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Magnetic dipole response in nuclei at the N=28 shell closure: a new look1
10 December 1998
Physics Letters B 443 Ž1998. 1–6
Magnetic dipole response in nuclei at the N s 28 shell closure: a new look 1 P. von Neumann-Cosel a , A. Poves b, J. Retamosa c , A. Richter
a
a
Institut fur ¨ Kernphysik, Technische UniÕersitat ¨ Darmstadt, D-64289 Darmstadt, Germany ´ Teorica, Departamento de Fisica UniÕersidad Autonoma de Madrid, 28049 Madrid, Spain ´ ´ ´ Atomica Departamento de Fisica y Nuclear, UniÕersidad Complutense, 28040 Madrid, Spain ´
b c
Received 5 August 1998 Editor: J.-P. Blaizot
Abstract The magnetic dipole response of N s 28 isotones is obtained for 48 Ca, 50 Ti, 52 Cr and 54 Fe by means of shell model calculations in the full pf-shell. Spin quenching factors derived in all four nuclei by comparison to experimental strengths from electron scattering agree well with each other. The deduced effective g-factor g seff s 0.75Ž2. g sfree is consistent with findings for the Gamow-Teller strength in this mass region. The ability of state-of-the-art shell-model calculations to describe the gross properties and the fragmentation of the complex experimental B Ž M 1. strength distributions is investigated. q 1998 Elsevier Science B.V. All rights reserved. PACS: 23.20.Js; 21.60.Cs; 27.40.q z Keywords: BŽM1. strength in N s 28 isotones; shell-model calculations; Quenching
The magnetic dipole response of nuclei represents a subject of intense interest to experimentalists and theorists for a variety of reasons. The properties of spin and convection currents in M 1 excitations serve as a thorough test of models aiming at a description of nuclear dynamics. Žsee e.g. w1–3x and references therein.. Furthermore, M 1 transitions provide a unique source to elucidate the role of non-nuclear degrees of freedom in complex nuclei w4–7x. Some famous examples are the neutron spin-flip excitations in 48 Ca, where the M 1 strength is essentially
1
Work supported by the DFG under Contract Ri 242r12-1 and by research grants from the DGES ŽSpain., Nos. PB96-53 and PB96-604.
concentrated in a single strong transition w8x, or the enhancement of the total M 1 strength in sd-shell nuclei due to meson exchange currents w9–11x. Finally, the degree of quenching and the details of the distributions of M 1 and of the closely related Gamow-Teller ŽGT. transitions in pf-shell nuclei strongly influence the late stages of a collapsing massive star as well nucleosynthesis processes induced by a supernova outburst w12x. The properties of magnetic dipole excitations in nuclei along the N s 28 shell gap provide a particularly intriguing example where the interplay of proton and neutron degrees of freedom can be studied in great detail w13x. Experimental data are available for 48 Ca, 50 Ti, 52 Cr and 54 Fe from electron scattering w14,16x. When moving from the doubly closed-shell
0370-2693r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 8 . 0 1 2 9 8 - 2
2 48
P. Õon Neumann-Cosel et al.r Physics Letters B 443 (1998) 1–6
Ca nucleus to an open proton 1 f 7r2 shell a complex picture of the total B Ž M 1. strength and its fragmentation emerges. Attempts to describe this fine structure in shell-model calculations Žrestricted at that time to two particle – two hole excitations from 1 f 7r2 to pf-shell configurations. failed completely w17x. Further, the total B Ž M 1. strengths were severely overestimated and it could not be decided whether this discrepancy was entirely due to the model space restrictions or implied unusually large quenching due to non-nucleonic degrees of freedom. Clearly, the problem remained unsolved w18x. In recent years considerable progress has been made in shell-model calculations, partly by the development of new methods such as a Monte Carlo approach w19x, but also because of the dramatic increase of computing power. For example, it is possible now to perform calculations for nuclei up to mass A f 52 allowing for the full pf model space w20,21x with great success in the description of basic structure features such