Correlations between magnetic moments and β decays of mirror nuclei

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1 PHYSICAL REVIEW C, (8) Correlations between magnetic moments β decays of mirror nuclei S. M. Perez,, W. A. Richter, B. A. Brown, M. Horoi Department of Physics, University of Cape Town, Private Bag, Rondebosch, South Africa ithemba LABS, P. O. Box, Somerset West 9, South Africa Department of Physics, University of the Western Cape, Private Bag X, Bellville, South Africa Department of Physics Astronomy, National Superconducting Cyclotron Laboratory, Michigan State University, East Lansing, Michigan 88-, USA Department of Physics, Central Michigan University, Mount Pleasant, Michigan 889, USA (Received March 8; published June 8) We examine the magnetic moments β-decay lifetimes of light T = mirror nuclei obtain very tight correlations between these quantities by making use of shell-model estimates for small quantities. Using the information thus obtained, we predict values for some unknown magnetic moments of heavier T = mirror nuclei. Correlations for the magnetic moments of some T = mirror nuclei are also included. The effective operators required to reproduce the data are discussed compared to those obtained with other methods. DOI:./PhysRevC.. PACS number(s):..hc,..ky,..bw,..+z I. INTRODUCTION The data on magnetic moments of the ground states of light mirror pairs, as well as on the corresponding superallowed β-decay lifetimes, summarized in Ref. [], have increased in quantity quality in recent years [ ]. These improvements are, however, becoming more difficult to make as the mass increases the mirror pairs fall away from the line of stability. It is thus of interest to have accurate predictions of missing information, conventionally obtained from large scale shell-model calculations [ ]. We employ here a different method of generating these predictions [,] by making a novel use of shell-model estimates for small quantities. This results in plots that show strong linear relationships between the data for magnetic moments Gamow-Teller β-decay matrix elements of mirror nuclei with T = /. When isospin is conserved the magnetic moments of mirror nuclei (in units of µ N ) can be written as [] µ p = g p J + (G p g p )(S o S e ) + [(G p g p + G n g n )S e (g p g n )J e ] () µ n = g n J + (G n g n )(S o S e ) + [(G p g p + G n g n )S e + (g p g n )J e ], () where µ p µ n are the magnetic moments of the oddproton odd-neutron members of the odd-even mirror pair. The free-nucleon values of the g factors are g p =.,g n =.,G p =.8,G n =.8. S e/o J e/o are the contributions from the even/odd type of nucleon to the z components of the total spin S total angular momentum J of the mirror pair. Because the quantities S e J e are small [], we rewrite Eqs. () () in terms of corrections to the gyromagnetic ratios /n = µ p/n /J in the form ( + ) = g p + (G p g p )(S o S e )/J () ( + ) = g n + (G n g n )(S o S e )/J, () where = [(G p g p + G n g n )S e (g p g n )J e ]/J () = [(G p g p + G n g n )S e + (g p g n )J e ]/J. () Eliminating (S o S e )fromeqs.() () we find ( + ) = α( + ) + β, () with α = (G p g p )/(G n g n ) β = g p αg n. For T = mirror pairs, the quantity (S o S e ) is related to the Gamow-Teller matrix element for the cross-over β decay obtained from the ft value: S o S e J = γ β R, (8) where ( γ β = ) ft J (J + ), (9) with a sign that can be determined from systematics [,]. The free-nucleon value for R is R = C A /C V =.. Thus for T = mirror pairs, Eqs. () () can be written as ( + ) = g p + G p g p γ β () R ( + ) = g n + G n g n γ β. () R We assume that for nucleons in nuclei we may replace the free-space g factors the ratio R = C A /C V byaset of effective values denoted by G, g, R. (The effective operator may also include terms proportional to the rank-one operator [Y σ ]. But based on the previous analysis of the size of these contributions [] for magnetic moments, they -8/8/()/() - 8 The American Physical Society

