(gs-g,) The electric quadrupole moment for both isotopes, *4Spm and 2*~ has several diagonal as well as offdiagonal terms therefore the complete equation ( 2) should be used.The corresponding one-body matrix elements given by ( 4) can be written Here e eee is the E2 effective charge of the singleparticle orbital.

Analysis and Results
The nuclear moments were evaluated for all the sets of coefficients C_, listed in Table 2. Two sets of Cp, were used for *4~Pm, these are listed as Set I and II in Table3 of Ref. El].The numerical values are quoted in Table2 as Set A and B, respectively 9 Each set corresponds to one of the two solutions of a quadratic equation solved in Ref. [9].Three sets of Cp, were used for 2*~ and these are listed as Case A, B and C in Table 3 of Ref. [2].The numerical values are quoted here in Table2 as Set A, B and C, respectively.These three sets correspond to different approaches used in Ref. [2] for the evaluation of the radial integrals of the single-particle fl-operators.
For Set A harmonic oscillator wave functions were used.Sets B and C were obtained from Woods-Saxon wave functions derived from parameters reported by Rost [14] and Batty and Greenlees [15], respectively (for more details see Ref. [2]).The radial integrals were evaluated using the harmonic oscillator parameter hooo=41A -1/3 MeV which yields a length parameter bo=l.OlOA*/6fm.
For Sets B and C the radial integrals were calculat- a These results correspond to the pure Schmidt-model [12] b An effective sping-factor g~ff=(2/3)g s was used c In addition tO gs elf used in case CP the effective orbital g-factors: gTff(z~)=l.13 and g~ff(v)= -0.02 from Ref. [18] were used a These results were taken from Kisslinger-Sorensen [24] 9 The experimental data were taken from the survey of Ref. [21] f An average of the results obtained in Ref. [24] for the v2fv/2 orbit and the value for #(v lh9/2) quoted in column CP + MEC were used to calculate/@48Pro) g Averages of the experimental single-particle data and the value #(vlh9/2) from column CP+MEC were used to calculate the total #(148pm) ed taking into account the corresponding Woods-Saxon wave functions.The occupation factors Vj for 148pm were taken from the solutions of the gap equations obtained with the parameters quoted in Ref. [3].For both nuclides the magnetic dipole moments were calculated using three different approaches for the single-particle contributions #(j) as given by (7).
The first choice was to use the g-factors corresponding to the pure Schmidt model [12] mentioned above.The second choice was to keep the orbital gfactors g~ at the Schmidt model values and to assume a renormalization, equal for protons and neutrons, of the spin g-factors, g~.For this purpose, an effective g-factor equal to g]ff=(2/3)g~ was used.This value is an average of the amount of renormalization due to the core polarization observed for spherical as well as for deformed nuclei (see Ref. [13] and p. 304 of Bohr and Mottelson in Ref. [16]).
For the third choice, in addition to the core polarization, the fact that the orbital g-factors, gl, are modified by the mesonic exchange current as suggested by Chemtob [17] was taken into account.The values gTff0z)=1.13_+0.02and g~ff(v)=-0.02++0.03 published by Yamazaki [18] were used.It should be noted that the mesonic effect on the gfactor gs is very small, see e.g.Arima [19].Therefore, it was neglected in our analysis.Although the values for g~fe taken from Ref. [18] have been obtained from the analysis of the g-factors in the 2~ region we have also used them for 148Pm since following Nagamiya and Yamazaki [20] the renorrealization of the orbital g-factor is similar for all mass number A.
All values of the single-particle #(j) obtained under the abovementioned approaches are listed in Tables 3 and 4. In addition, as a guide to the quality of the calculated single-particle #(j) we also quote the available data for the neighboring odd nuclei [8, 21-