as the GT strength in 48 Ca w22x or the buildup of a rotational band in the openshell nucleus 48 Cr w23x. These dramatic improvements ask for a new look at the problem of the magnetic dipole response in the N s 28 isotones with state-of-the-art shell-model calculations. We have performed such calculations in the full pf shell, using the code ANTOINE w24x and the effective interaction KB3. A detailed discussion of the shell-model techniques used can be found in w20x and a critical assessment of the interaction KB3 in w25x. While inclusion of the full pf shell is affordable now for the calculation of the ground state energies, wave functions and total magnetic strengths of the four nuclei investigated here, the determination of the strength functions in the full space is, at present, possible only for 48 Ca and 50 Ti. For 52 Cr and 54 Fe we have to limit ourselves to a calculation in which at most four particles are allowed to be excited from the 1 f 7r2 to the remaining pf-shell orbits Ža t s 4 truncation.. The total B Ž M 1. strengths in the full space, using bare g-factors, are 8.96 m2N , 12.95 m2N , 15.60 m2N and 18.90 m2N for 48 Ca, 50 Ti, 52 Cr and 54 Fe, respectively. We have computed separately the total spin and orbital contributions in order to confirm that the latter are very small Ž0.0. 0.7, 1.0 and 1.1 m 2N , respectively.. The strength going to states with isospin T) s T0 q 1, where T0
denotes the isospin of the ground state, increases with decreasing T0 Ž0.0, 0.30, 1.92 and 6.50 m2N . and becomes significant for 54 Fe. These results will be important for the discussion that follows. Note that a t s 4 truncation is already a very solid approximation to the final result, i.e. the resulting total strengths in 52 Cr and 54 Fe Ž15.81 and 19.25 m2N . are already very close to the results of calculations in the full model space. Experimentally, M 1 strength in the N s 28 isotones has been studied with high-resolution electron scattering under backward angles. Data are available for 48 Ca in an excitation energy range Ex s 8–13 MeV where a summed B Ž M 1.≠ strength of 5.3Ž6. m2N was found w14x. Investigations towards higher energies up to 17 MeV w15x reveal very little additional strength Žat most 0.5 m2N .. Results for 50 Ti have been reported in an interval Ex s 8–12 MeV and for 52 Cr, 54 Fe in the energy range Ex s 7–12 MeV. The data and the method for the extraction of the summed M 1 strengths are described in w16x. One finds B Ž M 1.≠ s 4.5Ž5. m2N Ž50 Ti., B Ž M 1.≠ s 8.1Ž8. m2N Ž52 Cr. and B Ž M 1.≠ s 6.6Ž4. m2N Ž54 Fe.. In a first step we use the shell-model results for an extraction of the quenching factors by normalizing the calculations to the data in the respective excitation energy ranges. As will be demonstrated below, such a procedure leads to meaningful results although limited energy intervals are used, because these contain the main parts of the total strength. It is assumed that the reduction of the calculated strength is due to the spin part of the magnetic dipole operator. The resulting effective spin g-factors are shown in Fig. 1 for all four nuclei. A consistent description can be achieved with g seff s 0.75Ž2. g sfree. Within the given uncertainty Žhatched area in Fig. 1. all data agree with this value. Spinrorbit interference plays a little role only although the summed cross terms are not small. Assuming the M 1 strength to be of pure spin character the extracted g seff would only marginally change to 0.74. Similar to what has been observed in lighter nuclei w26x the phases of the interference terms are distributed such that they tend to cancel each other. The deduced quenching factor is in remarkable agreement with the latest analysis w27x of GT b-decay transitions in the lower fp-shell which finds a quenching factor of 0.744Ž15. for the axial-vector
P. Õon Neumann-Cosel et al.r Physics Letters B 443 (1998) 1–6
Fig. 1. Effective spin g-factor of the M 1 operator deduced from the comparison of the shell-model calculations and data w14,16x for the total B Ž M 1. strengths in the stable even-mass N s 28 isotones.