2 S. M. PEREZ, W. A. RICHTER, B. A. BROWN, AND M. HOROI PHYSICAL REVIEW C, (8) are much smaller than the terms we do consider are not included.) We use the linear relations of Eqs. () () to determine the parameters g ( G g)/ R for protons neutrons from least-square fits to γ + γ γ β. In addition we use Eq. () to make a fit for the parameters α β. Three levels of approximations for the γ terms are considered: (A) when the γ are set to zero, (B) when the γ are evaluated with Se SM obtained with shell-model calculations with free-nucleon Je SM values for G g, (C) with Se SM Je SM together with G SM g SM obtained from fits of the shell-model matrix elements to experimental magnetic moments in the sd-shell, discussed in the next section. II. CORRECTIONS FROM SHELL-MODEL CALCULATIONS In Table I we list the T = / mirror nuclei in the mass range A for which a complete set of data on,, γ β exists, together with the values of the corrections obtained from hω shell-model calculations based on the Hamiltonians from Refs. [8 ] free-nucleon values for the g factors. We have omitted nuclei with A in the analysis as the various coupling constants may not have reached their fully quenched values for these nuclei []. For the p fp shells the corrections were based, respectively, on the (8 )CKPOT interaction [9] the GXFP interaction []. For the sd (sd) shell we use the recent USDB Hamiltonian [8]. The pf -shell calculations for A = were calculated with a truncated model space (t) that allowed at most six nucleons to be excited from the f / orbitals to any of the f /,p / or p / orbitals. To check the error from this truncation we ( + ) ( + ) ( ) ( ) (B) (A) FIG.. (Top) ( + )vsγ β (solid circles) ( + ) vs γ β (open circles). The lines are the result of fit (B). (Bottom) vs γ β (solid circles) vs γ β (open circles). The lines are the result of fit (A). The single-particle model using free-nucleon values for the coupling constants is shown by the dashed line. γ β TABLE I. Values of γ β,, [] obtained using data on magnetic dipole moments β-decay lifetimes [ 8]. The contributions have been estimated from hω shell-model calculations [8 ] freenucleon values for the g factors. A, J π γ β Nucleus Nucleus, / +.9() B +.9 C.() +.8.8, /.() N.(8) C , /.() N. O +.9().., / + +.() F +.888() O... 9, / + +.9() 9 F +. 9 Ne.8().9 +., / + +.8() Na +.99() Ne , / + +.() Na +.8 Mg.() +.., / + +.() Al +.8() Mg , / + +.(8) Al +. Si.() , / + +.() 9 P +.98() 9 Si , / + +.() P +.() S.99() +.., / +.(8) Cl +.() S , / +.8() Cl +.9 Ar +.() +.8., / +.() K +. Ar +.() +.. 9, / +.() 9 K +. 9 Ca +.8().., / +.() Sc +.8() Ca..., / +.() Sc +.() Ti.8()

3 CORRELATIONS BETWEEN MAGNETIC MOMENTS AND... PHYSICAL REVIEW C, (8) TABLE II. Result of a linear fit of Eq. () to the data of Table I with the three prescriptions described in the text for the even-spin correction terms. The last column gives the rms deviation between the left-h right-h sides of Eq. (). g p ( G p g p )/ R rms Free-nucleon. Fit (A).9 ±.. ±.9. Fit (B). ±.. ±..8 Fit (C).9 ±.9. ±..9 performed a full-space calculation for A =. The results with free-nucleon g factors for the truncated (full) space calculations are =.8(.) =.(.8). The difference is an order of magnitude smaller that the average difference between experiment theory showing that the t truncation is adequate. The values for G SM g SM for fit (C) are obtained from a least-squares fit to magnetic moments M γ -decay matrix elements in the sd shell []: G SM p =. ±., g p SM =. ±., () G SM n =. ±., g n SM =. ±.. In addition, from a fit to Gamow-Teller β-decay matrix elements in the sd shell [], R SM =.9 ±.. () The errors on these parameters include the variation between the fits with the USDA USDB Hamiltonians [8]. Similar results were obtained with the older USD Hamiltonian []. III. RESULTS Tables II IV show the results of fitting the data of Table I. We note an improvement of approximately % in the rms obtained when the even-spin corrections are included. Figure shows a plot of Eqs. () () with assumptions (A) (B) for the even-spin terms. Comparison of the bottom (case A without even-spin terms) top (case B with even-spin terms) shows that the overall trend is determined by the dominant odd-spin contribution as assumed in the original work of Buck, Merchant, Perez [] Buck Perez []. However, the results of Tables II to IV show that the inclusion of the even-spin terms has a TABLE III. Caption as for Table II with Eq. () Eq. (). g n ( G n g n )/ R rms Free-nucleon. Fit (A). ±..99 ±.9.9 Fit (B). ±.. ±.. Fit (C).9 ±.9. ±.. TABLE IV. Caption as for Table II with Eq. () Eq. (). α β rms Free-nucleon.99 Fit (A). ±.. ±.. Fit (B). ±.. ±.. Fit (C). ±.. ±.. non-negligible effect on the linear coefficients extracted in the analysis. Figure shows the plot of vs from Eq. (). For this plot the contribution of the even-spin terms has the effect of moving the points along the line resulting in essentially the same slope the similar linear coefficients for cases (A) (B) (B) (A) FIG.. (Top) + versus +. The line is the result of fit (B). (Bottom) versus. The line is the result of fit (A). The single-particle model using free-nucleon values for the coupling constants is shown by the dashed line. -