233.
A glance at Tables 3 and 4 indicates large corrections in the direction towards experiment when the effective g-factors are used.In particular, for the A=209 nuclei, 2~ and 2~ the agreement between theoretical results and experimental data is excellent.In the case of the neighbors of 14Spm the agreement is not so good.This latter result can be attributed to the fact that such nuclear states are not pure single-particle states but contain admixtures of phonon states.Kisslinger and Sorensen [24] have  [8] calculated these magnetic dipole moments in the framework of the pairing-plus-quadrupole model, we have included their results in Table 3.It is clear that the inclusion of phonons improves the agreement with experimental data especially for the ~ 2d5/2 and v 2f7/2 states.
The calculated total magnetic dipole moments are also included in Tables 3 and 4. For all approaches, the theoretical results for #(148pm) have the opposite sign to the experimental data.For the sake of completeness we also evaluated the magnetic dipole moment using: (a) the 'single-particle' values obtained in Ref. [24] and (b) the straight averages of the experimental 'single-particle' values listed in Table 3.These results are included in Table3.Even with these realistic 'single-particle' values the calculated results show large discrepancies with the measured data, though they exhibit a trend towards the experiment.In all the cases the sign is fixed by the sign of #j=l(zt2ds/2,v2fT/a) which always makes the main contribution.It is clear that to reverse the sign one needs those other components in the wave function which heretofore have been neglected in the analysis as performed in Ref. [1].To facilitate the comparison we list in Table 1 the results for #(~4SPm) obtained with the experimental 'single-particle' values.We should stress that the magnetic dipole moment of 21~ is very small, namely of the order of few hunderdths of #u.Therefore, considering that the theoretical #(Zl~ is obtained from incoherent admixtures of several contributions mostly larger than the experimental data, we can say that the agreement with the measured value achieved for the approach CP+MEC is good.To illustrate this statement it is helpful to estimate the theoretical errors due to the uncertainties in the effective g-factors.To. this end we used the errors of g~ff mentioned above and took into account an uncertainty of 20 ~o in gseff, this assumption agrees with that adopted by Baba et al. [8].It was found that the error is mainly given by the uncertainty in g~ff.To make possible a direct comparison, we include in Table 1 the results for the magnetic dipole moments obtained in the approach CP+ MEC together with the evaluated errors.A glance at Table 1 indicates that our predictions for #(21~ in Cases A and B are in agreement with the experimental data within the theoretical uncertainties, while in Case C the measured value lies slightly outside of the prediction.Moreover, our results are similar to that obtained in Refs.[7,8] using the wave functions of Refs.[9,10].However although we used exactly the same assumption to estimate the error of the magnetic dipole moment our result is much larger than that of Refs.[7,8].It might be that the error was underestimated in Ref. [8].As a final remark we note that #(21~ is very sensitive to the magnitude of the component [~ lh9/2 v2gT/2), which makes a contribution to (9) linear in the mixing amplitude and, together with the main configuration I~zlhg/2V2g9/2) , is the largest contributor to the magnetic dipole moment.In the discussion of the magnetic dipole moments we ignored the tensor correction [Y2" s]l induced by the residual interactions.This new operator gives the sole contribution in the case of M1 /-forbidden matrix elements.The inclusion of this term would complicate the analysis without changing the main conclusions.The electric quadrupole moments for both nuclides were calculated using one set, different for each nucleus, of effective charges for the proton and the neutron.For 148pm we used: e;ff=(2+_l)e for protons and e~ff=(1.0_+0.5)e.These values reflect the large uncertainties in the E2 effective charges in this transitional region (see, e.g., in Ref. [21] the large errors of the measured Q values).In the case of 21~ the values e~ff=(1.50_+0.06)efor protons and een ff =(0.42 +0.01)e for neutrons from the compilation of Blomqvist [-25] were adopted.The results are listed in Table 1, where the quoted errors due to the uncertainties in e elf are also indicated.The calculated value of the electric quadrupole moment of 148Pm is not inconsistent with the experimental one, while the agreement for Zl~ is quite good.Only in Case C does the prediction differ slightly from the measured value.For 14sPin the total electric quadrupole moment is given by an incoherent admixture of several contributions whereas for 2t~ it is basically determined by the main component of the wave function.

Conclusions
The main results obtained in the present work are summarized in Table 1.On the basis of these results we can state that the phenomenological wave functions derived for 21~ in Ref. [2] are very good.They not only provide a satisfactory explanation for all experimental data on /?-decay but, in addition, yield good results for the magnetic dipole and electric quadrupole moments.Moreover these predictions are completely equivalent to those obtained using the wave functions derived from nuclear forces [-9, 10].On the other hand the quality of the phenomenological wave function singled out for the ground state of 14~Pm in Ref. [-1] is much worse than that for 21~ Only the calculated value of the electric quadrupole moment is not inconsistent with the experimental one, the experimental magnetic dipole moment cannot be reproduced.Let us now inquire into the possible reasons for this feature.The nuclear wave functions of both nuclides were determined from the study of the spectrum shape factor, C~(W), and the longitudinal polarization of the electrons, PC(W).It is well known that when the experimental data of a first-forbidden/?-decay follow the characteristic behavior of an allowed/?-transitionthen, to obtain meaningful information about the nuclear structure, it is necessary to analyze many different kinds of/?-observables [-26].The spectrum shape factors exhibit in both cases a large deviation from the statistical shape.However, while the longitudinal electron polarization of 21~ shows a large departure from -v/c [-2] that of 148Pm, Pc(W), agrees with a full -v/c polarization [-1, 3].This latter fact allows one to determine a more complete wave function for 21~ than for 14SPin since more components can be fitted to the experimental data in order to reproduce the deviations from the allowed patterns.Thus we can state that the very simple wave function of the ground state of l*SPm is not capable of accounting for the experimental magnetic dipole moment and some properties of the fl-decay to the first excited state 2 + of ~4SSm.These failures can be attributed, at least partially, to the use of too small a configuration space for the expansion of the wave function.To improve this wave function it is necessary to include in the analysis more observables than those taken into account in Ref. [1].

Table l .
Experimental and theoretical nuclear moments of the ground state J = 1 -of 148Pm and 21~

Table 3 .
Single-particle contributions and total results for the magnetic dipole moment of the ground state of t48pm Table4.Single-particle contributions and total results for the magnetic dipole moment of the ground state of 2~~