coupling constant g A . Monte Carlo shell-model studies w19x find a slightly larger renormalization factor of 0.8 from a comparison to GT strengths in various fp-shell nuclei w28x extracted from charge-exchange reactions, however with uncertainties of about 20%. Thus, it becomes clear that the quenching of the M 1 response can be properly described in shell-model calculations if the model spaces are sufficiently large enough. The close agreement also indicates that MEC
3
contributions, which considerable enhance the M 1 over the GT strength in lighter nuclei w9–11x, are already small for nuclei with mass numbers around 50. This finding is consistent with theoretical analyses w29,30x of the form factor of the prominent M 1 transition in 48 Ca where only small effects due to MEC are predicted at low momentum transfer. The total magnetic dipole response in the N s 28 isotones has also been studied in the Monte Carlo shell-model work of w28x. These results are generally confirmed by the exact diagonalization although 20% higher and 15% lower total strengths are observed in 52 Cr and 54 Fe, respectively. However, without the information on the strength distributions it is not possible to extract spin quenching factors for the finite experimental energy ranges. The limited excitation energy range investigated in the experiments represents a major uncertainty in the above analysis. The deduced quenching factors are meaningful only if the calculations are capable to reproduce the gross features of the experimental M 1 distributions within these energy intervals. This can be tested by the so-called running sum, i.e. the cumulative M 1 strength plotted as a function of excitation energy. Fig. 2 compares the data and the
Fig. 2. Runnings sums of the B Ž M 1. strengths in the stable even-mass N s 28 isotones. Data Žhatched. are from w14x for 48 Ca and from w16x for 50 Ti, 52 Cr and 54 Fe. The shell-model calculations Ždotted. are performed with an effective spin g-factor g seff s 0.75 g sfree. The long-dashed curve for 52 Cr corresponds to a calculation with a slightly modified interaction, see text.
4
P. Õon Neumann-Cosel et al.r Physics Letters B 443 (1998) 1–6
shell-model results using the above-derived spin quenching factor. For 48 Ca and 50 Ti very good agreement is observed, while 52 Cr and 54 Fe exhibit a systematic shift of the theoretical running sums of about one MeV towards higher excitation energies. The origin of the shift is related to the fact that KB3 produces a too strong N s 28 shell gap when approaching N s Z, a defect that can be repaired without harming the accumulated good results of KB3. Thus, we have have repeated the calculations for the example of 52 Cr with a 100 keV more attractive average interaction between nucleons in the 1 f 7r2 shell and in the remaining orbits. The results are displayed as long-dashed line in Fig. 2. The total B Ž M 1. strength is unaffected but a shift towards lower excitation energies is observed which brings the calculations into very good agreement with the data.
The above discussion changes none of the conclusions of the paper because of the important observation that, in the investigated energy ranges, the data and as well as the model results for the B Ž M 1. strength saturate. If the calculations are extended up to Ex s 35 MeV, only a small fraction of about 20% of the total B Ž M 1. response is found above the experimental upper limits. An exception is 54 Fe, where significant excitation strength of T) states is predicted at Ex f 12–13 MeV, partly outside the investigated excitation energy range. Therefore, the strength predicted at higher energies amounts to about 35%. In passing we note the the calculations predict about 7% of the total B Ž M 1. strength in 50 Ti to lie at energies E x - 8 MeV, i.e. below the experimentally investigated range. This should be easily accessible in transverse electron scattering at low momentum transfer or photon scattering Žsee e.g.
Fig. 3. Comparison of experimental B Ž M 1. distributions in the stable even-mass N s 28 isotones with the shell-model calculations described in the text.
P. Õon Neumann-Cosel et al.r Physics Letters B 443 (1998) 1–6
w31x.. Overall, the experiments seem indeed to have detected the major part of the M 1 spin-flip resonance in the N s 28 nuclei. Finally, one may ask to what extent the shellmodel results are capable to describe the detailed B Ž M 1. strength distributions experimentally extracted in the N s 28 isotones. A comparison is plotted in Fig. 3, where the spin quenching factor deduced above is taken into account. Again, one must clearly distinguish between 48 Ca, 50 Ti calculated in the full fp-space and 52 Cr, 54 Fe which could only be treated with restrictions on the number of active particles. For 48 Ca the strength is concentrated in a single transition whose position and strength can be reproduced. However, in the calculations a splitting into two close-lying states is found. This must be due to minute details of the interaction. Actually, we have repeated the calculation using the FPD6 interaction of Ref. w32x and find the same total strength, but now the dominant peak is surrounded by four satellite peaks amounting to 1.5 m2N . The main concentration of M 1 strength in 50 Ti between 10 and 11 MeV is well accounted for by the calculations. Overall, less fragmentation is seen than experimentally observed. This is partly due to the limitation in the number of Lanczos iterations that we can carry on in the very large basis Žabout two million states. of the full pf-shell calculation, although one cannot exclude that mixing with intruder states, absent in the shell-model calculation, could be responsible for some of the extra fragmentation seen in the experimental results. In the description of 52 Cr and 54 Fe with restricted valence spaces the discrepancies to the experiments are more pronounced. We have already discussed the reason for the shift of strength to higher energies. Besides that, the degree of fragmentation is clearly underestimated. This is actually a severe problem, because in order to be able to describe in detail the experimental fragmentation, we need to approach the real level density and that would demand a tremendous number of Lanczos iterations in a huge basis. This exceeds our computational limits in many cases. However, recent developments permit significant improvements, using the new coupled code NATHAN w33x. E.g., for the example of 52 Cr it has been possible to increase the number of Lanczos iterations from 100 to 300. The resulting B Ž M 1.