4 S. M. PEREZ, W. A. RICHTER, B. A. BROWN, AND M. HOROI PHYSICAL REVIEW C, (8) TABLE V. Values of γ β,, [] obtained using data on magnetic dipole moments β-decay lifetimes [9,,]. The contributions have been calculated from hω shell-model calculations [] free-nucleon values for the g factors. A, J π γ β Nucleus Nucleus, / +.8() V Ti. a +..9, / +.(8) V Cr +.. 9, / +.9() 9 Mn 9 Cr.9() a +..9, / +.() Mn +.() Fe +.8., / +.8() Co Fe +.., / +.8() Co +.(9) Ni +.9.8, / +.() Cu.() Ni. +.. a Signs for are experimentally undetermined. The fits based on Eqs. () () only give g the combination of parameters ( G g)/ R. The g values for the orbital-g factors from Tables II III (fit C) are g p =.9 ±.9 g n =.9 ±.9, respectively. These results are not inconsistent (the error bars just touch) with the global magnetic moment fit values from Eq. () of g p SM =. ±. g n SM =. ±., respectively. We can use the G SM g SM values from Eq. () in ( G g)/ R from fit (C) in Tables II III to obtain R =.99() from the proton data in Table II R =.98() from the neutron data in Table III. These values are nicely consistent with each other with the earlier result R =.() []. These results are also consistent with the quenching of Gamow-Teller strength obtained from the direct comparison of calculated experimental Gamow-Teller β-decay matrix elements with USD [] USDA USDB [Eq. ()]. The implication is that C A /C V is quenched from its freenucleon value of. to its stard-model value of unity in nuclei. Taking R = for the smaller orbital part of ( G g)/ R, we find from Tables II III (fit C) G p / R =.8() G n / R =.9(). The difference ( G p G n )/ R = 8.9(8) can be compared to the free-nucleon value of.. The nucleonic operator associated with G p G n R is the same (the isovector spin operator). Hence, the % enhancement of experiment over the free-nucleon value must be attributed to mesonic exchange currents []. IV. PREDICTIONS FOR HEAVIER NUCLEI For completeness we show in Fig. the results for all data related to the vs plot. In addition to the points for the A =,T = / mirror pairs it includes the points for the A = A = mirror pairs, the recent result for the A = mirror pair [9], the four T = / mirror pairs [,]. Whenever one of the quantities γ β,,or is known the remaining ones can be deduced from Eqs. (), (), () by using appropriate values for the even-spin corrections the effective g factors from Tables II to IV. As an example we use the data of Table V, approximation B for γ, Eqs. () () to predict values of from the known values of γ β for several nuclei in the mass range A = to. The results are shown in Table VI. We note that whenever have in fact been measured a comparison of experimental predicted values from Tables V VI, respectively, gives an indication of the reliability of the method. We note here the relatively large deviation of the predicted value of from the measured value of =. ±. for Cu. As a further example we use the data on A =,, in Table V to predict the values of using Eq. (). The values of α β are taken from Table IV, fit (B). This gives =. ±. ( Fe), =. ±. ( Ni), =+.98 ±. ( Cu). The relatively large deviation between our TABLE VI. Values of deduced from the data of Table V, using Eqs. () () using methods (A) (B). A, J π Nucleus (A) (B) Nucleus (A) (B), / V +.8() +.() Ti.().(), / V +.8() +.() Cr.().() 9, / 9 Mn +.() +.() 9 Cr.().(), / Mn +.(8) +.9(8) Fe.8().8(), / Co +.9() +.88() Fe.8().(), / Co +.9(8) +.(8) Ni.8().9(), / Cu +.(8) +.(8) Ni.().() -