5
distribution, using the modified KB3 interaction discussed above, is shown in Fig. 4. The calculations now go a long way in reproducing the experimental fragmentation of the strength. In summary, we have investigated whether stateof-the-art shell-model calculations can resolve the failure of previous attempts to describe the complex evolution of the magnetic dipole response at the N s 28 shell closure. The gross properties of the strength distributions are well described in all cases, except for a shift of strength upwards in energy for the heavier nuclei 52 Cr and 54 Fe that can be easily understood. In the nuclei with larger dimensionalities, to reproduce the experimentally observed degree of fragmentation requires a major computational effort at the edge of – or in many cases beyond – the present possibilities. However, the model spaces used here are sufficiently large to achieve a systematic description of the total B Ž M 1. strengths in the N s 28 chain and the problem of the dramatic quenching suggested by the previous calculations could be resolved. These results represent a major step forward in our understanding of the magnetic dipole response in medium-mass nuclei. The deduced quenching factor g seff s 0.75Ž2. g sfree coincides with the renormalization of the axial-vector coupling constant in GT transitions. This close agreement indicates that contributions from meson exchange currents, which lead to an enhancement of the M 1 over the GT strength in sd-shell nuclei, are already small for nuclei around A , 50.
Fig. 4. Shell-model calculation of the B Ž M 1. distribution in 52 Cr. Compared to the result shown in Fig. 3 the number of Lanczos iterations was increased from 100 to 300, and the KB3 interaction was modified as described in the text.
P. Õon Neumann-Cosel et al.r Physics Letters B 443 (1998) 1–6
6
Acknowledgements We thank E. Caurier and G. Martinez-Pinedo for their help in different steps of this work.
w15x w16x w17x w18x w19x w20x
References w21x w1x w2x w3x w4x w5x w6x w7x w8x w9x w10x w11x w12x w13x w14x
I.S. Towner, Phys. Reports 155 Ž1987. 263. F. Osterfeld, Rev. Mod. Phys. 64 Ž1992. 491. A. Richter, Prog. Part. Nucl. Phys. 34 Ž1995. 261. I.S. Towner, F.C. Khanna, Nucl. Phys. A 399 Ž1983. 334. A. Arima, K. Shimizu, W. Bentz, H. Hyuga, Adv. Nucl. Phys. 18 Ž1987. 1. B.A. Brown, B.H. Wildenthal, Nucl. Phys. A474 Ž1987. 290. B. Reitz et al., submitted to Phys. Rev. Lett. W. Steffen et al., Phys. Lett. B 95 Ž1980. 23. A. Richter, A. Weiss, O. Hausser, B.A. Brown, Phys. Rev. ¨ Lett. 65 Ž1990. 2519. C. Luttge et al., Phys. Rev. C 53 Ž1996. 127. ¨ P. von Neumann-Cosel, A. Richter, Y. Fujita, B.D. Anderson, Phys. Rev. C 55 Ž1997. 532. C. Luttge, P. von Neumann-Cosel, F. Neumeyer, A. Richter, ¨ Nucl. Phys. A 606 Ž1996. 183. A. Richter, Prog. Nucl. Part. Phys. 13 Ž1984. 1. W. Steffen et al., Nucl. Phys. A 404 Ž1983. 413.
w22x w23x w24x w25x w26x w27x w28x w29x w30x w31x w32x w33x
A. Richter, Nucl. Phys. A 522 Ž1991. 139c. D.I. Sober et al., Phys. Rev. C 31 Ž1985. 2054. G. Eulenberg et al., Phys. Lett. B 116 Ž1982. 113. A. Richter, Phys. Scr. T 5 Ž1983. 67. S.E. Koonin, D.J. Dean, K. Langanke, Annu. Rev. Nucl. Part. Sci. 47 Ž1997. 403. E. Caurier, A.P. Zuker, A. Poves, G. Martinez-Pinedo, Phys. Rev. C 50 Ž1994. 225. J. Retamosa, E. Caurier, E. Nowacki, A. Poves, Phys. Rev. C 55 Ž1997. 1266. E. Caurier, A. Poves, A.P. Zuker, Phys. Rev. Lett. 74 Ž1995. 1517. E. Caurier et al., Phys. Rev. Lett. 75 Ž1995. 2466. E. Caurier, code ANTOINE, Strasbourg 1989, unpublished. G. Martinez-Pinedo et al., Phys. Rev. C 55 Ž1997. 187. M.S. Fayache et al., Nucl. Phys. A 627 Ž1997. 14. C. Martinez-Pinedo, A. Poves, E. Caurier, A.P. Zuker, Phys. Rev. C 53 Ž1996. R2602. K. Langanke et al., Phys. Rev. C 52 Ž1995. 718. K. Takayama, K. Shimizu, A. Arima, Nucl. Phys. A 481 Ž1988. 313. J.E. Amaro, A.M. Lallena, Phys. Lett. B 261 Ž1991. 229. J. Enders et al., Nucl. Phys. A 636 Ž1998. 139. W.A. Richter, M.G. van der Merwe, R.E. Julies, B.A. Brown, Nucl. Phys. A 523 Ž1991. 325. E. Caurier, F. Nowacki, code NATHAN, Strasbourg 1995, unpublished.
Physics Letters B 443 Ž1998. 1–6
Magnetic dipole response in nuclei at the N s 28 shell closure: a new look 1 P. von Neumann-Cosel a , A. Poves b, J. Retamosa c , A. Richter
a
a
Institut fur ¨ Kernphysik, Technische UniÕersitat ¨ Darmstadt, D-64289 Darmstadt, Germany ´ Teorica, Departamento de Fisica UniÕersidad Autonoma de Madrid, 28049 Madrid, Spain ´ ´ ´ Atomica Departamento de Fisica y Nuclear, UniÕersidad Complutense, 28040 Madrid, Spain ´
b c
Received 5 August 1998 Editor: J.-P. Blaizot
Abstract The magnetic dipole response of N s 28 isotones is obtained for 48 Ca, 50 Ti, 52 Cr and 54 Fe by means of shell model calculations in the full pf-shell. Spin quenching factors derived in all four nuclei by comparison to experimental strengths from electron scattering agree well with each other. The deduced effective g-factor g seff s 0.75Ž2. g sfree is consistent with findings for the Gamow-Teller strength in this mass region. The ability of state-of-the-art shell-model calculations to describe the gross properties and the fragmentation of the complex experimental B Ž M 1. strength distributions is investigated. q 1998 Elsevier Science B.V. All rights reserved. PACS: 23.20.Js; 21.60.Cs; 27.40.q z Keywords: BŽM1. strength in N s 28 isotones; shell-model calculations; Quenching
The magnetic dipole response of nuclei represents a subject of intense interest to experimentalists and theorists for a variety of reasons. The properties of spin and convection currents in M 1 excitations serve as a thorough test of models aiming at a description of nuclear dynamics. Žsee e.g. w1–3x and references therein.. Furthermore, M 1 transitions provide a unique source to elucidate the role of non-nuclear degrees of freedom in complex nuclei w4–7x. Some famous examples are the neutron spin-flip excitations in 48 Ca, where the M 1 strength is essentially
1
Work supported by the DFG under Contract Ri 242r12-1 and by research grants from the DGES ŽSpain., Nos. PB96-53 and PB96-604.