5 CORRELATIONS BETWEEN MAGNETIC MOMENTS AND... PHYSICAL REVIEW C, (8) FIG.. versus. Points are shown for the A =,T = / mirror pairs (solid circles); the A =,T = / mirror pair [open circle near (-,)]; the A = A =,T = / mirror pairs (open circles on the upper left-h side); the A = 9,,,, T = / mirror pairs (crosses). The line is the result of fit (A). results experiment for Cu has also been noted above. V. CONCLUSIONS In conclusion we have extended previous analyses of the linear correlations found between the magnetic dipole moments β-decay lifetimes of light T = mirror pairs. This has been done by explicitly including the contributions S e J e to the total spin total angular momentum generated by the even type of nucleon in these odd-even nuclei. The inclusion of these contributions via a hω shell model has led to improved linear fits. The results we present can be used to make predictions. For example, if the β-decay matrix element is known for a heavy T = mirror pair, then Fig. Tables II III can be used to predict the odd-proton odd-neutron magnetic moments. If the magnetic moment of one member of a mirror pair is known, then Fig. Table IV can be used to predict the other member. This can be used for any T value as illustrated by the results for T = shown in Fig.. ACKNOWLEDGMENTS This work is partly supported by NSF Grant PHY-, NSF/MRI Grant PHY-9, the National Research Foundation of South Africa under Grant,,. [] B. Buck, A. C. Merchant, S. M. Perez, Phys. Rev. C, (). [] S. Raman et al., At. Data Nucl. Data Tables, (98). [] M. Fukuda et al., Phys. Lett. B, 8 (99). []W.F.Rogerset al., Phys. Lett. B, 9 (98). [] J. Vanhaverbeke et al., Hyperfine Interact., (988). [] K. Matsuta et al., Hyperfine Interact. 8, (99). [] P. Raghaven, At. Data Nucl. Data Tables, 89 (989). [8] N. J. Stone (private communication). [9] K. Minamisono et al., Phys. Rev. Lett. 9, (). [] W. Geithner et al., Phys. Rev. C, (). [] T. J. Mertzimekis, P. F. Mantica, A. D. Davies, S. N. Liddick, B. E. Tomlin, Phys. Rev. C, 8 (). [] I. S. Towner F. C. Khanna, Nucl. Phys. A99, (98). [] B. A. Brown B. H. Wildenthal, Phys. Rev. C 8, 9 (98). [] B. A. Brown B. H. Wildenthal, At. Data Nucl. Data Tables, (98). [] B. Buck S. M. Perez, Phys. Rev. Lett., 9 (98). [] C. F. Clement S. M. Perez, Phys. Lett. B8, 9 (99). [] B. A. Brown B. H. Wildenthal, Nucl. Phys. A, 9 (98). [8] B. A. Brown W. A. Richter, Phys. Rev. C, (). [9] S. Cohen D. Kurath, Nucl. Phys., (9). [] M. Honma, T. Otsuka, B. A. Brown, T. Mizusaki, Phys. Rev. C, (). [] B. Buck S. M. Perez, Phys. Rev. Lett., 9 (98). [] B. A. Brown W. A. Richter (unpublished). []T.W.Burrows,J.W.Olness,D.E.Alburger,Phys.Rev.C, 9 (98). [] D. R. Semon, M. C. Allen, H. Dejbakhsh, C. A. Gagliardi, S. E. Hale, J. Jiang, L. Trache, R. E. Tribble, S. J. Yennello, H. M. Xu, X. G. Zhou, B. A. Brown, Phys. Rev. C, 9 (99). -

Nuclear structure input for rp-process rate calculations in the sd shell

Nuclear structure input for rp-process rate calculations in the sd shell W A RICHTER ITHEMBA LABS UNIVERSITY OF THE WESTERN CAPE B A BROWN NSCL, MICHIGAN STATE UNIVERSITY This work is supported by the