concentrated in a single strong transition w8x, or the enhancement of the total M 1 strength in sd-shell nuclei due to meson exchange currents w9–11x. Finally, the degree of quenching and the details of the distributions of M 1 and of the closely related Gamow-Teller ŽGT. transitions in pf-shell nuclei strongly influence the late stages of a collapsing massive star as well nucleosynthesis processes induced by a supernova outburst w12x. The properties of magnetic dipole excitations in nuclei along the N s 28 shell gap provide a particularly intriguing example where the interplay of proton and neutron degrees of freedom can be studied in great detail w13x. Experimental data are available for 48 Ca, 50 Ti, 52 Cr and 54 Fe from electron scattering w14,16x. When moving from the doubly closed-shell
0370-2693r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 8 . 0 1 2 9 8 - 2
2 48
P. Õon Neumann-Cosel et al.r Physics Letters B 443 (1998) 1–6
Ca nucleus to an open proton 1 f 7r2 shell a complex picture of the total B Ž M 1. strength and its fragmentation emerges. Attempts to describe this fine structure in shell-model calculations Žrestricted at that time to two particle – two hole excitations from 1 f 7r2 to pf-shell configurations. failed completely w17x. Further, the total B Ž M 1. strengths were severely overestimated and it could not be decided whether this discrepancy was entirely due to the model space restrictions or implied unusually large quenching due to non-nucleonic degrees of freedom. Clearly, the problem remained unsolved w18x. In recent years considerable progress has been made in shell-model calculations, partly by the development of new methods such as a Monte Carlo approach w19x, but also because of the dramatic increase of computing power. For example, it is possible now to perform calculations for nuclei up to mass A f 52 allowing for the full pf model space w20,21x with great success in the description of basic structure features such as the GT strength in 48 Ca w22x or the buildup of a rotational band in the openshell nucleus 48 Cr w23x. These dramatic improvements ask for a new look at the problem of the magnetic dipole response in the N s 28 isotones with state-of-the-art shell-model calculations. We have performed such calculations in the full pf shell, using the code ANTOINE w24x and the effective interaction KB3. A detailed discussion of the shell-model techniques used can be found in w20x and a critical assessment of the interaction KB3 in w25x. While inclusion of the full pf shell is affordable now for the calculation of the ground state energies, wave functions and total magnetic strengths of the four nuclei investigated here, the determination of the strength functions in the full space is, at present, possible only for 48 Ca and 50 Ti. For 52 Cr and 54 Fe we have to limit ourselves to a calculation in which at most four particles are allowed to be excited from the 1 f 7r2 to the remaining pf-shell orbits Ža t s 4 truncation.. The total B Ž M 1. strengths in the full space, using bare g-factors, are 8.96 m2N , 12.95 m2N , 15.60 m2N and 18.90 m2N for 48 Ca, 50 Ti, 52 Cr and 54 Fe, respectively. We have computed separately the total spin and orbital contributions in order to confirm that the latter are very small Ž0.0. 0.7, 1.0 and 1.1 m 2N , respectively.. The strength going to states with isospin T) s T0 q 1, where T0
denotes the isospin of the ground state, increases with decreasing T0 Ž0.0, 0.30, 1.92 and 6.50 m2N . and becomes significant for 54 Fe. These results will be important for the discussion that follows. Note that a t s 4 truncation is already a very solid approximation to the final result, i.e. the resulting total strengths in 52 Cr and 54 Fe Ž15.81 and 19.25 m2N . are already very close to the results of calculations in the full model space. Experimentally, M 1 strength in the N s 28 isotones has been studied with high-resolution electron scattering under backward angles. Data are available for 48 Ca in an excitation energy range Ex s 8–13 MeV where a summed B Ž M 1.≠ strength of 5.3Ž6. m2N was found w14x. Investigations towards higher energies up to 17 MeV w15x reveal very little additional strength Žat most 0.5 m2N .. Results for 50 Ti have been reported in an interval Ex s 8–12 MeV and for 52 Cr, 54 Fe in the energy range Ex s 7–12 MeV. The data and the method for the extraction of the summed M 1 strengths are described in w16x. One finds B Ž M 1.≠ s 4.5Ž5. m2N Ž50 Ti., B Ž M 1.≠ s 8.1Ž8. m2N Ž52 Cr. and B Ž M 1.≠ s 6.6Ž4. m2N Ž54 Fe.. In a first step we use the shell-model results for an extraction of the quenching factors by normalizing the calculations to the data in the respective excitation energy ranges. As will be demonstrated below, such a procedure leads to meaningful results although limited energy intervals are used, because these contain the main parts of the total strength. It is assumed that the reduction of the calculated strength is due to the spin part of the magnetic dipole operator. The resulting effective spin g-factors are shown in Fig. 1 for all four nuclei. A consistent description can be achieved with g seff s 0.75Ž2. g sfree. Within the given uncertainty Žhatched area in Fig. 1. all data agree with this value. Spinrorbit interference plays a little role only although the summed cross terms are not small. Assuming the M 1 strength to be of pure spin character the extracted g seff would only marginally change to 0.74. Similar to what has been observed in lighter nuclei w26x the phases of the interference terms are distributed such that they tend to cancel each other. The deduced quenching factor is in remarkable agreement with the latest analysis w27x of GT b-decay transitions in the lower fp-shell which finds a quenching factor of 0.744Ž15. for the axial-vector
P. Õon Neumann-Cosel et al.r Physics Letters B 443 (1998) 1–6
Fig. 1. Effective spin g-factor of the M 1 operator deduced from the comparison of the shell-model calculations and data w14,16x for the total B Ž M 1. strengths in the stable even-mass N s 28 isotones.
coupling constant g A . Monte Carlo shell-model studies w19x find a slightly larger renormalization factor of 0.8 from a comparison to GT strengths in various fp-shell nuclei w28x extracted from charge-exchange reactions, however with uncertainties of about 20%. Thus, it becomes clear that the quenching of the M 1 response can be properly described in shell-model calculations if the model spaces are sufficiently large enough. The close agreement also indicates that MEC
3
contributions, which considerable enhance the M 1 over the GT strength in lighter nuclei w9–11x, are already small for nuclei with mass numbers around 50. This finding is consistent with theoretical analyses w29,30x of the form factor of the prominent M 1 transition in 48 Ca where only small effects due to MEC are predicted at low momentum transfer. The total magnetic dipole response in the N s 28 isotones has also been studied in the Monte Carlo shell-model work of w28x. These results are generally confirmed by the exact diagonalization although 20% higher and 15% lower total strengths are observed in 52 Cr and 54 Fe, respectively. However, without the information on the strength distributions it is not possible to extract spin quenching factors for the finite experimental energy ranges. The limited excitation energy range investigated in the experiments represents a major uncertainty in the above analysis. The deduced quenching factors are meaningful only if the calculations are capable to reproduce the gross features of the experimental M 1 distributions within these energy intervals. This can be tested by the so-called running sum, i.e. the cumulative M 1 strength plotted as a function of excitation energy. Fig. 2 compares the data and the
Fig. 2. Runnings sums of the B Ž M 1. strengths in the stable even-mass N s 28 isotones. Data Žhatched. are from w14x for 48 Ca and from w16x for 50 Ti, 52 Cr and 54 Fe. The shell-model calculations Ždotted. are performed with an effective spin g-factor g seff s 0.75 g sfree. The long-dashed curve for 52 Cr corresponds to a calculation with a slightly modified interaction, see text.
4
P. Õon Neumann-Cosel et al.r Physics Letters B 443 (1998) 1–6
shell-model results using the above-derived spin quenching factor. For 48 Ca and 50 Ti very good agreement is observed, while 52 Cr and 54 Fe exhibit a systematic shift of the theoretical running sums of about one MeV towards higher excitation energies. The origin of the shift is related to the fact that KB3 produces a too strong N s 28 shell gap when approaching N s Z, a defect that can be repaired without harming the accumulated good results of KB3. Thus, we have have repeated the calculations for the example of 52 Cr with a 100 keV more attractive average interaction between nucleons in the 1 f 7r2 shell and in the remaining orbits. The results are displayed as long-dashed line in Fig. 2. The total B Ž M 1. strength is unaffected but a shift towards lower excitation energies is observed which brings the calculations into very good agreement with the data.
The above discussion changes none of the conclusions of the paper because of the important observation that, in the investigated energy ranges, the data and as well as the model results for the B Ž M 1. strength saturate. If the calculations are extended up to Ex s 35 MeV, only a small fraction of about 20% of the total B Ž M 1. response is found above the experimental upper limits. An exception is 54 Fe, where significant excitation strength of T) states is predicted at Ex f 12–13 MeV, partly outside the investigated excitation energy range. Therefore, the strength predicted at higher energies amounts to about 35%. In passing we note the the calculations predict about 7% of the total B Ž M 1. strength in 50 Ti to lie at energies E x - 8 MeV, i.e. below the experimentally investigated range. This should be easily accessible in transverse electron scattering at low momentum transfer or photon scattering Žsee e.g.
Fig. 3. Comparison of experimental B Ž M 1. distributions in the stable even-mass N s 28 isotones with the shell-model calculations described in the text.
P. Õon Neumann-Cosel et al.r Physics Letters B 443 (1998) 1–6
w31x.. Overall, the experiments seem indeed to have detected the major part of the M 1 spin-flip resonance in the N s 28 nuclei. Finally, one may ask to what extent the shellmodel results are capable to describe the detailed B Ž M 1. strength distributions experimentally extracted in the N s 28 isotones. A comparison is plotted in Fig. 3, where the spin quenching factor deduced above is taken into account. Again, one must clearly distinguish between 48 Ca, 50 Ti calculated in the full fp-space and 52 Cr, 54 Fe which could only be treated with restrictions on the number of active particles. For 48 Ca the strength is concentrated in a single transition whose position and strength can be reproduced. However, in the calculations a splitting into two close-lying states is found. This must be due to minute details of the interaction. Actually, we have repeated the calculation using the FPD6 interaction of Ref. w32x and find the same total strength, but now the dominant peak is surrounded by four satellite peaks amounting to 1.5 m2N . The main concentration of M 1 strength in 50 Ti between 10 and 11 MeV is well accounted for by the calculations. Overall, less fragmentation is seen than experimentally observed. This is partly due to the limitation in the number of Lanczos iterations that we can carry on in the very large basis Žabout two million states. of the full pf-shell calculation, although one cannot exclude that mixing with intruder states, absent in the shell-model calculation, could be responsible for some of the extra fragmentation seen in the experimental results. In the description of 52 Cr and 54 Fe with restricted valence spaces the discrepancies to the experiments are more pronounced. We have already discussed the reason for the shift of strength to higher energies. Besides that, the degree of fragmentation is clearly underestimated. This is actually a severe problem, because in order to be able to describe in detail the experimental fragmentation, we need to approach the real level density and that would demand a tremendous number of Lanczos iterations in a huge basis. This exceeds our computational limits in many cases. However, recent developments permit significant improvements, using the new coupled code NATHAN w33x. E.g., for the example of 52 Cr it has been possible to increase the number of Lanczos iterations from 100 to 300. The resulting B Ž M 1.
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distribution, using the modified KB3 interaction discussed above, is shown in Fig. 4. The calculations now go a long way in reproducing the experimental fragmentation of the strength. In summary, we have investigated whether stateof-the-art shell-model calculations can resolve the failure of previous attempts to describe the complex evolution of the magnetic dipole response at the N s 28 shell closure. The gross properties of the strength distributions are well described in all cases, except for a shift of strength upwards in energy for the heavier nuclei 52 Cr and 54 Fe that can be easily understood. In the nuclei with larger dimensionalities, to reproduce the experimentally observed degree of fragmentation requires a major computational effort at the edge of – or in many cases beyond – the present possibilities. However, the model spaces used here are sufficiently large to achieve a systematic description of the total B Ž M 1. strengths in the N s 28 chain and the problem of the dramatic quenching suggested by the previous calculations could be resolved. These results represent a major step forward in our understanding of the magnetic dipole response in medium-mass nuclei. The deduced quenching factor g seff s 0.75Ž2. g sfree coincides with the renormalization of the axial-vector coupling constant in GT transitions. This close agreement indicates that contributions from meson exchange currents, which lead to an enhancement of the M 1 over the GT strength in sd-shell nuclei, are already small for nuclei around A , 50.
Fig. 4. Shell-model calculation of the B Ž M 1. distribution in 52 Cr. Compared to the result shown in Fig. 3 the number of Lanczos iterations was increased from 100 to 300, and the KB3 interaction was modified as described in the text.
P. Õon Neumann-Cosel et al.r Physics Letters B 443 (1998) 1–6
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Acknowledgements We thank E. Caurier and G. Martinez-Pinedo for their help in different steps of this work.
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